Optimal Server Assignment in Multi-Server Queueing Systems with Random Connectivities

1 Optimal Serv er Assignment in Multi-Ser ve r Queueing Systems with Random Connecti vities Hassan Halabian, Student Member , IEEE, Ioannis Lambadaris, Member , IEEE, Y annis V i niotis, and Chung-Horng Lung, Member , IEEE, Abstract W e study th e p roblem of assigning K identical servers to a set of N parallel q ueues in a time- slotted queueing system. The c onnectivity of each queu e to each server is random ly chan ging with time; each server can serve at most o ne q ueue and each queue can be ser ved by at most one ser ver during each time slot. Such a q ueueing model has been used in addressing resou rce allocation problems in wireless networks. It has been previously proven that Maxim um W eighted Matching (MWM) is a thr ou ghpu t-optimal server assignment policy for such a queueing system. I n this paper, we prove that for a system with i.i.d. Bernou lli packet arriv als and connectivities, MWM minimizes, in stoch astic orderin g sense, a broad ran ge of cost f unctions of the queue lengths such as to tal queu e o ccupancy (which im plies min imization of a verage queueing d elays). Then, we extend the mod el by considering imperfect service s where it is a ssumed that the service o f a sched uled packet fails random ly with a certain pro bability . W e prove that the same policy is still optimal for the extended model. W e ﬁnally show that th e results ar e still valid for more gen eral conn ectivity and arriv al processes wh ich fo llow condition al p ermutation in variant distrib utions. I . I N T R O D U C T I O N Optimal stochasti c control is one of th e main obj ectiv es i n the design of emerging wireless networks. One of t he primary goals in stochast ic control and op timization of wireless networks is t o distribute the shared resources i n the phys ical (e.g., power) and MA C layers (e.g., radio H. Halabian, I. Lambadaris, C-H Lung are with the Department of Systems and C omputer Engineering, Carleton Univ ersity , Ottawa, ON, K1S 5B6 Canada (e-mail: hassanh@sce.carleton.ca; ioannis@sce.carleton.ca; chlung@sce.carleton.ca). Y . V iniotis is with the Department of El ectrical and Computer Engineering, North Carolina S tate Univ ersity , Raleigh, North Carolina (e-mail: candice@ncsu.edu). August 16, 2018 DRAFT 2 interfaces, relay s tations and orthogonal sub-channels) among multip le users such that certain stochastic performance attributes are optimized. While v arious p erformance criteria including the st able throughput region, power consum ption and uti lity functions of the admitted t raf ﬁc rates have b een s tudied in seve ral papers [1]–[19], av erage queueing delay has received l ess attention. The inherent random ness in wireless channels makes delay-optimal resource allo cation a challenging problem in wireless networks. In thi s p aper , we focus on d elay-optimal server assignm ent i n a time-slot ted, m ulti-queue, multi-server sys tem with random connectivities. Random connectivities can mod el unreliable and random ly varying wi reless chann els. Our queuein g model can be applied to study resource allocation in wireless access networks where th e w ireless users are modeled by t he queues; the sh ared resources are modeled by the servers and the wireless channels are model ed by the random connectivities between the queues and th e servers. Although thi s m odel is a simpli ﬁed representation of a real wireless system , ne vertheless it do es provide valuable intuition for the performance o ptimization of real s ystems. Sim ilar mo deling approaches hav e already appeared in [2], [3], [10], [15]–[17], [20]–[23]. A. Related W ork and Our Contri butions The probl em of th r ough put-optimal server allo cation i n mul ti-queue, singl e-server systems with random connectivities was add ressed in [2], [10], [20], [21]. In [2], the autho rs considered a time-slott ed, m ulti-queue s ingle-server sy stem with Bernoulli p acket arriv als and connectivities from each of the queues to a single server . They introdu ced LCQ (Longest Connected Queue) policy as a throughput-opt imal policy and also characterized the stability region b y a set of linear inequalities. Th e auth ors in [20] consid ered a continuo us-time version of t he model studied in [2] with ﬁnite buf fer sp ace and showed that under st ationary ergodic in put j ob ﬂow and modulatio n processes, LCQ policy maximizes the s table throughpu t region of this system. In [10], C-FES (Connected queue with the Fe west Em pty Spaces) policy , a policy that allocates the server to the conn ected queue wi th the fewest empt y spaces, was i ntroduced for this system. It was shown that C-FES s tochastically minimizes the lo ss ﬂow and maximizes the throug hput of the syst em. In [21], a model simi lar to the m odel of [2], [10] was studied and it was shown that the Best User (BU) policy maximi zes th e expected discount ed number of successful t ransmissions . While i n th roughput-optim al server allocation the objective is to ﬁnd a policy that maximizes August 16, 2018 DRAFT 3 the throughpu t region of the system and keeps the queues stabl e [1], [4], in d elay-optimal server allocation the goal is to d etermine a policy that mini mizes the ave rage queueing delay . Thus, the objective in delay opt imality is more s tringent than the objective in throug hput optimali ty . A server all ocation policy m ay be throughput-optim al but not delay-opt imal; howev er , a delay- optimal policy (for all the arriv al rates) is always throu ghput-optim al. In [2], the authors (other than p roving the throughput optimality of LCQ as mentioned earlier) p roved that for a multi- queue, single-server system with i.i.d . Bernoulli arri val and connectivity processes, the LCQ policy is also del ay-optimal. The extension of thi s result for non-i.i.d . case is s till an open problem. In generalizing the result s to multi-queue, mult i-server (MQMS) systems, various multi -server systems have been studied [3], [15], [16], [22 ]–[25]. In [22], Maximum W eight (MW) policy was proposed as a throughp ut-optimal server all ocation pol icy for an MQMS queueing system wit h general, stati onary channel processes. MW policy can b e considered as a sp ecial case of back- pressure algorithm which was proven in [1], [4] to be a throughput-optim al resource allocation algorithm in a general queueing system. In [15], the authors characterized th e network st ability region of multi-qu eue, multi-server systems with time-varying, independ ent connectivities . The results were furth er extended in [16] for more general, stationary channel d istributions (and not just ind ependent Bernoulli channels). In all the models stud ied in [15], [16], [22], there is no restriction on the number of servers that can be allocated to a queue. For ease of reference, we wil l call such an MQMS system as MQM S-T ype1. In [3], it was shown that for an MQMS system in which t he queues ar e r estricted to get s ervice from at most one server d uring each time slot , M aximum W eighted M atching (MWM) policy i s throughp ut-optimal. For ease of reference, we will call such an M QMS system with this extra assumpt ion as MQMS-type2. The authors also con sidered the ef fect of infrequent channel state m easurements o n the network stabil ity region of M QMS systems. Similar to MQMS-T ype1, for MQMS-T ype2 the through put-optimal policy (MWM) can be considered as a special case of the back-pressure algorit hm. In contrast to the sing le-server system (where LCQ was both throughput-optim al and delay- optimal), in MQ MS-T ype1 system the M W po licy is n ot necessarily delay-opti mal. More speciﬁ- cally , in [15] i t was also sho wn that although MW policy is throughput-optim al, e ven for a system with i.i.d. Bernoulli arri vals and connectivity processes, MW policy in its general form, is not delay-optimal. August 16, 2018 DRAFT 4 The delay-optimal server allocation problem in multi -server systems was addressed in [23]– [25]. The authors in [23] considered a queueing model w ith a set of parallel queues and i .i.d. Bernoulli packet arri vals that are competi ng to attract s ervice from K id entical servers formi ng a server-bank . The connectivities of the queues to th e entir e server-bank are ass umed to be i.i.d. Bernoulli processes. Each queue is restricted to receive service from at most on e s erver during each time slot . The authors proposed LCQ poli cy i n which t he servers of the s erver -bank are allocated to the K l ongest connected queues at each ti me sl ot. Using d ynamic coupling and stochastic ordering, they p roved the delay optimality of LCQ policy for such a system. In our work, the focus would be on delay opti mality of MWM policy in MQM S-T ype2 system in which the servers are not restricted to form a server -bank. Instead, we assume that each queue h as an independent connectivity to each individual server (as seen in Figure 1 ). Th e work in [24], [25] focuses on delay optimal server allo cation problem in the MQM S-T ype1 system. In [24], the authors introduced MTLB (M aximum-Throughp ut Load-Balancing) policy and us ing dynam ic programming s howed th at this policy minimizes a class of cost functions including total a verage delay for the case of two queues with i .i.d., Bernoulli-di stributed arriva ls and connectivities. In [24], no general ar gument was provided for the opti mality of MTLB for more th an two queues. The work in [25] considers th is problem for a general number of queues and s ervers. In [25], a class of Mo st Balancing (MB) policies was characterized among all work-conserving policies wh ich minim ize, in stochasti c ordering sense, a class of cost functi ons including t otal q ueue occupancy (and thus are delay-opt imal). Howe ver , this class of proposed MB policies is j ust characterized by a pr operty of thi s class ; the authors did not introduce an e xplicit implementation for the o ptimal pol icy . In t his paper , we focus on MWM policy and prove that this through put-optimal policy is also delay-optimal for an M QMS-T ype2 system with i.i. d. arriv al and connectivity processes. Our work extends the results derived in [2], [23]. In particular , th e researchers in [2], [23] h a ve consi dered queueing models w here a single server or a server-bank is random ly connected to a set of parallel queues. In this paper , we cons ider a mo re general mo del where each individu al s erver is randomly connected to the queues (as seen in Fig ure 1). Although the two m odels bear certain similariti es, extending the results from single-server (server- bank) sy stem to multi-server system is not a straightforward p rocedure. Our work is different from the work in [2], [23] from both th e modeling power and the di f ﬁculty in proof points of view . August 16, 2018 DRAFT 5 P S f r a g r e p l a c e m e n t s 1 1 2 2 K K λ λ λ λ λ λ a) The model studied in [2], [23] b) MQMS-T y pe2 system in t his paper . p p p p p p p p p X 1 ( t ) X 1 ( t ) X 2 ( t ) X 2 ( t ) X N ( t ) X N ( t ) Single server or server -bank Fig. 1: Previous models vs . our mo del. ( λ is the arriv al probabil ity and p is t he connectivity probability) For more information on optimal schedul ing and resource allocation p roblems in wireless networks the reader is encouraged t o als o consult with [4], [26]–[30]. Our cont ributions in this paper are su mmarized as foll ows: First , for an MQM S-T ype2 system we prov e that during each ti me slot, Maximum W eighted Matchin g (MWM ) policy wi ll result in the most balanced qu eue vector in the system, i.e., maximization of the matching weight a nd balancing o f th e queues ar e equival ent . Graph theoretic ar guments were applied to prove this result that is formally i ntroduced i n Lem ma 1 and L emma 2 later in the paper . Note t hat ou r approach to prove t his result is only applicable t o the MQM S-T ype2 model (due t o the structu re of the model and t he MWM policy) and cannot be easily extended to MQMS-T ype1 system. Second, using this result in conjunction with the n otions o f stochastic ordering and dynamic coupling, we prove the delay optim ality of MWM p olicy for an MQMS-T ype2 system with i.i.d. Bernoulli arri vals and connectivities. More speciﬁcally , we prove that MWM minimi zes, in s tochastic or dering sense, a range of cost functio ns of queue length s in cluding tot al queue occupancy 1 . Third, we then extend our model by consi dering imperfect services where it is assumed that th e service of a schedul ed packet fails randomly wi th a certain p robability . W e prove that MWM is still optimal for the extended model. W e ﬁnally show that the results are 1 The optimality of MWM is proven among all causal server assignment policies. August 16, 2018 DRAFT 6 still va lid for some more general connectivity and arriv al processes which follow conditional permutation in variant d istributions. The rest of this paper i s organized as foll ows. In Section II, we introduce the queueing model and the requi red notation. In Section III, we describe the Maximu m W eighted Matching (MWM) server assignm ent policy . In Section IV, we prove the delay optimalit y of MWM server assign ment poli cy . In Section V, we present si mulation results where we com pare the performance o f MWM pol icy wi th t he performance of two o ther server assignment policies in terms of a verage total queue occupancy (or equiva lently average queueing delay). Finall y , we summarize our conclusion s in Section VI. I I . M O D E L D E S C R I P T I O N Throughout the paper , random variables are represented by CAPIT AL letters and lo wer case letters are used t o represent sample values of the random variables. Moreover , we use boldface font to represent matrices and vectors. W e consider a time-sl otted, MQMS-T ype2 system consisting of a set of parallel queues N = { 1 , 2 , . . . , N } with inﬁnite buf fer space for each queue (see Figure 2 ). Packets in thi s s ystem are assumed to ha ve cons tant l ength and require o ne time sl ot to com plete service. The servi ce t o this set of queues is provided by a set of identical servers K = { 1 , 2 , . . . , K } . The connectivity of each queue n ∈ N to each server k ∈ K at each ti me slot t is random and varying across time slots. W e denote the connectivity o f queue n to server k at tim e slot t by C n,k ( t ) ∈ { 0 , 1 } . When C n,k ( t ) = 1 ( C n,k ( t ) = 0 ), queue n is conn ected to (disconnected from ) server k at time slot t . The connectivity v ariables C n,k ( t ) are assumed to be i.i.d. Bernoulli random variables with a ﬁxed p arameter p 2 . At any time slot, each server can serve at most one packet from a connected, non-emp ty queue. W e d o not allow server s haring in the system, i.e., a server can serve at m ost one queue per time slot. W e als o assu me t hat at most o ne server can be assigned to any connected queue during a time slot. Let A n ( t ) deno te the number of packet arriv als to q ueue n at ti me slot t . W e assume that new arriv als at each time slo t are added to the q ueues at the end of the time slo t. The arriv al 2 The actual value of p does not in volv e in our analysis. W e only rely on the fact that t he connecti vities are i.i. d. Bernoulli processes. August 16, 2018 DRAFT 7 2 K 1 P S f r a g r e p l a c e m e n t s C 1 , 1 ( t ) C 1 , 2 ( t ) C 1 ,K ( t ) C N , 1 ( t ) C N , 2 ( t ) C N ,K ( t ) X 1 ( t ) X 2 ( t ) X N ( t ) A 1 ( t ) A 2 ( t ) A N ( t ) Fig. 2 : Discrete-time MQMS-T ype2 system with N parallel q ueues and K servers. var iables A n ( t ) are assumed to be i .i.d. Bernoulli random variables wit h t he same p arameter λ for all n and t 3 . W e denote the length of queue n at the end of ti me slot t (i.e., after adding the new arriv als) by X n ( t ) . Hence, X n ( t ) represents the number of packets in the n th qu eue at the end o f time slot t (or beginning of time slot t + 1 ). A. Server As signment P olicy At each time slot t the server assignment p olicy has to decide about a bipartite (graph) matching 4 between sets N and K . W e assume that this decisi on is made in a causal fashion, i.e., based on the a vailable hist ory of arri val processes, service processes, queue states and t he connectivity states until tim e t . A poli cy π i s fully d etermined by i ts indicator variables M ( π ) n,k ( t ) ∀ n ∈ N , ∀ k ∈ K , t = 1 , 2 , . . . which are deﬁned as M ( π ) n,k ( t ) =      1 , if server k is assigned to queue n by policy π at time slot t , 0 , otherwise. (1) W e deﬁne the N × K matri x M ( π ) ( t ) = ( M ( π ) n,k ( t )) , ∀ n ∈ N , ∀ k ∈ K as the employed ma tching by policy π at tim e sl ot t . Hence, a s erver assignm ent policy π can be deﬁned as the set of all 3 The actual value of λ does not in volve in our analysis. W e only rely on the fa ct that the arriv als are i.i.d. Berno ulli processes. 4 A matching in a bipartite graph is a sub-graph of the original graph in which no two edges share a common vertex. August 16, 2018 DRAFT 8 the employed matchings by policy π at time s lots t = 1 , 2 , . . . , i.e., π = { M ( π ) ( t ) } ∞ t =1 . W e denote the matching spa ce containing all the possible assignments of the servers to the queues by M . The set M is equiv alent to the set of all the possible matchings i n an N × K complete bipartite graph 5 . W e can observe that X n ( t ) , the queue l ength random var iable, ev olves in ti me as follows: X n ( t ) = X n ( t − 1) − K X k =1 C n,k ( t ) M ( π ) n,k ( t ) ! + + A n ( t ) ∀ n ∈ N (2) The op erator ( · ) + returns the t erm i nside the parentheses if i t is non-negativ e and zero otherwise. The queueing model introduced in this section is useful in providin g int uition for mod eling resource assignment probl ems in var ious systems with s hared resources [3], [17]. In wireless communication s ystems, resou rces such as communi cation sub-channels, relay stations, etc. are shared among users. As an example, we can consider a relaying access network wit h N users and K shared relays. By modeling the cooperative wireless channel between each us er , each relay and th e base stati on as an erasure channel, the performance of such a system can be studi ed following our model in Figure 2. I I I . M A X I M U M W E I G H T E D M A T C H I N G ( M W M ) S E RV E R A S S I G N M E N T P O L I C Y A. MWM Opt imization Pr oblem In [1], [4], it was shown that bac k-pr essur e algorithm maximizes the stable throughput region of a general data n etwork, i .e., i t is throughp ut-optimal. Th e reader may refer to [1], [4] for m ore information about back-pressure algo rithm. For the model introduced in Section II, the b ack- pressure algorithm reduces to th e following opti mization problem at each time s lot t [3]. In thi s integer programmi ng problem, M n,k ( t ) variables are the optimization variables and X n ( t − 1) 5 A complete bipartite graph is a bipartite graph in which each verte x in each part is connected to all the vertices in t he other part. An N × K bipartite graph has N K edges. August 16, 2018 DRAFT 9 and C n,k ( t ) are known parameters. Maximize: M n,k ( t ) , ∀ n, k N X n =1 X n ( t − 1) K X k =1 M n,k ( t ) C n,k ( t ) Subject to: K X k =1 M n,k ( t ) ≤ 1 ( n = 1 , 2 , . . . , N ) , N X n =1 M n,k ( t ) ≤ 1 ( k = 1 , 2 , . . . , K ) , M n,k ( t ) ∈ { 0 , 1 } ( k = 1 , 2 , . . . , K ) , ( n = 1 , 2 , . . . , N ) (3) Finding the solut ion of problem (3) i s equiv alent t o ﬁnding a maxim um weighted matching in t he N × K bi partite graph G t = ( N , K , E ) shown i n Figure 3. Hence, th e back-pressure algorithm for the queueing model of Figure 2 is also known as Maximum W eighted Matchin g (MWM) alg orithm. In G t , N and K are th e two sets of vertices in each part of the graph and E = { e n,k , ∀ n ∈ N , ∀ k ∈ K } is the set of edges between these two parts. In G t , the associated weight to ea ch edge e n,k is X n ( t − 1) C n,k ( t ) . A matching in graph G t is a sub-graph of G t in which no t wo edg es share a commo n vertex. Any matching M ( π ) ( t ) at any time slo t t is corresponding to a su b-graph of G t namely G ( π ) t = ( N , K , E ( π ) ) i n which e n,k ∈ E ( π ) if and on ly if M ( π ) n,k ( t ) = 1 . Th ere are sev eral algorithm s to ﬁnd the maximum weighted matching in bipartite graphs. The mos t w ell-known on e is the Hun garian algorithm with O ((min { N , K } )(max { N , K } ) 2 ) complexity [31]. B. MWM P oli cy Assume th at M ( MWM ) ( t ) = ( M ( MWM ) n,k ( t )) ∀ n ∈ N , ∀ k ∈ K is the matching whose in dica- tor variables are the soluti on o f the op timization problem (3). M ( MWM ) ( t ) has the following properties: (a) M ( MWM ) ( t ) always exists at al l time slots . (b) The maximum weighted matching in a bipartit e graph may not be unique , i.e., there may be more than one matching M ( MWM ) ( t ) for the graph of Figure 3 at each time slot. Deﬁnition 1: A M aximum W eighted Matching (MWM) server assign ment policy is deﬁned as a poli cy that employs m aximum weighted matching M ( MWM ) ( t ) at all tim e slots, i.e., π ( MWM ) = { M ( MWM ) ( t ) } ∞ t =1 . August 16, 2018 DRAFT 10 P S f r a g r e p l a c e m e n t s 1 2 N 1 2 K X 1 ( t − 1) C 1 , 1 ( t ) X 1 ( t − 1) C 1 , 2 ( t ) X 2 ( t − 1) C 2 , 2 ( t ) X 2 ( t − 1) C 2 ,K ( t ) X N ( t − 1) C N , 1 ( t ) X N ( t − 1) C N , 2 ( t ) X N ( t − 1) C N ,K ( t ) Fig. 3: Bipartite graph for the Maxim um W eighted Matching (MWM) policy . An M WM poli cy at each t ime slot observes the qu eue length s X n ( t − 1 ) and the connectiv- ity variables C n,k ( t ) and d etermines a maximum weighted matching (th e matching in dicator var iables) in the bi partite g raph of Fig ure 3. Note that, by const ruction, the MWM pol icy is causal. Deﬁnition 2: W e deno te the set of all poli cies that employ maxi mum weighted matching at all time slots by Π MWM . According t o prop erty (a) above, the set Π MWM is not empty . Moreover , according to property (b) , we conclude that Π MWM may contain an inﬁnite number of p olicies. I V . D E L AY O P T I M A L I T Y O F M W M P O L I C Y In th is section, we prove the delay optimalit y of an M WM poli cy π ∈ Π MWM . Th is resul t is formally present ed in Theorem 2. M ore speciﬁcally , we show t hat i n an M QMS-T ype2 system with i.i.d . Bernoulli arri val and connectivity processes, any M WM pol icy is opti mal in min imizing, in stochastic or dering s ense , a class o f cost functions of q ueue length processes including average qu eueing delay . For brevity we will use the term “ delay opt imality ” to refer to the optim ality of MWM in this s ense. A. Delay Op timality of MWM P olicy-Outli ne of th e Pr oof The proof of Theorem 2 p roceeds along the foll owing steps: First, W e will introduce th e notion of balanced qu eue vectors and the corresponding bal ancing server r eallocat ion at ti me slot t in Deﬁnition 4. For any given policy π and a ﬁxed t ime s lot August 16, 2018 DRAFT 11 P S f r a g r e p l a c e m e n t s MW index Maximization Queue Queue Balancing Balancing Lemma 1 Lemma 2 Policy Improvement (Delay Reductio n) Lemma 3 f o r a n y π / ∈ Π M W M Theorem 1 Delay-optimal pol icy belongs t o Π MWM All MWM pol icies result in the same av erage delay Lemmas 4 and 5 Any MWM p olicy is delay optim al Fig. 4: Out line of th e proof t , we wil l also deﬁne the Matching W eight in dex MW π ( t ) in Deﬁnitio n 5. Note that this ind ex is not directly related to (aver age) delay; it is , howe ver , a crucial link in com paring arbitrary policies to the MWM ones in the set Π MWM deﬁned in the pre vious section . W e show t hen in Lemmas 1 and 2 that the notio ns o f “maxim izing the M atching W eight index” and “producing balanced queue vectors” via balancing server reallocations are equiv alent . This property allows us to characterize MWM po licies as ones that p roduce the most b alanced queue size vectors possible. Second, we use the balanced queue si ze property to show in Lemm a 3 t hat for any arbit rary policy π o utside th e set Π MWM , we may construct a policy i n t he set Π MWM that improves π in terms of delay . In the words of Theorem 1, we prove that the delay-optim al policy b elongs to the set of MWM poli cies Π MWM . Third, in Lemmas 4 and 5 we will sho w that all policies in the set Π MWM result in the same cost (and hence a verage delay). Fin ally , by using Theorem 1 and Lemma 4 we conclude Theorem 2 where we show that t he policies in the set Π MWM are all delay-optimal. Graph theoretic analysis is applied i n the proof o f Lemmas 2 and 5 and s tochastic ordering and dyn amic coupling ar gument s are used to prove Lemmas 3 and 4. B. Equivalence of Queue Length Balancin g and Ma ximum W eighted Matching W e start this section b y introduci ng the intermediat e queue state in the following d eﬁnition. Deﬁnition 3: L et X ′ ( t ) = ( X ′ 1 ( t ) , X ′ 2 ( t ) , . . . , X ′ N ( t )) denote the queue length vector at time slot t exactly after serving th e queues accor di ng to a server assi gnment policy π and befor e August 16, 2018 DRAFT 12 adding the new ar rivals of time sl ot t , i.e., X ′ n ( t ) = X n ( t − 1) − K X k =1 C n,k ( t ) M ( π ) n,k ( t ) ! + . (4) W e call this vector as a the intermediate queue s tate. Recall that the ﬁnal state o f queue n at time slot t is determined after adding t he new arri vals. Giv en x ′ ( t ) as a sampl e value of random vector X ′ ( t ) , we deﬁne a balan cing server r eal lo- cation at time slot t as follows. Deﬁnition 4: A ssume that the empl oyed m atching at time sl ot t (assignment of servers t o the q ueues at time slot t ) will result in the intermedi ate queue vector x ′ ( t ) . A balancing s erver reallocation at this time slo t is a new matching resulting in in termediate vector ˜ x ′ ( t ) such that one of the following conditi ons is satis ﬁed. ( C1 ) ˜ x ′ n ( t ) ≤ x ′ n ( t ) for all n = 1 , 2 , . . . , N and there exists an m ∈ { 1 , 2 , . . . , N } such that ˜ x ′ m ( t ) < x ′ m ( t ) . ( C2 ) ˜ x ′ ( t ) and x ′ ( t ) are different in o nly two elements n and m such that x ′ n ( t ) < ˜ x ′ n ( t ) ≤ ˜ x ′ m ( t ) < x ′ m ( t ) and the following const raints are satisﬁed: ˜ x ′ n ( t ) = x ′ n ( t ) + 1 and ˜ x ′ m ( t ) = x ′ m ( t ) − 1 . A balancing server reallo cation is a crucial too l in deﬁning new policies that improve the delay performance of an arbitrary poli cy as we will see in t he proo f later . Example: Consider a system with three queues and three s ervers. Assume that x ( t − 1) = (3 , 2 , 5) is the qu eue length vector right at the end of time s lot t − 1 (or at th e beginning of ti me slot t ). W e consider t wo distinct examples to s how the deﬁnition of balancing s erver reallocations corresponding to each of the cases C1 and C2 i n Deﬁnition 4. Figures 5a and 5b show these examples of b alancing s erver reallocatio ns. In each case, we also show the weight of each edge ( n, k ) which is equal to c n,k ( t ) x n ( t − 1) . In these ﬁgures, since none of th e queues is empty , the edges wi th weight 0 are the ones which are discon nected. W e hav e speciﬁed t he original allocations by solid lines and the balancing o nes by dashed lines. For t he syst em i n Figure 5 a, the original allocation wil l result i n th e intermediate vector x ′ ( t ) = (3 , 1 , 4) while the balancin g server reallocation wil l result in the in termediate vector ˜ x ′ ( t ) = (2 , 1 , 4) . The vectors x ′ ( t ) and ˜ x ′ ( t ) satisfy Condition C1 . For the sys tem in Figure 5b, the orig inal allocation wi ll result in the i ntermediate vector x ′ ( t ) = (2 , 1 , 5) whil e the balancing server reallocation wil l resul t in ˜ x ′ ( t ) = (3 , 1 , 4) . The vectors x ′ ( t ) and ˜ x ′ ( t ) satisfy Conditio n C2 . August 16, 2018 DRAFT 13 P S f r a g r e p l a c e m e n t s x 1 ( t − 1) = 3 x 2 ( t − 1) = 2 x 3 ( t − 1) = 5 1 2 3 0 3 2 2 5 5 x 1 ( t − 1 ) = 3 x 2 ( t − 1 ) = 2 x 3 ( t − 1 ) = 5 3 0 2 2 5 0 1 2 3 (a) S atisfying condition C1 P S f r a g r e p l a c e m e n t s x 1 ( t − 1 ) = 3 x 2 ( t − 1 ) = 2 x 3 ( t − 1 ) = 5 1 2 3 0 3 2 2 5 5 x 1 ( t − 1) = 3 x 2 ( t − 1) = 2 x 3 ( t − 1) = 5 3 0 2 2 5 0 1 2 3 (b) S atisfying condition C2 Fig. 5: Examples of balancin g server reallocati ons (the weight c n,k ( t ) x n ( t − 1) of each edge ( n, k ) is also shown) Deﬁnition 5: For a s erver ass ignment po licy π with t he allocation variables { M ( π ) n,k ( t ) } ∞ t =1 , ∀ k ∈ K and ∀ n ∈ N , w e deﬁne Matching W eight ( MW ) index at time slot t by MW π ( t ) = N X n =1 X n ( t − 1) K X k =1 C n,k ( t ) M ( π ) n,k ( t ) . (5) MW index is exactly the ob jectiv e of th e opt imization problem (3). MW π ( t ) is an index associated w ith policy π at t ime s lot t whose value is dependent on the s tate of the syst em (queue lengths and connectivities) as well as the matching em ployed by policy π at time slot t . In the following lemmas (Lemmas 1 and 2), we relate the notions of balanci ng server reallocation and Mat ching W eight in dex and we p rove that maximization of MW π ( t ) in dex and balancing of the queues are equiv alent. More s peciﬁcally , we show that if the MW π ( t ) index for policy π is not maxim ized at time slot t ( π is not using a maximum weighted m atching), then there exists a balancing server reallocation (i.e., a new matching that satisﬁes either C1 or C2 ) that results i n a lar ger MW index. Furthermore, if π is using a maximum weighted matching, th en there exists no bal ancing server reallocation at th at time sl ot, i.e., no m atching can be found that sati sﬁes either C1 or C2 . These facts are formally stat ed in the foll owing two lemmas . Lemma 1: For a giv en policy π employing matching M ( π ) ( t ) at tim e slo t t , by applying a balancing server reallocation at time slot t (if there exists any), we can create a new poli cy ˜ π (diffe ring from π on ly at time slot t ) such that MW π ( t ) < MW ˜ π ( t ) . The detail ed proof o f the l emma is giv en in Append ix I-A. Based on Lemma 1, we can s tate the following corollary . August 16, 2018 DRAFT 14 Cor oll ary 1: For a giv en policy π at time slot t , if MW π ( t ) i s maxim ized, i.e., policy π employs a maximum weighted matching at time slot t , t hen there exists no balancing server reallocation at that time slot. Lemma 1 st ates that any balancing server reallocatio n strictly increases the matching weig ht index. Howe ver , it does not imply t he existence of a balancing server reallocation when MW π ( t ) is not m aximized. In the foll owing, we p rove the existence result i.e., the in verse of Lemma 1. Lemma 2: For a give n po licy π at time s lot t , if MW π ( t ) is not maximi zed, i.e., if M W π ( t ) < MW MWM ( t ) , then there exists a balancing server reallo cation at that tim e sl ot. For the detailed proof, pl ease refer to Appendix I-B. Using Lemmas 1 and 2, we can conclude that maxim izing the m atching weight is equiv alent to balancing the qu eues (in a sense that there is no further matching that can satisfy C1 or C2 in Deﬁnit ion 4). Hence, an MWM matching will result in the most balanced intermediate queue state where no balancing server reallocation is possible. This property of an MWM matching will be crucial in the proo f of L emma 3. C. Ba c kgr ound on Stochastic Or dering and Dynamic Coupling In this s ection, we brieﬂy revie w the concepts of st ochastic orderin g (stochastic do minance) and dynamic cou pling techniques. These concepts are needed i n the proof of delay opti mality of MWM poli cy in the rest of our discussion. The reader is encouraged to consult [32]–[34] for more details about stochast ic ordering and dynamic coup ling. Deﬁnition 6: Cons ider two real-valued, discrete-tim e sto chastic processes A = { A ( t ) } ∞ t =1 and B = { B ( t ) } ∞ t =1 in R . W e say A is stochastically smaller than B and we write A ≤ st B if Pr( A ( t ) > r ) ≤ Pr( B ( t ) > r ) for all t = 1 , 2 , . . . and all r ∈ R [32], [33]. The fol lowing two properties of stochastic ordering are useful: if A ≤ st B , t hen (a) E [ A ( t )] ≤ E [ B ( t )] (b) f ( A ) ≤ st f ( B ) for all non-decreasing function s f . Process A i s st ochastically smaller than B , if there exists a process ˜ A = { ˜ A ( t ) } ∞ t =1 deﬁned on t he same p robability space as B , has the same probabilit y di stribution as A and satisﬁes ˜ A ( t ) ≤ B ( t ) alm ost surely (a.s.) for ever y t = 1 , 2 , . . . [23]. The l ast statement is known as coupling of A and ˜ A . When applying coupl ing technique, give n the process A , we cons truct a coupled process ˜ A with the same dis tribution as A and ˜ A ( t ) ≤ B ( t ) a.s. for all t . This gives August 16, 2018 DRAFT 15 us a tool for comparing the process es A and B st ochastically when it is infeasibl e to derive the d istributions o f A and B (e.g., i n our queueing model when comparing the total occupancy process for different server assignment policies). D. Delay Opt imality of MWM In this subs ection, w e will elaborate on proving t he delay opt imality of any MWM policy . W e ﬁrst introd uce some deﬁnitions. W e d enote by Z + the set of non-negativ e integers and by Z N + the N dimensi onal Cartesian space of n on-negati ve integers. W e d eﬁne the relati on “  ” o n Z N + as fol lows. Deﬁnition 7: For two vectors x , ˜ x ∈ Z N + , we writ e ˜ x  x if one of the following relation s holds: D1 : ˜ x n ≤ x n for all n = 1 , 2 , . . . , N . D2 : ˜ x is obtained by permutat ion of two di stinct elements of x , i.e., ˜ x and x are d iff erent in only two elements n and m such that ˜ x n = x m and ˜ x m = x n . In thi s case, we say ˜ x and x are equal in permutation and we write ˜ x p = x . D3 : ˜ x and x are different in only two element s n and m such that x n < ˜ x n ≤ ˜ x m < x m and the fol lowing constraints are satisﬁed: ˜ x n = x n + 1 and ˜ x m = x m − 1 . The three relations D1 , D2 and D3 are mutually exclusive. In D3 , we s ay that ˜ x is mo re balanced than x and can be obt ained by decreasing a lar ger element of x (i.e., m ) by one and in creasing a smaller element (i.e., n ) by one. W e call such an interchange as a balancing i nter change on vector x . Thus , the result of a balancing interchange on a vector x would be a vector ˜ x such that ˜ x  x . According to Deﬁnition 4, a balancing s erver reallocation sati sfying Condit ion C2 , will result in a balancin g i nterchange between x ′ ( t ) and ˜ x ′ ( t ) . W e deﬁne t he partial order “  p ” on Z N + as the transit iv e closure of relation“  ” [35]. In other words, ˜ x  p x i f and only if ˜ x i s obtained from x by performing a sequ ence of reductions (i.e., reducing an element o f the vector x such that x and ˜ x satisfy D1 ), permutatio ns of two elements (permutation of two elements of th e vector x such t hat x and ˜ x satisfy D2 ) and/or balancing int erchanges (su ch that x and ˜ x satisfy D3 ). When x and ˜ x are two queue l ength vectors, we write ˜ x  p x i f and only i f queue length vec tor ˜ x is ob tained from x by applying a sequence of packet remov als, two-queue permutati ons and b alancing interchanges. August 16, 2018 DRAFT 16 Deﬁnition 8: W e deﬁne F as the class of real-valued functions on Z N + that are mono tone and non-decreasing with respect to the partial o rder  p , i.e., f ∈ F ⇐ ⇒ ˜ x  p x ⇒ f ( ˜ x ) ≤ f ( x ) . (6) W e can easil y check that function f ( x ) = P N n =1 x n belongs to F . This function represents the total queue occupancy of t he system. Deﬁnition 9: W e deﬁne Π t , t = 1 , 2 , . . . , as the set of all poli cies that employ maximum weighted matching in ev ery time slot τ = 1 , . . . , t . W e o bserve that Π t − 1 ⊇ Π t and Π MWM = T ∞ t =1 Π t . Consider a policy π ∈ Π t − 1 which is using an arbitrary matching M ( π ) ( t ) at tim e slot t . If M ( π ) ( t ) is not a maximum weighted matching , th en from Lemmas 1 and 2 we conclude t hat by applying a sequence of balanci ng server reallocatio ns 6 we can create a poli cy π ⋆ ∈ Π t . Let h π t denote the number of balancing server reallocations required to con vert the empl oyed m atching in policy π at t ime slot t to a m aximum weighted matching. Deﬁnition 10: W e deﬁne the dist ance of policy π ∈ Π t − 1 from the set Π t to be h π t balancing server reallocations . According to Lemmas 1 and 2, s ince by applying each server reallocation, the matchi ng weight index s trictly increases, the num ber of balancing server reallocations needed to con vert π to a maximum weight ed matching is b ounded, i.e., h π t ≤ H < ∞ for all t, π . Hence, after applying the ﬁrst balancing server reallocation at time slot t we reach a policy ˜ π 1 whose distance from Π t is h π t − 1 balancing server reallocations. By repeatin g this procedure we ﬁnally identify a policy w hose d istance to Π t is zero, i.e., it belongs to Π t . Figure 6 i llustrates t he deﬁnition of th e distance h π t and how balancing server reallocations resul t in identi fying a policy that employs a maximum weighted matchin g at time slot t . In this ﬁgure, x ′ π ( t ) is the intermediate queue state due to the emp loyed m atching at time sl ot t and x ′ ˜ π 1 ( t ) , x ′ ˜ π 2 ( t ) , . . . , x ′ ˜ π h π t ( t ) are the int ermediate queue states after applying the b alancing server reallocations . Deﬁnition 11: By Π h t ( 0 ≤ h ≤ H ) we denote the set of all server assign ment policies in Π t − 1 whose distance from Π t is h balancing server reallocati ons. Recall that Π 0 t = Π t . 6 According to L emma 1, each balancing server reallocation strictly increases the matching weight index. August 16, 2018 DRAFT 17 P S f r a g r e p l a c e m e n t s employed matching π ﬁrst balancing reallocation ˜ π 1 second balancing reallocation ˜ π 2 h π t th balancing reallocation ˜ π h π t h π t MW π ( t ) < MW MWM ( t ) MW π ( t ) < MW ˜ π 1 ( t ) < MW MWM ( t ) MW ˜ π 1 ( t ) < MW ˜ π 2 ( t ) < MW MWM ( t ) MW ˜ π h π t ( t ) = MW MWM ( t ) x ( t ) time slot t x ( t − 1) x ( t − 1) x ( t − 1) x ( t − 1) x ( t − 1) x ′ π ( t ) = ⇒ x ′ ˜ π 1 ( t ) = ⇒ x ′ ˜ π 2 ( t ) = ⇒ x ′ ˜ π h π t ( t ) = ⇒ This balancing reallocation creates maximum weight matchin g Fig. 6: h π t balancing server reallocation s are required to create a policy in Π t from pol icy π ∈ Π t − 1 . M W i ndices giv en the state of t he system at time slot t are also com pared. Deﬁnition 12: For any two policies π and ˜ π with qu eue length processes X = { X ( t ) } ∞ t =1 and ˜ X = { ˜ X ( t ) } ∞ t =1 , respectively , we say ˜ π d ominates π , if f ( ˜ X ) ≤ st f ( X ) , f ∈ F , i .e., the queue length cost (delay) of pol icy ˜ π is stochast ically less than that of poli cy π . If ˜ π dom inates π we hav e E [ f ( ˜ X )] ≤ E [ f ( X )] 7 . In the following lemm a, we will interconnect the notio ns of “maxim izing the matching weight index” and “delay optim ality” and show that maximizatio n of the matching weight index (at any giv en time t ) will improve t he delay performance (wil l decrease the queue length cost function f ( X ) stochasti cally). The key element in t he interconnection is t he notion of balancing server reallocatio n. In particul ar , we show that, for any giv en pol icy π ∈ Π h t , h = h π t that does not employ a m aximum weighted matching at time s lot t (i.e., h > 0 ), there exists a balancing server reallocation at time slot t . In the following lemma, we show that by using such a balancing server reallocatio n at time slot t we can construct a n e w pol icy ˜ π that domi nates the original pol icy π . For the detailed proof, please refer to Appendi x II-A . W e used stochastic ordering and dy namic coupling to prove this lemma. Lemma 3: For any pol icy π ∈ Π h t where h = h π t > 0 , we can construct a po licy ˜ π ∈ Π h − 1 t such that ˜ π dom inates π . T hus, ˜ π outperforms π i n t erms of ave rage queueing delay . 7 Choosing f ( x ) = P N n =1 x n , we conclude that the expected t otal queue occupancy (or equi valen tly average queueing delay) of policy ˜ π is smaller than that of policy π in every time slot. August 16, 2018 DRAFT 18 Using Lemm a 3, we can prove the following theorem which states that any MWM pol icy outperforms any non-M WM policy in terms of average queueing delay . Theor em 1: For any s erver assi gnment policy π / ∈ Π MWM , there exists an M WM policy π ∗ ∈ Π MWM such that π ∗ dominates π . Pr oo f: Let π be any ar bitrary non-MWM policy . Then π ∈ Π H 1 1 where H 1 = h π 1 . By a pplyi ng Lemma 3 repeatedly , we can construct a sequ ence of poli cies such that each policy domin ates the previous one. Th us, we obtain policies that belong to Π H 1 1 , Π H 1 − 1 1 , Π H 1 − 2 1 , . . . , Π 0 1 = Π 1 . The last policy is called π 1 for which we have π 1 ∈ Π H 2 2 where H 2 = h π 1 2 . By continui ng su ch an ar gument, we ob tain a sequence of poli cies π t ∈ Π t , t = 1 , 2 , . . . such that π j dominates π i for j > i . This sequence of policies deﬁnes a lim iting policy π ∗ that agrees with MWM at all time slots. T hus, π ∗ is an MWM policy that dominates all the pre vious policies, including the startin g policy π . This proves t hat t he delay-optimal policy is an MWM policy in Π MWM . As we mentioned before, t he set Π MWM may contain an inﬁnite number of poli cies. In the following, we sh ow that any MWM policy is delay-optimal. T o achieve this, we need to prove the following lemma 8 . Lemma 4: The qu eue length costs of all the maximum wei ghted matching policies in Π MWM are equal in dis tribution, i.e., for any two MWM po licies π 1 , π 2 ∈ Π MWM , we have f ( X ( π 1 ) ) D = f ( X ( π 2 ) ) wh ere X ( π 1 ) and X ( π 2 ) are the queue length processes un der π 1 and π 2 , respectively . The proof of this lemm a is provided in Appendix II-D. Using Th eorem 1 and Lemma 4, we can conclude t he main result of this section in the following theorem. Theor em 2: Any Maximum W eighted Matching policy dominates any server assig nment pol- icy , i.e., any MWM poli cy is delay-optim al. E. Extensions 1) Imperfect Services: W e can extend Theorems 1 and 2 for the case where the service of a scheduled packet by a connected s erver fails rando mly wit h a certain probability . This can model the operation of realistic wireless networks where service failures usuall y occur du e to unexpected and unpredictable eff ects of noise, interference, etc. In the case of a packet service 8 As part of the proof f or this lemma, we need preliminary Lemma 5 presented and proven in Appendix II-C August 16, 2018 DRAFT 19 failure, the packet will be kept in the qu eue and will be rescheduled and retransmitted in futu re time slots. By the random variable Q n,k ( t ) ∈ { 0 , 1 } , we denote th e successful/ unsuccessful service o f queue n provided by server k at time slot t ; a value of 1 (resp. 0 ) denotes that the service is successful (resp. unsuccessful). W e assume that Q n,k ( t ) , ∀ n ∈ N , ∀ k ∈ K are i.i.d. Bernoulli random v ariables with the same success probabil ity q . The parameter q (simi lar to parameters λ and p ) is not explicitl y in volved in our analysis other t han the fact that E [ Q n,k ( t )] = q , ∀ n, k , t . The queue lengths are then updated at t he end of each tim e sl ot by t he following rule. X n ( t ) = X n ( t − 1) − K X k =1 C n,k ( t ) M ( π ) n,k ( t ) Q n,k ( t ) ! + + A n ( t ) ∀ n ∈ N (7) The network scheduler (that performs server assignment process) cannot observe the variables Q n,k ( t ) and from its perspectiv e they are assumed t o be rando m. The random vector X ′ ( t ) is deﬁned similar to equation (4). Hence, X ′ ( t ) represents the queue lengths before adding the new arri vals of ti me s lot t as if all the s ervices at t hat t ime slot are successful. For such a syst em, we can verify t hat Lemmas 1 and 2 are valid. W e can extend Lemma 3 for the system with service failures by considering the random var iables Q n,k ( t ) in our dyn amic coupling argument. The proof is followed by usi ng the same approach as i n Lemm a 3. The detailed analysis is broug ht in Appendix II-B. By applying the same approach as in the proof of Theorem 1 and Lemma 4, we can s imilarly prove the delay optimality of MWM policy for the system with imperfect services. 2) Extension s f or Connectivity and Arrival P r ocesses: The ar gument s i n Lem mas 3 and 4 and Theorem 1 remain valid if the i. i.d. assumption for connecti vity and arri val proce sses is relaxed as follows; we will consider connecti vity and ar riv al processes which follow con ditional permutation in variant distributions. Given eve nt H (which is used to denote the history of the s ystem), we deﬁne a condi tional multivariate probabilit y distribution f ( y 1 , y 2 , . . . , y n | H ) to be permutatio n in variant if for any permutation of t he variables y 1 , y 2 , . . . , y n namely y ′ 1 , y ′ 2 , . . . , y ′ n we have f ( y 1 , y 2 , . . . , y n | H ) = f ( y ′ 1 , y ′ 2 , . . . , y ′ n | H ) . W e can readil y see that for all the connectivity and arri val processes who se joi nt distributions at each time slot giv en the history of th e system 9 9 By history of the system we mean all the channel states, arriv als and matchings of the previous time slots up to time slot t . August 16, 2018 DRAFT 20 (i.e., f A ( t ) ( a 1 , a 2 , .., a N | H ) and f C ( t ) ( c 1 , 1 , c 1 , 2 , . . . , c N ,K − 1 , c N ,K | H ) ) are p ermutation in variant, Lemmas 3, 4 and Theorem 1 are still valid and t herefore MWM is delay-optim al. W e also consider the generalization of Th eorems 1 and 2 for non-Bernoull i arri val processes. Suppose t hat the number of arriv als to each queue can be represented b y the summati on o f som e i.i.d. Bernoulli random variables, i.e., has Binom ial dist ribution. Also suppose that A n ( t ) ≤ A max for all n ∈ N and al l t . In this case, w e can create a new (virtual) sys tem in which after each time slot we append A max − 1 virtual time slo ts and put the connectivities all equal to zero, i.e., for each virt ual time slot t , C n,k ( t ) = 0 , ∀ n ∈ N , ∀ k ∈ K . W e then distribute the arri vals of t he actual time slot among these A max time slots (one actual ti me s lot and A max − 1 virtual time slots ) randomly s uch that at each ti me slot at mos t one packet arriv al occurs. Since the connectivities and the arri vals in both sys tems are permutat ion in variant , we can s till prove Theorems 1 and 2 for the virtual sy stem. W e observe that the operation of the two systems (the original system and the virtual sys tem) are the same. Therefore, we can conclude that Theorem 1 is also valid for a multi-s erver sy stem with Binomial arriv al p rocesses. V . S I M U L A T I O N R E S U LT S W e have compared the del ay performance of M WM p olicy with two alternative server assign- ment policies described in the fol lowing. • Maximum Matching (MM) policy appl ies the maximum matching on matrix C ( t ) . The maximum matching p olicy at each time s lot t empl oys a server assign ment (or matching) M ( MM ) ( t ) which is obtained by solving t he following problem (equiv alent to ﬁnding the maximum matching in the connectivity matrix). Maximize: N X n =1 K X k =1 M n,k ( t ) C n,k ( t ) Subject to: K X k =1 M n,k ( t ) ≤ 1 , ( n = 1 , 2 , . . . , N ) , N X n =1 M n,k ( t ) ≤ 1 , ( k = 1 , 2 , . . . , K ) . (8) The MM maximizes t he ins tantaneous throughput at each time slot withou t cons idering the queue l ength information in its server assignment decisions. August 16, 2018 DRAFT 21 • A heuristic policy that assigns the server s to th e queues at each tim e sl ot according to the following rule: It selects a server randomly and assigns it to its longest connected queue. Then, u pdates t he set of s ervers by removing the selected server from K and the set of queues (i.e., N ) b y removing the queue to which the selected server was assi gned. This procedure is repeated K times. For some servers t he updated set N may be empty (e.g., when K > N ) and therefore tho se servers are no t ass igned to any queue. Algorithm 1 Heuristic Poli cy Pseudocode input : N , K , c ( t ) and x ( t − 1 ) initialize : M ( H ) ( t ) = ( 0 ) N × K f or i = 1 to K do Choose a server k ⋆ ∈ K random ly if N 6 = ∅ n ⋆ ← − arg max n ∈N c n,k ⋆ ( t ) x n ( t − 1) M ( H ) n ⋆ ,k ⋆ ( t ) ← − 1 N ← − N − { n ⋆ } endif K ← − K − { k ⋆ } end Return M ( H ) ( t ) The mo tiv ation for the heuris tic pol icy i s com ing from Longest Connected Queue (LCQ) policy which was prov en in [2] t o be optimal for a single-server sys tem. For mul ti-server sy stem, we will use the same principl e for each s erver . Howe ver , the order in which servers are selected for assignment is random. W e have preformed a comprehensive set of simul ations in which we in vestigate the ef fects of the number of servers K , the probabilit y of connectivity p and the probabil ity of servi ce s uccess q on the p erformance of the aforementioned poli cies. In all t he simulation s, we set N = 8 and the arri vals are i.i.d. Binom ial di stributed which is the su mmation of 10 Bernoulli random variables. W e us e log-scale for the y-axis in the ﬁgures so that we can easily compare t he performance of different poli cies in low arriv al rates (where the av erage queue lengths are very close for August 16, 2018 DRAFT 22 0.05 0.1 0.15 0.2 0.25 0.3 10 0 10 1 10 2 10 3 Arrival Rate per Queue (Packets/time slot) Average Queue Occupancy (Packets) Maximum Weighted Matching Policy Maximum Matching Policy Heuristic Policy (a) p = 0 . 2 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 10 0 10 1 10 2 10 3 Arrival Rate per Queue (Packets/time slot) Average Queue Occupancy (Packets) Maximum Weighted Matching Policy Maximum Matching Policy Heuristic Policy (b) p = 0 . 5 Fig. 7: A verage total queue occupancy , N = 8 , K = 4 , q = 0 . 8 diffe rent policies). Figures 7-9 il lustrate the simu lation results. In all the cases, the conﬁdence i nterval is very small and is n ot visi ble i n th e graphs. As we can see in all cases, MWM exhibits im proved performance with respect t o t he other policies i n t erms o f av erage queue o ccupancy or ave rage queueing delay . Figure 7 shows the simulati on results for K = 4 , q = 0 . 8 and p = 0 . 2 , 0 . 5 . In th ese cases, si nce the numb er of servers is relativ ely l ow , server assignm ent will be m ore competitive. As s hown earlier , the M WM minimizes the queue imbalance. The heuristi c policy August 16, 2018 DRAFT 23 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 10 0 10 1 10 2 10 3 Arrival Rate per Queue (Packets/time slot) Average Queue Occupancy (Packets) Maximum Weighted Matching Policy Maximum Matching Policy Heuristic Policy (a) p = 0 . 2 0.6 0.62 0.64 0.66 0.68 0.7 0.72 0.74 0.76 0.78 10 2 10 3 Arrival Rate per Queue (Packets/time slot) Average Queue Occupancy (Packets) Maximum Weighted Matching Policy Maximum Matching Policy Heuristic Policy (b) p = 0 . 5 Fig. 8: A verage total queue occupancy , N = 8 , K = 8 , q = 0 . 8 follows th e same p rinciple. H owe ver , since the s election o f servers for assignment is random, i n certain cases it may h appen that two or more servers hav e the s ame lon gest conn ected queue. In such cases, the order o f selecting the server s for ass ignment does hav e an effect on the system performance. Maxim um Matching pol icy howe ver , does not try to b alance the queues since by construction it does not consid er the queue lengt hs in its assi gnments and that is why it p erforms worse than the oth er two p olicies. W e o bserve that as the connectivity p robability g ets larger , the performances of MWM and the heuristic policy get clos er . It is worth menti oning that the August 16, 2018 DRAFT 24 0.1 0.2 0.3 0.4 0.5 0.6 10 0 10 1 10 2 10 3 Arrival Rate per Queue (Packets/time slot) Average Queue Occupancy (Packets) Maximum Weighted Matching Policy Maximum Matching Policy Heuristic Policy (a) q = 0 . 8 0.02 0.04 0.06 0.08 0.1 0.12 0.14 10 0 10 1 10 2 10 3 Arrival Rate per Queue (Packets/time slot) Average Queue Occupancy (Packets) Maximum Weighted Matching Policy Maximum Matching Policy Heuristic Policy (b) q = 0 . 2 Fig. 9: A verage tot al qu eue occupancy , N = 8 , K = 6 , p = 0 . 5 heuristic policy introduced here performs the same as MWM for K = 1 (which is equiv alent to LCQ whose optim ality has been p re viously shown in [2]). Figure 8 shows the results for 8 servers. In this case, since the n umber of servers i s relatively lar ge and comparable to the numb er of queues, in MWM and MM policies each queue gets service with hig h probabili ty wh en the prob ability of server connectivities in creases. As the connectivity probabili ty g ets smaller , the differe nce in performance of MWM and MM b ecomes more apparent. In t his case, the heuristic pol icy performs worse than the other t wo policies si nce August 16, 2018 DRAFT 25 it is more probable to lead to cases where two or mo re s ervers hav e th e same longest conn ected queue. As th e numb er of servers increases, we expect MM t o perform the same as MWM as in this case the probabi lity of s erving all th e queues i ncreases. Therefore, i n the lim iting case where K becom es very large, M M and MWM result in very close performance. In Figure 9 we have in vestigated the ef fect of service success probability . As we can see in the ﬁgures, the on ly effect of this parameter is to change the stabili ty point (the arri val rate at which queue occupancy tends t o inﬁnity). In t his case, again we can see th at for bot h q = 0 . 2 , 0 . 8 , MWM policy outperforms the other policies. V I . C O N C L U S I O N S In t his paper , we consi dered the problem of assi gning K identical servers to a set of N parallel q ueues in a t ime-slotted, multi-server queueing syst em with random connectivities. For such sys tems, it has b een previously s hown that MWM is throughput-opt imal, i .e., has the maximum stabi lity region. In this p aper , we s howed that for a s ystem wit h i.i.d. Bernoulli arriv al and connectivity processes, MWM is als o opt imal for m inimizing a class of cost function s of queue l engths includin g the a verage queueing delay . W e ﬁrst proved th at MWM and qu eue length balancing are equiv alent . Th en, usi ng t his result and by applying the noti ons of stochastic ordering and dynamic coup ling techniques, we proved the delay o ptimality of M WM. Finall y , we considered extensions of the mod el in which we hav e imp erfect packet services or more general packet arriv al and server connectivity processes. W e hav e shown t he opti mality o f MWM in these cases as well. A P P E N D I X I P R O O F O F L E M M A 1 A N D L E M M A 2 A. Pr oo f of Lemma 1 Pr oo f: Let M ( ˜ π ) ( t ) denote the employed matching after applyi ng the balancin g server reallocation. According to the deﬁniti on of balancing server reallocati on, a server reallocation at ti me slot t results in an i ntermediate queue leng th vector ˜ x ′ ( t ) that satisﬁes eith er conditio n C1 or C2 . Therefore, we con sider t he following two cases: Case 1 : Condition C1 is satisﬁed at time slot t . Thus, ˜ x ′ i ( t ) ≤ x ′ i ( t ) for all i = 1 , 2 , . . . , N and there exists at least one m ∈ { 1 , 2 , . . . , N } , such that 0 ≤ ˜ x ′ m ( t ) < x ′ m ( t ) . W e deno te August 16, 2018 DRAFT 26 the (sub )set of queues for which we ha ve 0 ≤ ˜ x ′ i ( t ) < x ′ i ( t ) by Q . Therefore, there exists no queue that was served by po licy π but not by policy ˜ π . Also t he queues in s ubset Q which were not recei ving service by policy π at ti me sl ot t , are n ow receiving service after app lying the balancin g server reallocation. Therefore, for all i ∈ Q , x i ( t − 1) P K k =1 c i,k ( t ) M ( π ) i,k ( t ) = 0 , x i ( t − 1) P K k =1 c i,k ( t ) M ( ˜ π ) i,k ( t ) = x i ( t − 1) > 0 . Thus, MW π ( t ) = X i / ∈Q x i ( t − 1) K X k =1 c i,k ( t ) M ( π ) i,k ( t ) + X i ∈Q x i ( t − 1) K X k =1 c i,k ( t ) M ( π ) i,k ( t ) < X i / ∈Q x i ( t − 1) K X k =1 c i,k ( t ) M ( ˜ π ) i,k ( t ) + X i ∈Q x i ( t − 1) K X k =1 c i,k ( t ) M ( ˜ π ) i,k ( t ) = MW ˜ π ( t ) , (9) and the result follows. Case 2 : Conditi on C2 is sat isﬁed at tim e t . In this case, by using policy π at ti me sl ot t queue n is receiving s ervice but queue m i s not. In contrast , by using policy ˜ π , at time slot t qu eue m is receiving service but queue n is not . The service of other qu eues is n ot disturbed, i.e., the other queues which were recei ving service by p olicy π st ill recei ve a service by policy ˜ π at t ime slot t and the ones that were not receiving s ervice under policy π still do n ot get service under policy ˜ π . Therefore, MW ˜ π ( t ) − MW π ( t ) = N X i =1 i 6 = m,n x i ( t − 1) K X k =1 c i,k ( t ) M ( ˜ π ) i,k ( t ) + x m ( t − 1) − N X i =1 i 6 = m,n x i ( t − 1) K X k =1 c i,k ( t ) M ( π ) i,k ( t ) − x n ( t − 1) = x m ( t − 1) − x n ( t − 1) > 0 (10) Therefore, MW π ( t ) < MW ˜ π ( t ) . B. Pr oo f of Lemma 2 Pr oo f: W i thout loss o f generality , we may con vert th e bipartit e graph G t to a complete weighted bipartite graph G ′ t with max { N , K } vertices i n each part. This is don e by adding some vertices and edges of zero weight as necessary . In parti cular , if N > K , we will add N − K servers on th e right hand side with edges of weig ht zero to each queue (each vertex on the left hand si de). If N < K , we will add K − N qu eues on th e l eft hand si de with edges of weight zero to each server (each vertex on the right hand side). Th is w ill not change t he operation of t he system since the added queues and servers are discon nected from the whol e August 16, 2018 DRAFT 27 system. W e denote the sets of vertices on each part of G ′ t by N ′ and K ′ , respective ly and the set of edges by E ′ . Consequ ently , a poli cy π is deﬁned as π = { M ( π ) ( t ) } ∞ t =1 where M ( π ) ( t ) is a perfect matching 10 in t he compl ete biparti te graph G ′ t . W e can easily verify t hat a maxim um weighted perfect matching M ( MWM ) ( t ) in the com plete bipartite graph G ′ t is the same as the maximum weighted matching in graph G t if we remove th e added edges of weig ht zero from matching M ( MWM ) ( t ) . Consider a policy π which is employing p erfect m atching M ( π ) ( t ) at time sl ot t . Suppos e that M ( π ) ( t ) is no t a maxim um wei ghted perfect matching on graph G ′ t , i.e., MW π ( t ) < MW MWM ( t ) . Also, consider a maxim um weight ed perfect matching M ( MWM ) ( t ) at time slot t . Now , consider these two matchings on G ′ t = ( N ′ , K ′ , E ′ ) . Each o f M ( π ) ( t ) and M ( MWM ) ( t ) corresponds to a distinct sub-graph of G ′ t namely G ′ ( π ) t = ( N ′ , K ′ , E ′ ( π ) ) and G ′ ( MWM ) t = ( N ′ , K ′ , E ′ ( MWM ) ) , respectiv ely . W e now build two di rected , weighted sub -graphs D ( π ) t and D ( MWM ) t as fol lows: D ( π ) t is the same as G ′ ( π ) t with all th e edges directed from N ′ to K ′ with the same edge weights as G ′ ( π ) t . D ( MWM ) t is the same as G ′ ( MWM ) t with all the edg es directed from K ′ to N ′ with edge weights equal to the ne gati ve of edge wei ghts of G ′ ( MWM ) t . Now , consider graph U = D ( π ) t S D ( MWM ) t , i.e., the union of the sub-graphs D ( π ) t and D ( MWM ) t . The graph U can be seen as the unio n of a number of ev en cycles 11 denoted by L . This is directly concluded from the fact that D ( π ) t and D ( MWM ) t are each perfect matchi ngs of G ′ t and t hus each vertex is incident to an incoming edge and an outgoi ng edge. Furthermore, for the weight of U shown by w ( U ) , we have w ( U ) = X ℓ ∈ L w ( ℓ ) = MW π ( t ) − MW MWM ( t ) < 0 . (11) In (11), w ( ℓ ) is the weight of edge ℓ in L . Th erefore, th ere m ust exist a negati ve cycle 12 in U . W e denote th is negati ve cycle by ℓ ⋆ . The cycle ℓ ⋆ is an even cycle and contains an ev en number of nodes and edges. W e assume that ℓ ⋆ contains 2 W nodes ( W nodes from N ′ and W nodes from K ′ ) and also 2 W edges. Let u s denot e the no des of sets N ′ and K ′ that form ℓ ⋆ by n 1 , n 2 , . . . , n W and k 1 , k 2 , . . . , k W , respectiv ely . Thus, the cycle ℓ ⋆ can be represented by the sequence of its edges as ℓ ⋆ = e n 1 ,k 1 , e k 1 ,n 2 , e n 2 ,k 2 , e k 2 ,n 3 , . . . , e k W − 1 ,n W , e n W ,k W , e k W ,n 1 (see Figure 10). 10 A perfect matching is a matching that matches all vertices of t he graph. 11 A cycle w ith ev en number of vertices. 12 A cycle w hose total edge weight is negati ve. August 16, 2018 DRAFT 28 P S f r a g r e p l a c e m e n t s n 1 n 2 n W k 1 k 2 k W Fig. 10: The negative cycle ℓ ⋆ The edges e k 1 ,n 2 , e k 2 ,n 3 , . . . , e k W ,n 1 belonging to D ( MWM ) t hav e negati ve weight s whil e the edges e n 1 ,k 1 , e n 2 ,k 2 , . . . , e n W ,k W belonging t o D ( π ) t hav e pos itive weights. W e can also represent the edges of ℓ ⋆ by pairing the edges incident to each node n 1 , n 2 , . . . , n W as follows; ℓ ⋆ = ( e k W ,n 1 , e n 1 ,k 1 ) , ( e k 1 ,n 2 , e n 2 ,k 2 ) , . . . , ( e k W − 1 ,n W , e n W ,k W ) . W e label each pair of edges as follows: r n 1 = ( e k W ,n 1 , e n 1 ,k 1 ) , r n 2 = ( e k 1 ,n 2 , e n 2 ,k 2 ) , . . . , r n W = ( e k W − 1 ,n W , e n W ,k W ) . All r n 1 , r n 2 , . . . , r n W pairs i n ℓ ⋆ hav e th e following property: The weights associated t o the edges of each pair r n i , 1 ≤ i ≤ W is either (0 , 0) , (0 , x n i ( t − 1)) , ( − x n i ( t − 1) , 0) or ( − x n i ( t − 1) , x n i ( t − 1)) . A ccordingly , we will specify three ty pes o f edge pairs as follows: • T ype 0 (T0) : pairs wi th edge weights (0 , 0) or ( − x n i ( t − 1) , x n i ( t − 1)) , 1 ≤ i ≤ W . • T ype 1 (T1) : pairs wi th edge weights ( − x n i ( t − 1) , 0) , 1 ≤ i ≤ W and x n i ( t − 1) > 0 . • T ype 2 (T2) : pairs wi th edge weights (0 , x n i ( t − 1)) , 1 ≤ i ≤ W and x n i ( t − 1) > 0 . Since ℓ ⋆ is a negativ e cycle, there must exist a node n j with edge pairs ( e k j − 1 ,n j , e n j ,k j ) = ( − x n j ( t − 1) , 0) and x n j ( t − 1) > 0 (if j = 1 , ( e k W ,n 1 , e n 1 ,k 1 ) = ( − x n 1 ( t − 1) , 0) and x n 1 ( t − 1) > 0 ). Obviously , the edge pair associ ated with n ode n j is of type T 1 . W e now consi der the follo wing three, exhaustiv e cases: Case 1 : All the edge p airs oth er than r n j are of type T0 . In this case, by replacing th e edges of D ( π ) t by t he edges of D ( MWM ) t in t he negative cycle ℓ ⋆ we obtain a s erver reallocation at tim e slot t . This server reallocation is balancing as in π queue n j was not receiving service whi le after the server reallocation it is. Therefore, if ˜ x ′ ( t ) is the queue leng th vector after server reallocation we hav e ˜ x ′ ( t ) ≤ x ′ ( t ) and ˜ x ′ n j ( t ) < x ′ n j ( t ) (satisfying condit ion C1 ). Case 2 : If we trace backward the edge pairs of cycle ℓ ⋆ from r n j and th e ﬁrst no n- T0 pair is a T1 pair . Let r n j ′ be su ch a T1 edge pair . In oth er words, in cycle ℓ ⋆ the pairs r n j ′ and r n j are of typ e T1 while other pairs between them are of type T0 (see Figure 11a). In this case, by August 16, 2018 DRAFT 29 P S f r a g r e p l a c e m e n t s 0 x n j ′ ( t − 1 ) − x n j ( t − 1 ) 0 k j ′ k j n j ′ n j T 2 T 0 T 1 − x n j ′ ( t − 1 ) 0 − x n j ( t − 1 ) 0 k j ′ k j n j ′ n j T 1 T 0 T 1 (a) Case 2 P S f r a g r e p l a c e m e n t s 0 x n j ′ ( t − 1 ) − x n j ( t − 1 ) 0 k j ′ k j n j ′ n j T 2 T 0 T 1 − x n j ′ ( t − 1 ) 0 − x n j ( t − 1 ) 0 k j ′ k j n j ′ n j T 1 T 0 T 1 (b) Case 3 Fig. 1 1: Cases 2 and 3 in the proof of Lem ma 2 replacing the edges of D ( π ) t by the edges of D ( MWM ) t just for nodes n j ′ +1 to n j of cycle ℓ ⋆ and allocating k j to n j ′ we obtain a server reallocation at time slot t . This server reallocation is balancing as in π queue n j does not receive service while after the server reallo cation it does. Furthermore, queue n j ′ was not being served in π but under t he new s erver reallocati on it may get service (depending on the weigh t of edge e k j ,n j ′ ) and the service of other queues i s not disturbed. Therefore, if ˜ x ′ ( t ) is the intermediate queue length vector after server reallocation, we hav e ˜ x ′ ( t ) ≤ x ′ ( t ) and ˜ x ′ n j ( t ) < x ′ n j ( t ) (satisfying condition C1 ). Case 3 : If we go backward on th e edge pairs of cycle ℓ ⋆ from r n j and th e ﬁrst non- T0 pair is a T2 pair . Let r n j ′ be such a T2 edge p air . In other words, in cycle ℓ ⋆ , the pair r n j ′ is of type T2 and r n j is of type T1 while other pairs between them are of type T0 (see Figure 11b). First of all, we claim that x n j ′ ( t − 1) ≤ x n j ( t − 1) . If x n j ′ ( t − 1) > x n j ( t − 1) , by replacing the edges of D ( MWM ) t by t he edges of D ( π ) t just for nodes n j ′ to n j − 1 of th e cycle ℓ ⋆ and not serving queue n j , without disturbi ng the service of other queues we obtai n a s erver reallo cation at time slot t with larger MW index than MW MWM ( t ) and this con tradicts the fact that MWM has the maximum MW in dex at tim e sl ot t . Accordingly , we cons ider t he following two s ub-cases: Sub-case 3 .1 : x n j ( t − 1) > x n j ′ ( t − 1) . In this case, we replace the edges of D ( π ) t by t he edges of D ( MWM ) t just for queues n j ′ +1 to n j of the cycle ℓ ⋆ and allo cate s erver k j to q ueue n j ′ . Server k j may or may not serve q ueue n j ′ . W e consi der the worst case where i t does not serve this queue ( w n j ′ ,k j = 0 ). Thus, wi thout di sturbing the service of o ther queues we obtain a server reallocation wit h the following property: all the queues other than n j and n j ′ hav e August 16, 2018 DRAFT 30 the same s ervice as before; queue n j which was receiving zero service in π , now receives one packet service and q ueue n j ′ which was receiving one packet service, now in the worst case loses its service. Therefore, this server reallocation is balancing as the queue l ength after server reallocation (denoted b y ˜ x ′ ( t ) ) and the queue length after applying poli cy π (denoted by x ′ ( t ) ) sati sfy conditi on C2 . So, ˜ x ′ ( t )  x ′ ( t ) and therefore th e applied server reallocation is a balancing one. Sub-case 3.2 : x n j ( t − 1) = x n j ′ ( t − 1) . In t his case, the sequence of edge pairs r n j ′ , . . . , r n j contribute “0” in the calculation of w ( U ) . Th erefore, we may treat them as a sequence o f edge pairs of type T0 . By doing so , we have a new n egati ve cycle ℓ ′ ⋆ which is t he same as ℓ ⋆ but the sequence of edge pairs r n j ′ , . . . , r n j are replaced by the same number of edge pairs of type T0 . Cycle ℓ ′ ⋆ is a negati ve cycle with the same weight as ℓ ⋆ . Therefore, th ere must exist an edge pair of type T1 in ℓ ′ ⋆ (In this case, it is not possible for ℓ ′ ⋆ to h a ve just edge pairs of typ e T0 as it results i n zero weight for the cycle) and all the cases of 1, 2 and 3 apply for ℓ ′ ⋆ as well . As long as case 3.2 is t rue, we always obtain a n e w negative cycle for which we will hav e one of the cases 1,2 and 3 satisﬁed where we can determine a balancing server reallocation. Cases 1, 2 and 3 cover all th e possible i nstances o f negative cycle ℓ ⋆ for all of which we prove d that there exists a balancing server reallocatio n. In summary , we proved that if t he policy π does not employ a maximum weighted matchin g at tim e slot t , there exists a negati ve cycle in U , the graph obtained by taking the union of the matchi ngs of π and MWM. W e p rove d th at by reallocation of t he servers in volved in the negative cycle, we always can ﬁnd a balancing server reallocation for policy π . A P P E N D I X I I P R O O F O F L E M M A S 3 A N D 4 A. Pr oo f of Lemma 3 Pr oo f: Fix any arbitrary policy π ∈ Π h t where h = h π t > 0 , and any arbitrary sample path ω = ( x (0) , c (1) , a (1) , x (1) , c (2) , a (2) , x (2) , . . . ) of the underlying random variables ( X (0 ) , C (1) , A (1) , X (1) , C (2) , A (2) , X (2) , . . . ) . W e apply the coupling metho d to con struct from ω a new sample path ˜ ω = ( ˜ x (0) , ˜ c (1) , ˜ a (1) , ˜ x (1) , ˜ c (2) , ˜ a (2) , ˜ x (2) , . . . ) resulting in a new sequence of random variables ( ˜ X (0 ) , ˜ C (1 ) , ˜ A (1) , ˜ X (1) , ˜ C (2) , ˜ A (2) , ˜ X (2) , . . . ) with August 16, 2018 DRAFT 31 X (0 ) = ˜ X (0 ) . Recall that X (0) is t he queue length vector in whi ch the sys tem starts. W e denote the policy deﬁned on the new sample path ˜ ω by ˜ π . In fact, we construct ˜ ω and ˜ π ∈ Π h − 1 t in such a fashion that for al l th e s ample paths we ha ve ˜ x ( t ′ )  p x ( t ′ ) for all t ′ = 1 , 2 , . . . . Therefore, for any f ∈ F we hav e f ( ˜ x ( t ′ )) ≤ f ( x ( t ′ )) for all t ′ . As it will be shown, the processes { ( C ( t ′ ) , A ( t ′ )) } ∞ t ′ =1 and { ( ˜ C ( t ′ ) , ˜ A ( t ′ )) } ∞ t ′ =1 are the same in distribution (these processes are permutation in variant). Thu s, the process f ( ˜ X ) = { f ( ˜ X ( t ′ )) } ∞ t ′ =1 obtained by applying policy ˜ π to the system is sto chastically small er than f ( X ) = { f ( X ( t ′ )) } ∞ t ′ =1 , i.e., f ( ˜ X ) ≤ st f ( X )) and ˜ π domi nates π . Therefore, in t he following, our goal w ill be to construct ˜ π and ˜ ω such that ˜ x ( t ′ )  p x ( t ′ ) for all time s lots. In the proof, we alw ays use the tilde n otation for all random variables that belong to the new system. The con struction o f ˜ π is done in two steps: Step 1: Construction of ˜ π for τ ≤ t : T o construct the new sample path ˜ ω we let the arriv al, connectivity and the policy be th e sam e as t he ﬁrst system until time slot t − 1 , i.e., ˜ c ( τ ) = c ( τ ) , ˜ a ( τ ) = a ( τ ) and M ( π ) ( τ ) = M ( ˜ π ) ( τ ) for τ ≤ t − 1 . Thus, the resulting queue lengths at the beginning of ti me slot t (or at the end of t ime slot t − 1 ) are equal, i.e., ˜ x ( t − 1) = x ( t − 1 ) . W e now cons ider the constructi on of ˜ ω and ˜ π for time slot t . Since π ∈ Π h t and h > 0 , according to Lemma 2 there exists a balancing server reallocation such that either C1 or C2 is satisﬁed. Thus, we consider two cases: Case 1 : After applying the balancing server reallocation, condition C1 is sati sﬁed. In other words, there exists a matching su ch that if applied on th e queue length ˜ x ( t − 1) = x ( t − 1) at time slot t , we get ˜ x ′ ( t ) such that ˜ x ′ ( t ) ≤ x ′ ( t ) . W e denote such a mat ching b y M ( ˜ π ) ( t ) . In this case, we let ˜ c ( t ) = c ( t ) and ˜ a ( t ) = a ( t ) and we apply M ( ˜ π ) ( t ) at time slot t , i.e., arri vals and connectivities are the same in both s ystems and policy ˜ π acts at time slot t . So, w e can easily check that ˜ x ( t ) ≤ x ( t ) and therefore ˜ x ( t )  x ( t ) . Case 2 : After applying the balancing server reallocation, condition C2 is sati sﬁed. In other words, there exists a m atching such that if applied on the syst em at tim e slo t t , we get ˜ x ′ ( t ) which is differ ent from x ′ ( t ) in two elem ents m and n such that x ′ n ( t ) < ˜ x ′ n ( t ) ≤ ˜ x ′ m ( t ) < x ′ m ( t ) and the following constraints are satisﬁed: ˜ x ′ n ( t ) = x ′ n ( t ) + 1 and ˜ x ′ m ( t ) = x ′ m ( t ) − 1 . W e call such a matching by M ( ˜ π ) ( t ) . In this case, we l et ˜ c ( t ) = c ( t ) and ˜ a ( t ) = a ( t ) and we apply M ( ˜ π ) ( t ) at t ime slot t , i .e., arriv als and connecti viti es are the same in both systems and policy ˜ π acts at tim e slot t . W e consider all the following conditions for arri vals to queues m and n as August 16, 2018 DRAFT 32 follows: • If th ere i s no arriv al or there i s an arriv al t o bo th queues m and n (i.e., a m ( t ) = a n ( t ) = 0 or a m ( t ) = a n ( t ) = 1 ), we conclu de that ˜ x ( t ) and x ( t ) satis fy c ondit ion D3 . Thus, ˜ x ( t )  x ( t ) . • If there is an arriv al to queue m but not n (i. e., a m ( t ) = 1 , a n ( t ) = 0 ), we conclude that ˜ x ( t ) and x ( t ) satis fy cond ition D3 . Thu s, ˜ x ( t )  x ( t ) . • If there is an arriv al to queue n but not m (i.e., a m ( t ) = 0 , a n ( t ) = 1 ) and ˜ x m ( t ) = ˜ x n ( t ) , we conclude that ˜ x ( t ) and x ( t ) satis fy cond ition D2 . Thu s, ˜ x ( t )  x ( t ) . • If there is an arriv al to queue n but not m (i.e., a m ( t ) = 0 , a n ( t ) = 1 ) and ˜ x n ( t ) < ˜ x m ( t ) , we conclude that ˜ x ( t ) and x ( t ) satis fy cond ition D3 . Thu s, ˜ x ( t )  x ( t ) . In all the cases we can see that ˜ x ( t )  x ( t ) . The obtained policy ˜ π belongs to Π h − 1 t since we applied a balancing server reallocati on t o the matching employed in π at time sl ot t . Step 2: Construction of ˜ π for τ > t : In t his step, we focus on constructio n of ˜ ω and ˜ π for τ > t . W e will emp loy mathemati cal indu ction to achie ve t his goal. In particular , we assum e that ˜ ω and ˜ π are constructed up to tim e slot τ ( τ ≥ t ) su ch that ˜ x ( τ )  x ( τ ) i.e., one of the condi tions D1 , D2 and D3 is satisﬁed for x ( τ ) and ˜ x ( τ ) . W e will prove that poli cy ˜ π and sample path ˜ ω can be constructed such that ˜ x ( τ + 1)  x ( τ + 1) (i. e., one of the condi tions D1 , D2 and D3 is satisﬁed fo r x ( τ + 1) and ˜ x ( τ + 1) ). Accordingly , we consider three cases corresponding to each condit ion D1 , D2 o r D3 at time slot τ : Case 1 : ˜ x ( τ ) ≤ x ( τ ) . In this case, the const ruction of ˜ ω and ˜ π at tim e slot τ + 1 is st raight- forward. W e let ˜ c ( τ + 1) = c ( τ + 1) and ˜ a ( τ + 1) = a ( τ + 1) and M ( ˜ π ) ( τ + 1) = M ( π ) ( τ + 1) . Thus, ˜ x ( τ + 1) ≤ x ( τ + 1) and therefore ˜ x ( τ + 1)  x ( τ + 1) . Case 2 : ˜ x ( τ ) is obtai ned from x ( τ ) by permu tation of two dist inct elements m and n . In this case, we l et ˜ c n,k ( τ + 1) = c m,k ( τ + 1) and ˜ c m,k ( τ + 1) = c n,k ( τ + 1) for k = 1 , 2 , . . . , K ; ˜ c i,k ( τ + 1) = c i,k ( τ + 1) for all i ∈ N , i 6 = n, m and k = 1 , 2 , . . . , K ; ˜ a n ( τ + 1) = a m ( τ + 1) , ˜ a m ( τ + 1) = a n ( τ + 1) and ˜ a i ( τ + 1) = a i ( τ + 1) for i ∈ N , i 6 = n, m . Supp ose that M ( π ) ( τ + 1) = ( M ( π ) n,k ( τ + 1)) ∀ n ∈ N , k ∈ K be the empl oyed m atching by policy π at tim e slot τ + 1 . W e construct M ( ˜ π ) ( τ + 1) as fol lows: Let M ( ˜ π ) i,k ( τ + 1) = M ( π ) i,k ( τ + 1) for i ∈ N , i 6 = n, m and also let M ( ˜ π ) n,k ( τ + 1) = M ( π ) m,k ( τ + 1) and M ( ˜ π ) m,k ( τ + 1) = M ( π ) n,k ( τ + 1) for k = 1 , 2 , . . . , K . As a result, ˜ x ( τ + 1) and x ( τ + 1) satisfy condi tion D2 at time slot τ + 1 and therefore ˜ x ( τ + 1)  x ( τ + 1 ) . Case 3 : ˜ x ( τ ) is obtained from x ( τ ) by p erforming a balancing interchange of t wo di stinct elements m and n as deﬁned in cond ition D3 . In particul ar , ˜ x ( τ ) and x ( τ ) are different in only August 16, 2018 DRAFT 33 two elements n and m such th at x n ( τ ) < ˜ x n ( τ ) ≤ ˜ x m ( τ ) < x m ( τ ) and the foll owing con straints are satisﬁed: ˜ x n ( τ ) = x n ( τ ) + 1 and ˜ x m ( τ ) = x m ( τ ) − 1 . In t his case, we consider the foll owing sub-cases: Sub-case 3.1 : ˜ x n ( τ ) < ˜ x m ( τ ) − 1 : In this case, we let ˜ c ( τ +1) = c ( τ +1) and ˜ a ( τ +1) = a ( τ +1) and we let M ( ˜ π ) ( τ + 1) = M ( π ) ( τ + 1) . Thus, if x n ( τ ) = 0 and queue n is served, condition D1 is satisﬁed at τ + 1 . Otherwise, ˜ x ( τ + 1) is obtain ed from x ( τ + 1) by performing a bal ancing interchange of elements m and n . Therefore ˜ x ( τ + 1)  x ( τ + 1) . Sub-case 3.2: ˜ x n ( τ ) = ˜ x m ( τ ) − 1 : In t his case again we let ˜ c ( τ + 1) = c ( τ + 1) and ˜ a ( τ + 1) = a ( τ + 1) and we let M ( ˜ π ) ( τ + 1) = M ( π ) ( τ + 1) . Thus, if x n ( τ ) = 0 and q ueue n is served, conditi on D1 is sati sﬁed at τ + 1 . If queue m get s service, q ueue n does not get service, there is an arriv al to queue n and no arriv al t o queue m , th en ˜ x n ( τ + 1) = x m ( τ + 1) and ˜ x m ( τ + 1 ) = x n ( τ + 1 ) . Therefore, ˜ x ( τ + 1) and x ( τ + 1) satisfy conditi on D2 and ˜ x ( τ + 1)  x ( τ + 1) . Otherwise, ˜ x ( τ + 1) and x ( τ + 1) satis fy condi tion D3 and ˜ x ( τ + 1)  x ( τ + 1) . Sub-case 3.3 : ˜ x n ( τ ) = ˜ x m ( τ ) : In this case, we let ˜ c ( τ + 1) = c ( τ + 1) and M ( ˜ π ) ( τ + 1) = M ( π ) ( τ + 1) . Now , we consid er t he following cases to determine the arriv als at time slot τ + 1 . • If x n ( τ ) > 0 and both queues m and n or none of them get service at t ime slot τ + 1 , we let ˜ a ( τ + 1) = a ( τ + 1) . Therefore, if a m ( τ + 1) = 0 and a n ( τ + 1) = 1 , ˜ x ( τ + 1) and x ( τ + 1) s atisfy condition D2 and thus ˜ x ( τ + 1)  x ( τ + 1) . Otherwis e, ˜ x ( τ + 1) and x ( τ + 1) satisfy condition D3 and thu s ˜ x ( τ + 1)  x ( τ + 1) . • If x n ( τ ) > 0 and qu eue n gets service at time slot τ + 1 and queue m does not get service at ti me slo t τ + 1 , we l et ˜ a ( τ + 1) = a ( τ + 1) . Therefore, ˜ x ( τ + 1) and x ( τ + 1) s atisfy condition D3 and thus ˜ x ( τ + 1)  x ( τ + 1) . • If x n ( τ ) > 0 and queue m gets service at time slot τ + 1 and qu eue n does n ot get service at time slot τ + 1 , we let ˜ a m ( τ + 1) = a n ( τ + 1) and ˜ a n ( τ + 1) = a m ( τ + 1) and ˜ a i ( τ + 1) = a i ( τ + 1) for i ∈ N and i 6 = m, n . Therefore, ˜ x ( τ + 1) and x ( τ + 1) satisfy condition D2 and thus ˜ x ( τ + 1)  x ( τ + 1) . • If x n ( τ ) = 0 and queue n gets service at ti me sl ot τ + 1 (although it does not ha ve any packet to be served), we let ˜ a ( τ + 1) = a ( τ + 1) . Therefore, ˜ x ( τ + 1) and x ( τ + 1) satis fy condition D1 and thus ˜ x ( τ + 1)  x ( τ + 1) . • If x n ( τ ) = 0 and queue m gets service at time slot τ + 1 and qu eue n does n ot get service at time slot τ + 1 , we let ˜ a m ( τ + 1) = a n ( τ + 1) and ˜ a n ( τ + 1) = a m ( τ + 1) and August 16, 2018 DRAFT 34 ˜ a i ( τ + 1) = a i ( τ + 1) for i ∈ N and i 6 = m, n . Therefore, ˜ x ( τ + 1) and x ( τ + 1) satisfy condition D2 and thus ˜ x ( τ + 1)  x ( τ + 1) . • If x n ( τ ) = 0 and neith er queue m nor n gets service at time slot τ + 1 , we let ˜ a ( τ + 1) = a ( τ + 1) . If a m ( τ + 1) = 0 and a n ( τ + 1) = 1 , ˜ x ( τ + 1) and x ( τ + 1) satisfy con dition D2 and thus ˜ x ( τ + 1)  x ( τ + 1) . Otherwise, ˜ x ( τ + 1) and x ( τ + 1 ) satisfy condi tion D3 and thus ˜ x ( τ + 1)  x ( τ + 1) . The above cases cover all the po ssible cases for all of which we constructed ˜ ω and ˜ π such that ˜ x ( τ + 1)  x ( τ + 1) . According to steps 1 and 2, from any sample path ω and any arbitrary policy π ∈ Π h t , h = h π t > 0 , we can construct a sample path ˜ ω and a policy ˜ π ∈ Π h − 1 t such that at all time slots we have ˜ x ( t ′ )  p x ( t ′ ) . Therefore, f ( ˜ x ( t ′ )) ≤ f ( x ( t ′ )) . Consequently , the process f ( ˜ X ) = { f ( ˜ X ( t ′ )) } ∞ t ′ =1 obtained by applying poli cy ˜ π to the system is stochasti cally smaller than f ( X ) = { f ( X ( t ′ )) } ∞ t ′ =1 , i.e., f ( ˜ X ) ≤ st f ( X ) and therefore ˜ π ∈ Π h − 1 t dominates π ∈ Π h t . B. Pr oo f of Lemma 3 fo r t he System with Random Service F ailur es Pr oo f: The only di f ference of the proof in th is case from the p roof of Lemma 3 i s t hat we have to cons ider random v ariables Q n,k ( t ) ∀ n ∈ N , ∀ k ∈ K in our dynamic coupli ng argu- ment. Therefore, for the arbitrary sample path ω = ( x (0) , c (1) , q (1) , a (1) , x (1) , c (2) , q (2) , a (2) , x (2) , ... ) , we appl y the coupling meth od to cons truct from ω a new sample path ˜ ω = ( ˜ x (0) , ˜ c (1) , ˜ q (1) , ˜ a (1) , ˜ x (1) , ˜ c (2) , ˜ q (2) , ˜ a (2) , ˜ x (2) , ... ) resulting in a new sequence of random variables ( ˜ X (0 ) , ˜ C (1) , ˜ Q (1) , ˜ A (1) , ˜ X (1 ) , ˜ C (2) , ˜ Q (2) , ˜ A (2) , ˜ X (2 ) , ... ) wit h X (0) = ˜ X (0 ) . W e will fol- low the same approach and consider t he same cases. Before we proceed to the d etails we introduce the following complementary notation. Suppose that s ( π ) n ( t ) denotes the index of the server assig ned t o qu eue n at time slot t by pol icy π . Note that s ( π ) n ( t ) ∈ K or it is empty . Step 1: Construction of ˜ π for τ ≤ t : In t his case, as in th e proof o f Lemma 3 we ﬁrst consider τ ≤ t − 1 . For those slots, w e let all ˜ c ( τ ) , ˜ q ( τ ) , ˜ a ( τ ) and M ( ˜ π ) ( τ ) to be the same in both sys tems i.e., ˜ c ( τ ) = c ( τ ) , ˜ q ( τ ) = q ( τ ) , ˜ a ( τ ) = a ( τ ) and M ( ˜ π ) ( τ ) = M ( π ) ( τ ) and therefore we hav e ˜ x ( t − 1) = x ( t − 1 ) . At time slot t , the distance of pol icy π to Π t is h balancing server reallocati ons. Since π ∈ Π h t and h > 0 , according to Lemma 2 there exists a balancing server reallocation such that either August 16, 2018 DRAFT 35 C1 or C2 is satisﬁed. Th us, we consider two cases: Case 1 : After applying the balancing server reallocation, condition C1 is sati sﬁed. In other words, there exists a matching M ( ˜ π ) ( t ) such th at if applied on the queue length s ˜ x ( t − 1) = x ( t − 1) , w e get ˜ x ′ ( t ) such that ˜ x ′ ( t ) ≤ x ′ ( t ) . Not e that if a non-emp ty queue is served u nder M ( π ) ( t ) , it should also get service under ˜ π . Otherwis e, the condition ˜ x ′ ( t ) ≤ x ′ ( t ) will be violated. In this case, we let ˜ c ( t ) = c ( t ) and ˜ a ( t ) = a ( t ) and we apply M ( ˜ π ) ( t ) at tim e slot t . F or any non-empty queue n that i s served under both policies π and ˜ π we let ˜ q n,s ( ˜ π ) n ( t ) ( t ) = q n,s ( π ) n ( t ) ( t ) and ˜ q n,s ( π ) n ( t ) ( t ) = q n,s ( ˜ π ) n ( t ) ( t ) . In other words, we let each non-empty queue n which was being served in both systems experience the sam e service failure. For other va riables ˜ q n,k ( t ) we let ˜ q n,k ( t ) = q n,k ( t ) . Then, we can easil y check t hat ˜ x ( t ) ≤ x ( t ) and therefore ˜ x ( t )  x ( t ) . Case 2 : After applying the balancing server reallocation, condition C2 is sati sﬁed. In other words, there exists a m atching such that if applied on the syst em at tim e slo t t , we get ˜ x ′ ( t ) which is differ ent from x ′ ( t ) in two elem ents m and n such that x ′ n ( t ) < ˜ x ′ n ( t ) ≤ ˜ x ′ m ( t ) < x ′ m ( t ) and the following constraints are satisﬁed; ˜ x ′ n ( t ) = x ′ n ( t ) + 1 and ˜ x ′ m ( t ) = x ′ m ( t ) − 1 . W e call such a matching by M ( ˜ π ) ( t ) . Note t hat in t his case, each q ueue (other than queue m and queue n ) w hich is (resp. is not) recei ving service under π is (resp. is not) also receiving service under ˜ π . Queue n is recei ving service under π and queue m is n ot. Qu eue n is not recei ving service under ˜ π and queue m is. In this case, we let ˜ c ( t ) = c ( t ) and ˜ a ( t ) = a ( t ) and we apply M ( ˜ π ) ( t ) at tim e slot t . W e let ˜ q i,s ( ˜ π ) i ( t ) ( t ) = q i,s ( π ) i ( t ) ( t ) and ˜ q i,s ( π ) i ( t ) ( t ) = q i,s ( ˜ π ) i ( t ) ( t ) for any queue i which is recei ving service under bot h matchings M ( π ) ( t ) and M ( ˜ π ) ( t ) . W e als o let ˜ q m,s ( ˜ π ) m ( t ) ( t ) = q n,s ( π ) n ( t ) ( t ) , ˜ q n,s ( π ) n ( t ) ( t ) = q m,s ( ˜ π ) m ( t ) ( t ) and for other failure variables we let ˜ q n,k ( t ) = q n,k ( t ) . By s uch coupling of the s ervice success/failure random variables we will consider the following cases for arriv als and service failures: • If q n,s ( π ) n ( t ) ( t ) = 0 , we l et ˜ a ( t ) = a ( t ) and th erefore, ˜ x ( t ) = x ( t ) which im plies ˜ x ( t )  x ( t ) . • If q n,s ( π ) n ( t ) ( t ) = 1 , th en we can use th e same coupling argument on the arriv al processes as what we did in the proof of Lemma 3 and conclud e that ˜ x ( t )  x ( t ) . W e omi t the ar gument to a void redundant discussion. Note that the obt ained policy ˜ π belo ngs to Π h − 1 t as we appl ied a balancing server reallocation to the matching employed in π at tim e sl ot t . Step 2: Co nstruction of ˜ π f or τ > t : In th is s tep, the same as the proof of Lemma 3 we use mathematical indu ction to construct policy ˜ π for τ > t . Th erefore, suppose that ˜ ω and ˜ π are August 16, 2018 DRAFT 36 constructed up t o time slot τ ( τ ≥ t ) such that ˜ x ( τ )  x ( τ ) . Therefore, one of the conditio ns D1 , D2 and D3 is sati sﬁed for x ( τ ) and ˜ x ( τ ) as follows. Case 1 : ˜ x ( τ ) ≤ x ( τ ) . In thi s case, the const ruction of ˜ ω and ˜ π at t ime slot τ + 1 is straightforward. W e let ˜ c ( τ + 1) = c ( τ + 1) , ˜ q ( τ + 1) = q ( τ + 1) , ˜ a ( τ + 1) = a ( τ + 1 ) and poli cy M ( ˜ π ) ( τ + 1) = M ( π ) ( τ + 1 ) . Thus, ˜ x ( τ + 1) ≤ x ( τ + 1) and th erefore ˜ x ( τ + 1)  x ( τ + 1) . Case 2 : ˜ x ( τ ) is obtained from x ( τ ) by permutatio n of t wo disti nct element s m and n . In this case, we couple the rando m variables c ( τ + 1) , a ( τ + 1) and construct matching M ( ˜ π ) ( τ + 1) the same as what we di d in Case 2 of step 2 in the p roof of Lemm a 3. In addition t o these settings, we let ˜ q m,k ( τ + 1) = q n,k ( τ + 1 ) and ˜ q n,k ( τ + 1 ) = q m,k ( τ + 1 ) for k = 1 , 2 , ..., K and also ˜ q i,k ( τ + 1) = q i,k ( τ + 1) for all i ∈ N , i 6 = n, m and k = 1 , 2 , ..., K . By doi ng such a coupling, we conclude t hat ˜ x ( τ + 1) and x ( τ + 1) sati sfy condi tion D2 at time slot τ + 1 and therefore ˜ x ( τ + 1)  x ( τ + 1) . Case 3 : ˜ x ( τ ) is obtained from x ( τ ) by p erforming a balancing interchange of t wo di stinct elements m and n as deﬁned in cond ition D3 . In particul ar , ˜ x ( τ ) and x ( τ ) are different in only two elements n and m such th at x n ( τ ) < ˜ x n ( τ ) ≤ ˜ x m ( τ ) < x m ( τ ) and the foll owing con straints are satisﬁed; ˜ x n ( τ ) = x n ( τ ) + 1 and ˜ x m ( τ ) = x m ( τ ) − 1 . In this case, wit h t he same argument as what we did in the proof of Lemma 3, we can check that ˜ x ( τ + 1) and x ( τ + 1) satisfy one of the conditions D1 - D3 . W e omit th e detail s t o av o id redundant discussion. C. St atement and Pr oo f o f Lemma 5 This lemma will be used in the proo f of L emma 4. Lemma 5: Suppose that, at a gi ven time slot t , multi ple, di stinct maximum weighted matchings exist in the graph of Figure 3. Any two of t hem w ill result in two intermediat e queue length vectors which are equa l i n permutati on , i.e., one is a permu tation of the other . Pr oo f: W e need to show that given the queue length vector x ( t − 1) at the beginning of time slot t , i f we apply two disti nct maximum weighted matchi ngs M ( MWM 1) ( t ) and M ( MWM 2) ( t ) (i.e., when M ( MWM 1) ( t ) 6 = M ( MWM 2) ( t ) ), then the intermediate qu eue length vectors x ′ (1) ( t ) and x ′ (2) ( t ) resulting from these t wo matchings are permut ations of each other , i.e., for each qu eue n ∈ N a) either queue n is being served by both matchin gs M ( MWM 1) ( t ) and M ( MWM 2) ( t ) . August 16, 2018 DRAFT 37 b) or if queue n is being served und er M ( MWM 1) ( t ) but not under M ( MWM 2) ( t ) , then there exists a queue m wi th x n ( t ) = x m ( t ) whi ch is bein g served under M ( MWM 2) ( t ) but not under M ( MWM 1) ( t ) . W e in voke the graph theory analysis we used in the proof of Lemma 2. Similarly we deﬁne a perfect graph G ′ t of si ze max { N , K } over which we deﬁne sub-graphs G ′ ( MWM 1) t and G ′ ( MWM 2) t corresponding t o M ( MWM 1) ( t ) and M ( MWM 2) ( t ) . W e build two directed sub-graphs D MWM 1 t and D MWM 2 t using sub -graphs G ′ ( MWM 1) t and G ′ ( MWM 2) t as foll ows: D MWM 1 t is the sam e as G ′ ( MWM 1) t with positive edges directed from q ueues to th e servers with posit iv e weights . D MWM 2 t is the same as G ′ ( MWM 2) t with ne gative edges directed from s ervers to th e queues. Similarly we deﬁne graph U as the unio n o f these two sub-graphs, i.e., U = D ( MWM 1) t S D ( MWM 2) t . In case “ a) ” above, if queue n is bein g served i n both MWM1 and MWM2 then x ′ (1) n ( t ) = x ′ (2) n ( t ) . Th erefore, we cons ider only case “ b) ” and we show t hat if queue n ∈ N is being served under M ( MWM 1) ( t ) but n ot under M ( MWM 2) ( t ) , then there exists a queue m wit h x n ( t ) = x m ( t ) which is b eing served under M ( MWM 2) ( t ) but not under M ( MWM 1) ( t ) . W e recall the deﬁnition of T ype 0 (T0) , T y pe 1 (T1) and T ype 2 (T2) edges pairs we h ad in the p roof of Lemma 2. If queue n is being served under M ( MWM 1) ( t ) but n ot under M ( MWM 2) ( t ) , then the edges incident to q ueue n make a T2 pai r r n = (0 , x n ( t − 1)) . Recall that th e graph U is the u nion of a number of even cycles. Assume that queue n belongs to cycle ℓ in graph U . W e now t race forward over cycle ℓ , as shown i n Figures 1 2a and 1 2b. The edge pairs after queue n cannot be all T0 pairs, since by using t he allocations of MWM 1 in MWM2 for the queues in ℓ , we can increase the matching weig ht index of MWM2 whi ch contradicts the fact that MWM2 is a maximum weighted matching. Thus, we consi der the following two cases: Case 1 : Assume t hat the ﬁrst non- T0 pair after r n is a T2 pair denoted by r m . This case is shown in Figure 12a. In this case, by us ing t he allocations used in MWM1 for queues n t o the one right before queue m in cycle ℓ in MWM2 and not serving queue m , we will obtain a matching wh ose m atching weigh t i ndex is large r than that of MWM2. This contradicts the fact that MWM2 is a m aximum w eighted matching. Case 2 : Assume t hat the ﬁrst non- T0 pair after r n is a T1 pair denoted by r m . This case is s hown i n Figure 12b. In this case, if x m ( t − 1) > x n ( t − 1) , by using t he allocati ons used in M WM2 for the queues right after n in the cycle ℓ to queue m in MWM 1 and not servin g queue n , we obt ain a m atching whose m atching weight index i s larger t han t hat of MWM1 . This August 16, 2018 DRAFT 38 P S f r a g r e p l a c e m e n t s 0 x n ( t − 1 ) − x m ( t − 1 ) 0 n m T 2 T 0 T 1 0 x n ( t − 1 ) x m ( t − 1 ) 0 n m T 2 T 0 T 2 (a) Case 1 P S f r a g r e p l a c e m e n t s 0 x n ( t − 1 ) − x m ( t − 1 ) 0 n m T 2 T 0 T 1 0 x n ( t − 1 ) x m ( t − 1 ) 0 n m T 2 T 0 T 2 (b) Case 2 Fig. 1 2: Cases 1 and 2 in the proof of Lem ma 5 contradicts the fact that MWM1 is a maxim um weighted matching. If x m ( t − 1) < x n ( t − 1) , by us ing the allocation s used in MWM1 for the queues n t o th e queue right before queue m in cycle ℓ in MWM 2 and not serving qu eue m , we will obt ain a m atching whos e matching weight index is lar ger than that of MWM2. Th is contradicts the fact that MWM2 is a maxim um weighted matching. Thus, the only valid case i s x m ( t − 1) = x n ( t − 1) . In th is case, either qu eue n and m are the same (we hav e covere d this case where queue n is served in both MWM1 and MWM2) or queue n and queue m are d iff erent in which case the resul t fol lows. D. Pr oo f of Lemma 4 T o show this lemma, we sh ow that any two MWM policy do minate each other , i.e., for any π 1 , π 2 ∈ Π MWM , we hav e f ( X ( π 1 ) ) ≤ st f ( X ( π 2 ) ) and f ( X ( π 2 ) ) ≤ st f ( X ( π 1 ) ) . Therefore, according to th e deﬁnitio n of “ ≤ st ”, we can concl ude that f ( X ( π 1 ) ) and f ( X ( π 2 ) ) are equal in distribution, i.e., f ( X ( π 1 ) ) D = f ( X ( π 2 ) ) . In order to do so, we will construct a sequence of policies ˜ π 1 , ˜ π 2 , . . . , ˜ π k such that lim k →∞ ˜ π k = π 2 and fo r a ll k = 1 , 2 , . . . , f ( X ( ˜ π k ) ) D = f ( X ( π 1 ) ) . Consider any arbitrary po licies π 1 , π 2 ∈ Π MWM . As sume th at tim e slot t 0 is th e ﬁrst tim e slot at which the t wo policies employ d iff erent matching matrices, i.e., w e have M ( π 1 ) ( τ ) = M ( π 2 ) ( τ ) , ∀ τ < t 0 and M ( π 1 ) ( t 0 ) 6 = M ( π 2 ) ( t 0 ) . For the sy stem using policy π 1 , for any sample path ω = ( x ( π 1 ) (0) , c (1) , a (1) , x ( π 1 ) (1) , c (2) , a (2) , x ( π 1 ) (2) , . . . ) of the underlying random variables ( X ( π 1 ) (0) , C (1) , A (1) , X ( π 1 ) (1) , C (2) , A (2) , X ( π 1 ) (2) , . . . ) , we can construct a ne w sample path ˜ ω = ( x ( ˜ π 1 ) (0) , ˜ c (1) , ˜ a (1) , x ( ˜ π 1 ) (1) , ˜ c (2) , ˜ a (2) , x ( ˜ π 1 ) (2) , . . . ) and a new MWM poli cy ˜ π 1 such that x ( ˜ π 1 ) ( t ) p = x ( π 1 ) ( t ) for all t = 1 , 2 , . . . and M ( ˜ π 1 ) ( τ ) = M ( π 2 ) ( τ ) , ∀ τ < t 0 + 1 . It is August 16, 2018 DRAFT 39 important to observe here t hat p olicy ˜ π 1 , un like poli cy π 1 , uses the same matchings as π 2 until time t 0 . Th e construction of ˜ π 1 is done as follows: Construction of ˜ π 1 f or τ < t 0 : W e l et the arriv al, connectivity and the matchings of the new system (which is working under ˜ π 1 ) be the same as those un der π 1 until time slot t 0 − 1 , i.e., ˜ c ( τ ) = c ( τ ) , ˜ a ( τ ) = a ( τ ) and M ( ˜ π 1 ) ( τ ) = M ( π 1 ) ( τ ) for τ < t 0 . Thus, we have x ( ˜ π 1 ) ( τ ) = x ( π 1 ) ( τ ) for τ < t 0 . Construction of ˜ π 1 f or τ = t 0 : At time slot t 0 , we let ˜ c ( t 0 ) = c ( t 0 ) and M ( ˜ π 1 ) ( t 0 ) = M ( π 2 ) ( t 0 ) . Since M ( π 1 ) and M ( π 2 ) are both maxi mum weighted matchings, according to Lemm a 5, t he intermediate qu eue lengths resul ted from M ( π 1 ) and M ( π 2 ) are permutatio n of each other , i.e., x ′ ( ˜ π 1 ) ( t 0 ) = ( x ′ ( ˜ π 1 ) 1 ( t 0 ) , x ′ ( ˜ π 1 ) 2 ( t 0 ) , . . . , x ′ ( ˜ π 1 ) N ( t 0 )) p = ( x ′ ( π 1 ) 1 ( t 0 ) , x ′ ( π 1 ) 2 ( t 0 ) , . . . , x ′ ( π 1 ) N ( t 0 )) = x ′ ( π 1 ) ( t 0 ) . In other words, for each element x ′ ( ˜ π 1 ) n ( t 0 ) in x ′ ( ˜ π 1 ) ( t 0 ) t here exists an element x ′ ( π 1 ) m ( t 0 ) in x ′ ( π 1 ) ( t 0 ) such that x ′ ( ˜ π 1 ) n ( t 0 ) = x ′ ( π 1 ) m ( t 0 ) . W e call queue m and queue n permuted queues. In the new system (whi ch is working under ˜ π 1 ), we let the arri vals of all the permut ed queues t o be the same at time slot t 0 , i.e., if q ueues n and m are two permuted queues we let ˜ a n ( t 0 ) = a m ( t 0 ) . Thus, for the queue length stat e of the system at time slot t 0 , we can easily observe that x ( ˜ π 1 ) ( t 0 ) p = x ( π 1 ) ( t 0 ) . Construction of ˜ π 1 f or τ > t 0 : For t ime slots τ > t 0 , we let the connectivities, arriv als and the allocation variables of all the perm uted queues to be the same. Thus, if queues n and m are two permu ted queues, we let ˜ c n,k ( τ ) = c m,k ( τ ) ∀ k = 1 , 2 , . . . , K , ˜ a n ( τ ) = a m ( τ ) and M ( ˜ π 1 ) n,k ( τ ) = M ( π 1 ) m,k ( τ ) ∀ k = 1 , 2 , . . . , K . Therefore, the service and the ev oluti on of all the permuted queues in the orig inal s ystem and the new system are the same. Since the employed matchings (allocatio n variables) in policy π 1 are all maximum weight ed matchings, th e allocation var iables i n th e new s ystem deﬁne a maximum weig hted m atching. W e can now conclude that the queue length vectors x ( ˜ π 1 ) ( τ ) and x ( π 1 ) ( τ ) , τ > t 0 , are permutati ons of each ot her and therefore, x ( ˜ π 1 ) ( τ ) p = x ( π 1 ) ( τ ) . According to Deﬁnition 7, we conclude that f ( x ( π 1 ) ( t )) = f ( x ( ˜ π 1 ) ( t )) for all t . T hus, t he two policies π 1 and ˜ π 1 are d ominating each oth er , i.e., f ( X ( π 1 ) ) ≤ st f ( X ( ˜ π 1 ) ) and f ( X ( ˜ π 1 ) ) ≤ st f ( X ( π 1 ) ) or f ( X ( π 1 ) ) D = f ( X ( ˜ π 1 ) ) . Recall that π 1 and π 2 are using simil ar matchings until time slot t 0 − 1 . W e const ructed a new MWM policy ˜ π 1 that agrees with π 2 until t ime s lot t 0 and it still results in the same queue length cost di stribution as π 1 , i.e., f ( X ( ˜ π 1 ) ) D = f ( X ( π 1 ) ) By usin g a mathematical i nduction approach, we can construct a sequence of p olicies ˜ π 2 , August 16, 2018 DRAFT 40 ˜ π 3 , . . . , whose queue length cost f ( X ) i s equal in distribution t o that of poli cy π 1 and are usi ng similar maximu m weighted matchin gs as π 2 is using at tim e slo ts t 0 + 1 , t 0 + 2 , . . . . The lim iting policy for this sequence o f policies is po licy π 2 . Therefore, f ( X ( π 1 ) ) D = f ( X ( ˜ π 1 ) ) D = f ( X ( ˜ π 2 ) ) D = · · · D = f ( X ( π 2 ) ) . R E F E R E N C E S [1] L . T assiulas and A. Ephremides, “Stability properties of constrained queueing systems and sc heduling policies for maximum throughpu t i n multihop radio networks, ” IEEE T rans. Auto. Contr ol , vol. 37, no. 12, pp. 1936–1949, Dec. 1992. 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Optimal Server Assignment in Multi-Server Queueing Systems with Random Connectivities

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