On the Stability Region of Multi-Queue Multi-Server Queueing Systems with Stationary Channel Distribution

On the Stabili ty Re gion of Mult i-Queue Multi-Serv e r Queueing Systems with Stati onary Channel Distrib ution Hassan Halabian, Ioannis Lambadaris, Chung-Horng Lung Department of Systems and Computer Engineering, Carleton Univ ersity , Ottawa, ON, Canada Email: { hassan h, ioannis, chlung } @sce. carleton.ca Abstract —In this paper , we charac terize the stability region of multi-queu e multi-server (MQMS ) q ueueing systems with stationary channel and packet arriva l processes. T oward t his, the necessary and suffi cient conditions for th e stability of th e system are derive d un der general arriva l processes with fini te first and second moments. W e show th at wh en th e arriva l p rocesses are stationary , th e stability region form is a polytope for which we explicitly find the coefficients of the linear inequ alities w hich characterize the stabil ity regio n polytope. I . I N T RO D U C T I O N Stability r e gion or network capacity re gion of a commu- nication network is the clo sure of the set o f all arriv al rate vectors f or which there exists a r esource allocation p olicy that can stabilize the system [1]–[3]. Having the stability region of a network ch aracterized by a set of con vex inequa lities, we can o btain a pr ecise solutio n f or n etwork fairness o ptimization problem s with the general form of (1). Maximize: N X n =1 f n ( r n ) (1) Subject to: r = ( r 1 , r 2 , ..., r N ) ∈ Λ 0 ≤ r n ≤ λ n ∀ n = 1 , 2 , ..., N In (1), N is the number of traffic sources. By r = ( r 1 , r 2 , ..., r N ) , we d enote the admitted rate vector, λ n is the input r ate of each source n and Λ denotes the s tability re - gion. Functions f n ( r n ) are no n-decre asing and concave utility function s. The ch oice of f n ( r n ) determin e the d esired fairne ss proper ties of the network. The g oal of this pa per is to c har- acterize the network capacity region of Mu lti-Queue Multi- Server (MQMS) queuein g systems with stationary channel dis- tribution (i.e., Λ for MQMS system). Su ch queueing systems are u sed to mode l multiuser wireless networks with multip le orthog onal subchannels such as OFDMA access network s. The reso urce allocation in suc h networks can be mode lled as the server allocation problem in multi-queue multi-server queuein g systems with time varying chann el condition s [ 4]– [9]. The stability problem in wireless queueing networks was mainly a ddressed in [1]– [3], [5], [ 9]–[11]. I n [3], th e au thors introdu ced the n otion of capacity region of a queueing net- work. They considered a tim e slotted system an d assumed that arrival proce sses ar e i.i.d. sequences an d the queue len gth process is a Markov pro cess. In [10], they charac terized the network capacity region o f multi- queue single- server system with time v ar ying ON-OFF connectivities. T hey also proved that for a symmetr ic system i.e., with the same arriv al and connectivity statistics for all the q ueues, LCQ (Longest Con- nected Queue) policy maximize s th e capacity region and also provides the op timal perfo rmance in terms of average q ueue occupan cy (or equiv alently a verage queue ing delay). In [1], [2] and [ 11], the notio n of network cap acity region of a wireless network was introduced for mor e general arriv al and queue length pro cesses. The problem of s erver allocation in multi-queue multi-server systems with time v arying connectivities was mainly addressed in [4]–[ 7], [9]. In [5], Max imum W eight (MW) policy was propo sed as a throug hput optimal server allocation policy for MQMS systems with station ary channel processes. Howe ver , in [5] the conditions on the arri val traffic to guarantee the stability of MW were not explicitly mentioned. In our previous work [9], we ch aracterized the network capacity region of multi-que ue multi-server systems with time varying ON -OFF channels. W e also obtained an upper bound for the a verag e queuein g delay of AS/LCQ (Any Server/Long est Connected Queue) policy which is the through put optimal server alloca- tion po licy for such systems. In this paper, we ch aracterize th e stab ility r egion of mu lti- queue multi-server queueing system with stationar y channel and arr i val processes. T oward this, th e nec essary an d sufficient condition s for the stability of th e system are derived for general arriv al p rocesses with finite first and seco nd momen ts. For stationary arri val pro cesses, these con ditions establish the network stability region of such systems. Our contribution in this work is to characterize the stability region as a conve x polytope specified by a finite set of linear inequalities that can b e n umerically tabulated an d used to solve network optimization problem (1). W e further introdu ce an upper bound for the average qu eueing delay o f MW po licy which is a throug hput optimal policy for MQMS systems. The rest of this pap er is organized as f ollows. Section II describes the mo del and notation requ ired throughou t the paper . In sectio n III, we d iscuss about som e preliminar y definitions (stability definition, p olytope , etc.) used in the paper . I n section IV, we will derive necessary and s ufficient condition s for the stability of ou r mode l and also fin d an u pper bound for the average queue occupancy (or av erage queuein g delay). Section V summarizes th e conc lusions of the pa per . I I . M O D E L D E S C R I P T I O N Before we proceed to describe the mo del, we intro duce basic notations often used throu ghout the paper . Other no- ) ( 1 t A ) ( 1 t X ) ( 2 t X ) ( 2 t A ) ( t X N ) ( t A N 2 K 1 ) ( 11 t C ) ( 12 t C ) ( 1 t C K ) ( t C NK ) ( 2 t C N ) ( 1 t C N Fig. 1: Multi- queue multi-server queueing system with station- ary ch annel distribution tations ar e introdu ced wh enever it is necessary . Th rough this paper, a ll the vectors ar e co nsidered to be row vectors. The time a verage of a fun ction f ( t ) is d enoted b y f ( t ) , i.e., f ( t ) = lim t →∞ 1 t P t τ =1 f ( τ ) . Th e oper ator “ ⊛ ” is used for entry -wise multiplication o f two matr ices. By 1 K ( 0 K ), we denote a r ow vector of size K who se elements are all identically equa l to “1” ( “0 ” ). T he card inality of a set is shown b y | · | . For any vector α = ( α 1 , α 2 , ..., α N ) and a n on- empty ind ex set U = { u 1 , u 2 , ..., u |U | } ⊆ { 1 , 2 , ..., N } and u 1 < u 2 < ... < u |U | , we defin e α U = ( α u 1 , α u 2 , ..., α u |U | ) . W e co nsider a time slotted queu eing system with equal length time slo ts an d eq ual le ngth p ackets. The mode l con sists of a set o f par allel queu es N = { 1 , 2 , ..., N } a nd a set of identical servers K = { 1 , 2 , ..., K } (Figu re 1) . E ach server can serve at most one queue at each time slot, i.e., we do not have server sharing in the system. At each time slot t , the capacity of each link b etween each qu eue n ∈ N an d server k ∈ K is assumed to be C n,k ( t ) packets/slot. W e a ssume that C n,k ( t ) ∈ M where M = { m ∈ Z + | m ≤ M } fo r given M . T herefor e, at each time slot t , the chan nel state may be expressed by an N × K matrix of C ( t ) = ( C n,k ( t )) , n ∈ N , k ∈ K , C n,k ( t ) ∈ M . The chann el pro cess is d efined a s { C ( t ) } ∞ t =1 with the state space S . Note tha t S is a finite set with |S | = ( M + 1) N K . W e label each element of S by a positive integer in dex s ∈ { 1 , 2 , ..., ( M + 1) N K } . Suppose that the cha nnel state matrix associated to the channel state s is denoted by C s . The channel state of the system is assumed to h av e stationar y distribution, with stationary prob abilities π s = Pr( C ( t ) = C s ) . Assume that the arri val process to each queue n ∈ N at time slot t (i.e., the number o f packet ar riv als during time slot t ) is represented by A n ( t ) . Thus, the arriv al vector at time slot t is deno ted by A ( t ) = ( A 1 ( t ) , A 2 ( t ) , ..., A N ( t )) . Assume tha t E [ A 2 n ( t )] < A 2 max < ∞ for all t . Each que ue has an infinite buf fer space. New arri vals ar e added to each queu e at the end of each time slot. Let X ( t ) = ( X 1 ( t ) , ..., X N ( t )) be the q ueue length p rocess vector at the end of time slot t after ad ding new arriv als to the que ues. A server scheduling p olicy at each time slot sho uld decide on how to allocate servers (from set K ) to th e queu es (in set N ). This must be acc omplished based on th e a vailable informa tion about the ch annel state of the system at time slot t ( which is C ( t ) ) a nd also the queu e length process at the beginning of time slot t (which is X ( t − 1) ). In o ther words, at each time slot t , the schedu ler has to determine an allocation matrix I ( t ) ∈ I where I = ( I N × K = ( I n,k ) , I n,k ∈ { 0 , 1 } | N X n =1 I n,k = 1 ∀ k ∈ K ) . W e ca n easily ob serve that th e qu eue leng th proce ss ev olves with time accord ing to th e f ollowing rule. X T ( t ) =  X T ( t − 1 ) − ( C ( t ) ⊛ I ( t )) 1 T K  + + A T ( t ) (2) where ( · ) + is defined as follows: For an arbitrary vector v o f size | v | , ( v ) + is a vector of th e same size whose i ’th elem ent ( v ) + i = v i if v i ≥ 0 and zero otherwise. I I I . B AC K G R O U N D A. Str ong S tability W e begin with intro ducing the definition of strong stability in queuein g systems [1], [2], [1 1]. Other definition s can be found in [3], [10], [1 2]. Con sider a discrete time single queu e system with an arriv al process A ( t ) and service process µ ( t ) . Assume th at th e arr iv als ar e ad ded to the system at the end of each time slot. Hence, the queue length process X ( t ) ev o lves with time accord ing to th e f ollowing rule. X ( t ) = ( X ( t − 1) − µ ( t )) + + A ( t ) (3) Strong stability is giv en by the fo llowing definition [1]. Definition 1 : A queue satisfyin g th e co nditions above is called str o ngly stable if lim sup t →∞ 1 t t − 1 X τ =0 E [ X ( τ )] < ∞ (4) Naturally f or a queu eing system we have th e fo llowing definition [1]. Definition 2 : A queueing system is called to be strongly stable if all the q ueues in the system are strong ly stable. In this paper , we u se th e stro ng stability de finition an d from now we use “stability” and “strong stability” in terchang eably . B. Som e Fund amental Concepts of P olyto pes In this part, we will have a b rief review o n the notion of conv ex poly topes an d some fundamental prop erties of th em. These concepts are needed when describing the stability region of MQMS system in section IV. Definition 3 : A con vex polytop e is defined as the co n vex hull of a finite set of points [1 3], [ 14]. According to the W eyl’ s Theorem [13], a polytop e in R N always can be expressed by a set  x ∈ R N | α ℓ x T ≤ β ℓ , ℓ = 1 , 2 , ..., L  for some positiv e integer L and α ℓ ∈ R N and β ℓ ∈ R . Dimension Th eor em [13]: For a polyto pe P ⊂ R N , d i- mension of P is equal to N minus the max imum number of linearly ind ependen t equatio ns satisfied by all the po ints in P . Definition 4 : An equality ax T ≤ b is called valid for polytop e P if for ev ery poin t x 0 ∈ P , ax T 0 ≤ b . Definition 5 : A face o f poly tope P is defined as F = { x ∈ P | a x T = b } where inequality ax T ≤ b is a valid ineq uality for P . W e call the hyp erplane ax T 0 = b associated to the valid inequality ax T 0 ≤ b a face defin ing hyperplane for P if its associated face is not e mpty . In other word s, ax T 0 = b is a face defining hy perplane fo r P if it in tersects with P on at least one poin t. No te that P has finitely m any faces. Howev er , the face defining h yperplan es of a polytope can be in finite. Definition 6 : A facet o f p olytope P is a maximal face distinct from P [1 4]. In other words, all faces of P with dimension dim( P ) − 1 are called facets of P . For polyto pe P =  x ∈ R N | α ℓ x T ≤ β ℓ , ℓ = 1 , 2 , ..., L  an inequality is redundant if polyto pe P is u nchang ed by removing the inequality . Redund ancy Theor em in P olytopes : Face d efining h yper- planes (inequalities) describ ing faces of d imension less than dim( P ) − 1 are r edund ant [13]. Redundan cy theo rem states th at to describe a polytope completely , on ly facet defining hy perplan es (inequ alities) are sufficient. I V . N E C E S S A RY A N D S U FFI C I E N T C O N D I T I O N S F O R T H E S TA B I L I T Y O F M Q M S S Y S T E M A. Necessary Conditions for the Stability of MQMS System W e introduce the departure matrix H N × K ( t ) = ( H n,k ( t )) , n ∈ N , k ∈ K in which H n,k ( t ) represents the total numb er of packets served b y server k from queue n at time slot t . Obviously H n,k ( t ) ≤ C n,k ( t ) . Thus, the depa rture p rocess f rom queu e n a t time slot t would be P K k =1 H n,k ( t ) . The following e quality illustrates the ev o lution of queue length pro cess with time. X n ( t ) = X n ( t − 1) − K X k =1 H n,k ( t ) + A n ( t ) ∀ n ∈ N (5) Let G be th e c lass of deter ministic p olicies in wh ich at each state of the system, each po licy g ∈ G allocates the servers accordin g to a predeter mined allocation matrix. Th erefore, each deter ministic policy g is specified by |S | allocatio n matrices I ( g ) s , s ∈ S . Note that set I is a finite set an d since the channel state space is also finite, set G is finite. A determ inistic policy g will re sult into an average tr ans- mission rate f or each qu eue n . Let R ( g ) n denote the time av eraged tr ansmission rate provided to qu eue n and R ( g ) := ( R ( g ) 1 , R ( g ) 2 , ..., R ( g ) N ) . I t fo llows that, R ( g ) = X s ∈S π s  C s ⊛ I ( g ) s  1 T K ! T (6) Each rate vector R ( g ) determines a single point in R N + . Now , consider the co n vex hull of all the poin ts R ( g ) , g ∈ G in R N + , i.e., R = conv .hul l ( R ( g ) ) . According to the above discussion and d efinition of co n vex polytop e, we can show th at the set of achiev able transmission rate vectors R is specified by a po lytope. Henc eforth, we denote the achievable transmission rate p olytope b y P . Note that conve x hull representation of the stability region may not b e suitab le to use in ne twork optim ization prob lems like (1). Since it does not provide mathematical statements in a fo rm to be used as the co nstraints in (1). I n the following, we will characterize the stability region by a set o f linear inequalities. T o introdu ce the n ecessary con ditions for the stability o f MQMS system we need to prove a series of Lemmas (Lemm as 1 to 5). In this pap er , the pro ofs were skipped du e to space lim itation. Lemma 1: If the MQMS system is st rong ly stable under a server allocatio n po licy ∆ , then for any vector α = ( α 1 , α 2 , ..., α N ) ∈ R N + we h av e lim t →∞ 1 t t X τ =1 αE [ A ( τ )] T = lim t →∞ 1 t t X τ =1 αE [ H ( τ )]1 T K . (7) Lemma 2: If the MQMS system is stron gly stable un der a server allocation policy ∆ , then E [ A ( t )] ∈ P . Lemma 3: If the MQMS system is stron gly stable un der a server allocation policy ∆ , then for all α ∈ R N + α E [ A ( t )] T ≤ X s ∈S π s max I ∈I  α ( C s ⊛ I ) 1 T K  (8) As we can see, the nu mber of inequ alities defined in ( 8) is infinite. No te that e ach ineq uality in ( 8) determin es a valid inequality fo r poly tope P . It is not h ard to show that th e hyperp lanes associated to the valid inequ alities of (8) are all face defining hyperp lanes of polytope P . T o see this, let I α denote a set o f allocatio n matrices of { I α s , s ∈ S } th at maximizes th e righ t hand side of (8), i.e., I α s = arg max I ∈I α ( C s ⊛ I ) 1 T K . (9) Note th at I α is n ot un ique and there may be more than o ne set of allocation m atrices of I α whose elements maximize α ( C s ⊛ I )1 T K . According to (6) and definition of polytope P ,  P s ∈S π s ( C s ⊛ I α s ) 1 T K  T ∈ P . On the other han d, P s ∈S π s α ( C s ⊛ I α s )1 T K = α P s ∈S π s ( C s ⊛ I α s )1 T K . Thus,  P s ∈S π s ( C s ⊛ I α s ) 1 T K  T is locate d on th e hyp erplane as- sociated to ( 8). In oth er words, the h yperplan e associated to (8) tou ches polyto pe P . So, the set of inequ alities of (8) are all face definin g hyperp lanes of po lytope P . A ccording to redund ancy theo rem in polytopes [13], not all the face defining hyperp lanes of a po lytope are requ ired to determine a polyto pe completely . Th e ineq ualities cor respond ing to the facets of a polytop e ar e just sufficient to characterize a p olytop e. I n the following, we will characterize the facets of po lytope P . Let V den ote the set of vectors α ∈ R N + with the following proper ty , i.e., V :=  α ∈ R N + | ∀ ( U ⊂ N , U 6 = ∅ , α U 6 = 0 |U | , α U c 6 = 0 |U c | ) ∃ ( i ∈ U , j ∈ U c , m, n ∈ M , α i , α j , m, n 6 = 0) : α i m = α j n } (1 0) In oth er words, a vector α belong s to set V if for any partitioning o f elem ents of vector α into two n on-emp ty disjoint (su b)vectors in which all th e elements of each vector are not id entically equ al to zero , the re exists at lea st one n on- zero eleme nt in eac h vector, the ratio of which is equal to ratio of two non- zero elemen ts of set M . T o clarify the definition of set V consider the fo llowing example. Example: Let M = { 0 , 1 , 2 } a nd N = 4 . Co nsider vector α 1 = (1 , 2 , 5 , 10 ) . W e can partition α 1 to α 1 { 1 , 2 } = (1 , 2) a nd α 1 { 3 , 4 } = (5 , 10) . Note that ther e exist no elements in α 1 { 1 , 2 } and α 1 { 3 , 4 } whose ratio is 1 or 2 an d therefor e, α 1 / ∈ V . Another examp le is the vector α 2 = (0 , 2 . 5 , 0 , 0 ) in wh ich for any partitioning of the vector into two non -empty disjoint parts, one part will always h av e all th e elements identically equal to zero . Therefo re, α 2 ∈ V . In the following we will show that any vector in R N + − V cannot fo rm a facet de fining hyp erplane for polytope P and therefor e the ine qualities d erived by the vectors of set R N + − V are red undan t to characte rize polytop e P . Lemma 4: The hyp erplane associated to the valid inequality of (8) is not a facet defining hy perplan e o f P if α ∈ R N + − V . Set V is an in finite set. Howev er , we can show that set V is finite up to multiplication of vectors by positive scalar s. T o show this, we define set W = n z ∈ Z + | z = Q N − 1 j =1 m j , m j ∈ M o . Th en, we conside r the set W N = { ( α 1 , α 2 , ..., α N ) | α n ∈ W } . W e can prove th e following Lemma. Lemma 5: There exists b V ⊆ W N such that any v ector α ∈ V can be written as α = q β for some β ∈ b V and q > 0 . W e can show that |W N | =   M + N − 2 N − 1  + 1  N . Since b V ⊆ W N , th erefore | b V | ≤ |W N | − 1 (exclud ing the vector 0 N ). W e now introdu ce the necessary conditions f or the stability of MQMS system with stationary chann el distributions b y the following theorem. Theor em 1: If ther e exists a server allo cation policy △ under wh ich the system is stable, then α E [ A ( t )] T ≤ X s ∈S π s max I ∈I  α ( C s ⊛ I ) 1 T K  , α ∈ b V . (11) The p roof fo llows from Lemm as 1 to 5. Note that ac cording to Theor em 1, in order to charac terize polytop e P , we have to specify all the elem ents of set b V which ca n be obtained after consid ering all possible v ectors α ∈ W N − 0 N and removin g redundancies f ollowing (10). This ca n be d one using numerical compu tations. W e may also use α ∈ W N − 0 N instead of α ∈ b V in (11). Although by usin g W N − 0 N we may o btain so me red undant ine qualities, since |W N − 0 N | < ∞ we still have a finite number of inequalities to describe poly tope P . T able I d epicts b V and |W N | − 1 f or some simple cases of N = 2 , 3 and M = 1 , 2 , 3 , 4 . T ABLE I: | b V | and |W N | − 1 for N = 2 , 3 and M = 1 , 2 , 3 , 4 M = 1 M = 2 M = 3 M = 4 | b V | , |W N | − 1 N = 2 3 , 3 5 , 8 9 , 15 12 , 24 N = 3 7 , 7 25 , 63 109 , 342 253 , 1330 Cor ollary 1: For an MQMS system with stationary arrival processes, we have E [ A ( t )] = λ = ( λ 1 , λ 2 , ..., λ N ) and the necessary c ondition s for the stability o f the system would be αλ T ≤ X s ∈S π s max I ∈I  α ( C s ⊛ I ) 1 T K  , α ∈ b V , (12) Cor ollary 2: For an MQMS system with ON- OFF chan- nels, we ha ve W = { 0 , 1 } an d therefo re th e necessary condition s for the stability of the system b ecome α E [ A ( t )] T ≤ X s ∈S π s max I ∈I  α ( C s ⊛ I ) 1 T K  , α n ∈ { 0 , 1 } (13 ) In a s pecial case considered in [9] where the channels are modelled by ind ependen t Bern oulli ran dom variables with E [ C n,k ( t )] = p n,k , the nece ssary condition s for th e stability of the system are g iv en by X n ∈ Q E [ A n ( τ )] ≤ K − K X k =1 Y n ∈ Q (1 − p n,k ) ∀ Q ⊆ N . (14 ) In this case, the total numb er of inequalities needed to describe the po lytope P is equal to 2 N − 1 . B. Sufficien t Conditions for the Sta bility of MQMS System Consider a server allocation policy which d etermines the allocation matr ix at eac h time slot t by solving th e following maximization pro blem. I ( t ) = a rg max I ∈I X ( t − 1)( C ( t ) ⊛ I )1 T K (15) This policy is called Maximum W eight (MW) and was first introdu ced in [3], [ 5]. Accord ing to the con straints o n al- location matrix I , we can easily con clude th at MW policy allocates the servers by the following ru le: At each time slot t , M W policy allocates each server k to queue n who has the max imum X n ( t − 1) C n,k ( t ) . In the special case where the chan nels a re ON-OFF , the p olicy is eq uiv alent to AS/LCQ (Any Server/Lo ngest Connected Queu e) intr oduced in [9]. In the fo llowing, we der i ve suf ficient cond itions for the stability of our mo del and prove that MW stabilizes the system as long as cond ition (1 6) is satisfied. An upp er bo und is also derived for the time averaged expected nu mber of packets in the system. Theor em 2: The MQMS sy stem is stable under MW if for all t αE [ A ( t )] T < X s ∈S π s max I ∈I  α ( C s ⊛ I ) 1 T K  , α ∈ b V (16) Furthermo re, the following bound f or the a verag e expe cted “aggregate” occupa ncy ho lds. lim sup t →∞ 1 t t − 1 X τ =0 N X n =1 E [ X n ( τ )] ≤ N A 2 max + ( M K ) 2 2 δ (17) where δ is the maximu m positive number such that for all t we h av e E [ A ( t )] + δ 1 N ∈ P . Cor ollary 3: For an MQMS system with stationary a rriv al processes, we have E [ A ( t )] = λ an d the sufficient con ditions for the stability o f th e system under MW po licy become αλ T < P s ∈S π s max I ∈I  α ( C s ⊛ I ) 1 T K  , α ∈ b V . According to Corollaries 1 an d 3 an d the definition of network capacity region, we can conclude that for an MQMS system with station ary cha nnel distribution and station ary arriv al processes, th e stability region is characterize d b y (12). Consider an MQMS system with ON-OFF channels. For such a system, the MW p olicy is equivalent to AS/LCQ p olicy introdu ced in [9]. In AS/LCQ the policy takes an arb itrary orderin g of servers (to be allocated to the queues) and then for each server , AS/LCQ allo cates it to its lon gest c onnected queue (LCQ). Note that in suc h a sy stem, W = { 0 , 1 } an d therefor e we will have the following coro llary . Cor ollary 4: The MQM S sy stem with ON-OFF chan nels is stable u nder AS/LCQ if for all t αE [ A ( t )] T < X s ∈S π s max I ∈I  α ( C s ⊛ I ) 1 T K  , α n ∈ { 0 , 1 } (18) In a special case where the channels a re modelled b y inde- penden t Bernoulli random variables with E [ C n,k ( t )] = p n,k , AS/LCQ stabilizes the system if X n ∈ Q E [ A n ( τ )] < K − K X k =1 Y n ∈ Q (1 − p n,k ) ∀ Q ⊆ N (19) C. Sta bility Region for Fluid Model MQMS Systems with Stationa ry Continuous Chan nel Distributions W e will consider a time slotted fluid model MQMS system with stationary chann el distrib utions. In this case, the amo unt of work that arr iv es into (and d eparts from) the q ueues is considered to be a continuou s p rocess. W e also assume that the chan nel state of th e link f rom each q ueue to ea ch server is modelled by a continu ous random v a riable. The channel state matrix in the fluid model M QMS system is d efined as C ( t ) = ( C n,k ( t )) , n ∈ N , k ∈ K , C n,k ( t ) ∈ R + . W e assume that th e channel state fo llows a stationar y distribution f C ( c ) . f C is the joint d istributions of all C n,k variables. W e c an easily check that in the flu id mod el MQMS system, set b V is equ iv alen t to R N + as in the fluid mo del MQMS system, set M is rep laced with R + . Theref ore, stability region for the fluid mo del system is cha racterized by the following set of inequalities. αλ T ≤ Z ∞ c n,k =0 Z ∞ c n,k − 1 =0 · · · Z ∞ c 1 , 1 =0 max I ∈I  α ( c ⊛ I )1 T K  × f C ( c ) dc 1 , 1 · · · dc n,k − 1 dc n,k α ∈ R N + (20) Note that in order to char acterize the stability region of the fluid model system we n eed to comp ute an infinite number of integrals in (20). In f act, the stability region of fluid model system is c haracterized by infinite nu mber of half sp aces which means that th e stability r egion is a conve x surface . In th is case, depend ing o n th e channel distribution and dimension of the system, we could characterize the stability region by a limited number of non-linea r inequ alities instead of in finite number of linea r inequ alities as we show in the following example. Example: Conside r a fluid model MQMS system with two queues and one server . Suppose that the channel state variables C 1 , 1 and C 2 , 1 are indepen dent and follow e xpon en- tial distrib ution with mean s µ 1 and µ 2 , r espectively . Such a modelling is used to model slow Rayleigh f ading channels with low SNR wh ere the approximation log(1 + x ) ≃ x is valid for small po siti ve x . W e can sho w that th e stability region for this examp le is described by λ 2 ≤ µ 2 s 1 − λ 1 µ 1 ! 2 − s 1 − λ 1 µ 1 ! (21) for λ 1 ≥ 0 and λ 2 ≥ 0 . The deriv atio n of (21) is skipped because of space lim itation. As we can see fro m equation (21), the stability region in this example is character ized by ju st one non-lin ear ineq uality (21) an d two linear ineq ualities λ 1 ≥ 0 and λ 2 ≥ 0 . The stability region for this examp le for µ 1 = 2 and µ 2 = 1 is illustrated in Figure 2. 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.2 0.4 0.6 0.8 1 λ 1 λ 2 λ 2 ≤ µ 2 ³q 1 − λ 1 µ 1 ´ ³ 2 − q 1 − λ 1 µ 1 ´ µ 1 = 2 a n d µ 2 = 1 Fig. 2: Stability region for µ 1 = 2 and µ 2 = 1 V . C O N C L U S I O N S In this pap er , we characterized the stability region polytop e of m ulti-queu e multi-server (MQMS) qu eueing systems with stationary chan nel and packet arr iv al pro cesses by a finite set of linear inequalities given by (12). W e also argued th at in gen eral it m ay be (computationally ) hard to quan tify set b V from set W N . In this case, we c an use set W N itself instead of b V in (12) as b V ⊆ W N , althou gh we may have some redund ant inequalities. W e also co nsidered the stab ility region for a fluid model MQMS. In this case, we determ ine the stability region by a n infinite set of lin ear inequ alities given by (20). Finally , by use of an example we sho wed that dep ending on the c hannel distribution and number of queues, we may characterize the stab ility region by a finite set of non- linear inequalities in stead of infinite set of linear inequalities. R E F E R E N C E S [1] L. Geor giadis, M. J. Neel y , and L. T assiulas, R esour ce Allocation and Cr oss Layer Contr ol in W irel ess Networks . No w Publisher , 2006. [2] M. J. Neely , “Dynamic power alloca tion and routing for satell ite and wireless networks with time varying channels, ” Ph.D. dissertatio n, Massachuset ts Inst itute of T echnology , LIDS, 2003. [3] L. T assiula s a nd A. Ephremi des, “Stability prope rties of const rained queuein g systems and scheduling polic ies for maximum throughput in multihop radi o netw orks, ” IEEE T rans. Auto. Cont r ol , v ol. 37, no. 12, pp. 1936–1949, Dec. 1992. [4] S. Kittipiyaku l and T . Javi di, “Delay-optima l server allocat ion in m ulti- queue multi -serve r systems with ti me-v arying connecti vities, ” IEE E T rans. Inform. Theory , vol. 55, no. 5, pp. 2319–2333, May 2009. [5] T . Javidi, “Rate stable resource allocatio n in ofdm systems: from wate rfilling to queue-balanci ng, ” in Pro c. Allerton Confer ence on Com- municati on, Cont r ol, and Computing , Oct. 2004. [6] S. Kittipi yakul and T . Javid i, “Resource allocat ion in ofdma with time- v arying channel and bursty arr i va ls, ” IEEE Commun. Lett. , vol. 11, no. 9, pp. 708–710, Sep . 2007. [7] H. A l-Zubaidy , I. Lambadaris, and I. V iniot is, “Optimal resource scheduli ng in wireless multi-servic e systems with random channel connec ti vity , ” in P r oc. of IEEE Global Communicat ions Confer ence (GLOBECOM 2009) , Honolulu, HI, USA, Nov . 2009. [8] A. Ganti, E. Modiano, and J. N. Tsitsiklis, “Opt imal transmissio n scheduli ng in symmetric communicat ion models with intermitte nt con- necti vity , ” IEEE T rans. Inform. T heory , vol. 53, no. 3, pp. 998–1008, Mar . 2007. [9] H. Halabia n, I. L ambadari s, and C.-H. Lung, “Network capacit y region of m ulti-qu eue multi-server queueing system with time var ying con- necti vities, ” in Proc. of IEEE Int. Symp. on Inform. Theory (ISIT’10) , Austin, T X, USA, June 2010. [10] L. T assiulas and A. Ephremides, “Dynamic serve r allocat ion to paral- lel queues with randomly varying connecti vity , ” IEEE T rans. Inform. Theory , vol. 39, no. 2, pp. 466–478, Mar . 1993. [11] M. J . Neely , E. Modiano, and C. E. Rohrs, “Dynamic power allocation and routing for time varyin g wire less net works, ” IEEE Journal on Selec ted Area s in Communicati ons, Special Issue on W ir eless Ad-hoc Network s , v ol. 23, no. 1, pp. 89–103, Jan. 2005. [12] S. Asmuss en, Applied Pr obabilit y and Queues . New Y ork: Spring- V erlag, 2003. [13] J. L ee, A Fi rst Course in Combinatorial Optimization . Cambridge, UK: Cambridge T exts in Applied Mathematics, Cambridge Univ ersity Press, 2004. [14] A. Schrijve r , Theory of Linear and Inte ger Pro gramming . Chichester: John W ile y and Sons, 1986.