Cone Schedules for Processing Systems in Fluctuating Environments

Cone Schedule s for Processing Systems in Fluctuat ing En vironmen ts KEVIN R OS S 1 NICHOLAS B AMBOS 2 GEORGE MICHAILIDIS 3 Abstract W e consider a generalized processing system ha ving se veral queues, where the a vailable service rate combinatio ns are fluctuating over time due to reliability and a vailability v ariations. The objectiv e is to allocate the a vailable resources, and correspo nding service r ates, in respon se to both workload an d service capacity considerations, in order to maintain th e long term stability of the system. The ser vice configur ations are completely arbitrary , includin g negative service rates which represent forwarding a nd service-ind uced cross tr affic. W e employ a trace-ba sed trajectory asymptotic te chnique, which req uires minimal assumptions about the arriv al dynamics of the s ystem. W e p rove tha t cone s chedules , which le verage the geometry of the q ueueing dynam ics, m aximize the system through put for a broad class of processing systems, even under adversarial arri val processes. W e study the impact of fluctu ating service av ailability , where resources are av ailable only some o f the time, and the schedule must dynamically respond to the chan ging av ailable service rates, establishing both the capacity of such systems and the class o f schedules which will stabilize the system at f ull capacity . The rich geom etry o f the system dynamics leads to impor tant insights for stability , perfo rmance and scalability , and substantially generalizes previous findings. The processing system studied here models a broad v ar iety of computer, comm unication and service networks, including v arying channel condition s an d cross-traffic in wireless networking, an d call centers with fluctuatin g capacity . The find ings have implications for b andwidth and processor allocation in commun ication network s and workfor ce sched uling in cong ested call centers. By establishing a broad class o f stabilizin g sched ules u nder ge neral con ditions, we find that a schedu ler can select the schedu le from within this class that best meets their load balancing and scalability requirem ents. Ke ywords: random en vironment, st ability , ad versa rial queue ing theory , dyna mic schedulin g, throu gh- put maximizatio n. 1 Introd uction W e consid er a processing system comprised of Q infinite capacity queues, index ed by q ∈ Q = { 1 , 2 , ..., Q } , operat ing in a time-va rying e n vironment whi ch fluctuates amongst en vironment s tates e ∈ E = { 1 , 2 , ..., E } . In each en viron ment state , only a subset of the service configuratio ns are av ailabl e. The process scheduler selects a service configuratio n vecto r S = ( S 1 , · · · , S Q ) from the en vironment-d ependent a vailab le set S e . Upon selection, if S q > 0 then queue q is emptied at rat e S q , and if S q < 0 then the queue is filled at the corres ponding rate. The a vail able service configurat ions can be complete ly arbitrary , including vecto rs with any combi nation of positi ve and negati ve component s. A ke y questi on ad dressed in this study is which of the av ailabl e serv ice configurations should be selecte d, giv en the system worklo ad and en vironment state histori es, so as to maximize its throughput. W e 1 School of Engineering, Univ ersity of California Santa Cruz; kross@soe.ucsc.edu ; 2 Electrical Engineering and Manage ment Science & Engineering, Stanford Univ ersity; bambos@stanford.edu 3 Statistics and Electrical Engineering & Computer Science, The Uni versity of Michigan; gmichail@umich .edu 1 introd uce a family of resource allocat ion policies - calle d Cone Schedul es - which ar e sho wn to stabiliz e the system under the maximal possible traffic load, ev en if that load is designe d by an advers ary to destabilize the system whene ver possible. This ca nonical process ing m odel ca ptures sev eral applications in comput ing and communicatio n sys- tems, includi ng wire less netwo rks, p acket switches and call center s. The main chara cteristic of these applica- tions is that the s ervice rates across multiple queue s are coup led through operationa l constr aints, gi ving rise to the a v ailable serv ice configura tions. Service rate a v ailabili ty (corr espondi ng to the en vironmen t states) is af fected by staf f schedulin g in call centers, congestion dynamics in w ireles s net works and schedul ed or unsch eduled outages due to maintenan ce or reliability issues in other processin g systems. 1.1 Related W ork The trace-b ased stability an alysis tec hnique emplo yed in this paper relates to the study of adversari al queue- ing netw orks ex emplified in [Andre ws et al., 2001] and [Borodin et al., 2001]. Thi s appro ach av oids im- posing a probab ilistic frame work on the arriv al traf fic, and instead analyze s the performanc e of a queueing netwo rk under and adve rsarial arriv al traffic trace, designed to stress the system as m uch as possible. They descri be a queue ing network as u niv ersally stable when they can sho w that the total wor kload of the syste m is bo unded under any det erministic or stochastic adv ersary ’ s arri val trace . This work is really finding the worst -case behavi or of a network by consid ering the network to be a game between the schedule (protocol ) and the worst possible arri v al trace (advers ary). They limit the absolute arriv al v olume within a fi nite inter - v al, but do not require it to follo w any stationa ry distrib ution or apply any further restrictio ns. This concept b uilds upon earlier work called leak y-b uc ket analysis in [Cruz, 1991a] and [Cruz, 1991b]. Adver sarial mo dels ha ve been us ed t o in pack et ne tworks before , such as [Borodin et al., 2001] wh ich consid ers a fi xed -path pack et network. Some m ore general queueing systems, includin g multicl ass queue- ing networks are studied in [Tsaparas, 2000], with genera lized service times and hetero geneous customers. Adver sarial methods hav e also been employed to study multi-hop network stability in [Kushne r , 2006 ]. In [Anshele vich et al., 2002], adv ersarial models ar e us ed to analy ze load-ba lancing algorit hms in a distrib uted setting based u sing a token- based system on a ne twork with limited de viations from the a verag e load . While none of these study the same netwo rk scheduling setting of this paper (to our knowle dge they ha ve only consid ered fixed -path networks und er time-in varia nt service en vironments ), e ach example presents a per- suasi ve ar gument for the v alue of network stability anal ysis in the absen ce of a well-defined probabilist ic frame work. A spec ial example of the system descri bed in this paper is a single cross bar p acke t/cell s witch with vir- tual outp ut queues, used in high spee d IP networ ks. T he switch paradigm is the focus of [Ross and Bambos, 200 9 ], and provide s a helpful context to dev elop th e cone alg orithms. In this switc h, cel ls arrivin g to each input p ort get buf fered in separate virtual queues, based on the output port the y are destin ed to. The switching fabric can be set to a dif ferent connecti vity mode in each time slot, matchi ng each input port with a corresp ond- ing output port for cell trans fer . In this context, Maximum W eigh t Matchin g (MWM) has been sho wn in [McK eown e t al., 1999] to maximize the throughpu t of input queued switches, emplo ying L yapunov m eth- ods for stability analysi s, as also in const rained queueing systems studie d in [T assiulas and Ephremides , 1992, T assiulas, 1995, T assiulas and Bhatta charya, 2000, Hung and Michailidis , 2011]. In our more gener al ser- vice m odel, MWM corresp onds to maximizing h S, X i = P q S q X q , where the weight X q is the cell work- load of queue q or a related congestio n measure, and the S vecto rs represent the crossbar configurations . More general results on the stabili ty of MWM algorithms, using fluid scalin g methods, were later 2 obtain ed in [Dai and Prabhak ar , 2000], and on a gen eralized switch model in [Stolyar , 2004]. Stabili ty in netwo rks of switches was studied in [Marsan et al., 2005] and [Leonardi et al., 2005]. [Dai and Lin, 2005] and [Dai and Lin, 200 8] conside red maximum press ure policies by modeling fluid flows for type s of pro- cessin g network s. Their work can be seen as a genera lization of the policie s which maximize P q S q X q where some of the service rates are negati ve becaus e the a vai lable configurati ons in volv e forwarding work- load from one queue to anoth er downst ream queue. [Neely et al., 2003] stud ied broader optimal con trols for generalized (wireless) netwo rk models that in volv e joint scheduling, routing and power allocatio n. A ll of these hav e significan tly adva nced the theory of the stabilit y of schedul ing rules which allocate service to queue s based on a weighted-matc hing approach, and u tilize a p robabili stic frame work to ap ply fluid l imit or hea vy-tra ffic analysis. Instea d of using fl uid scaling methods (primar ily analytic, in v olving passage to a limit reg ime) to establ ish th e r esults, we opt to use an alternati ve direct and pr imarily geometric approach in this work, which seems to ha ve broader applicabil ity to other queue ing systems and rev eals useful geometric in- sight regardi ng their dynamics. The trace-based asymptoti c analysis employed here was introduce d in [Armony and Bambos, 20 03], where the maximum weight matching algorithms were studi ed and it was sho wn tha t maintain maxi mal throu ghput is guara nteed under v ery gen eral arri v al proc ess assumption s. The method was also employ ed in [Bambos and Michailid is, 2004] where randomly fluctuat ing service le vels were stu died. In that case the service rate assignmen ts are made without full kno wledge of service av ail- ability , as opposed to the proces sing systems studie d here where service allocat ion decisions are made in respon se to av ailability . Like th e adver sarial queueing models, the re is no p robabilis tic frame work requir ed, b ut unlike the traditiona l adv ersarial models, there is also no short-te rm restriction on arriv al b ursts in finite time, but just a long-term traf fic load restriction . This leads to m ore general stability results, bu t eliminates the possibility of tighter b ounds on other pe rformance metrics . For ex ample under such ge neral assu mptions there can be n o guarant eed finite bound on t he total workl oad, or ev en the e xpected worklo ad in the system. 1.2 Results Overview W e classify the stability re gion for the se proc essing syste ms with fluctuating service a vailab ility . W e fi nd that rate stabili ty for the se gene ral processin g systems can be guarante ed by the clas s of cone sc hedule s , for an y arbitrar y arriv al proc ess that can pos sibly be stabil ized. Cone schedules use the a v ailable service vec tor with maximal projection h S, B X i = P p P q S p B pq X q on the pr oject ed workload vector B X , for e very matrix that is positive-d efinite , has ne gative or zer o of f-dia gonal elements, and is symmetric . This substa ntially generalizes a similar result in [Ross and B ambos, 200 9 ] , where the same class of algorit hms was sho w n to maximize throug hput for the special case of packe t switches. In classify ing the stability regio n, we sho w how the combination of service vectors in each en viron- ment impacts the ov erall capacity of the system, beyon d the long ter m a v ailabili ty of each service ve ctor . The geometric framewo rk for stability aids the intuition and analysi s significantly . B ecause of en vironment fluctuatio ns, one m ay e xpect that a scheduling rule needs to account for future and past states . Howe ver we find that cone sche dules, which respon d only to the curre nt workload, are able to guarante e stabilit y for any arri v al rate w ithin the stabi lity region. The service rates in this paper are allowed to be completely arbitrar y , in contrast to previ ous results using the trac e-based analysi s which onl y appli ed to positi ve-s ervice switches. This captur es cross-tr affic and forwardin g between queues, becaus e the selected service vect or m ay induce additiona l worklo ad to the system, in addition to the exte rnal arri val process. F urther , in this work time is continuo us, and arbitra rily lar ge arriv al bursts can be han dled at arbi trarily small time interv als. This is more general than pre vious 3 models where arri vals and decision s were restricted to timeslots. From an architectur al point of vie w , the geometric approach to the scheduli ng problem prov ides key practic al design leads. Specifically , the conic representati on (Section 4) of cone schedule s leads to scalable implementa tions in switching systems. Further , v arying the elements of m atrix B , we can generate a very rich family of cone schedules that implement a soft coupled priority scheme (and coupled load balancin g) across the var ious queues, managing delay tradeof fs between them. The schedule s are also rob ust to any sublin ear perturba tion such as delayed or flawed state informat ion. The remainder of the paper pro ceeds as fol lows. I n Section 2, we intro duce the model and system dynamic s. Section 3 descri bes the thr oughput capaci ty or stabili ty region o f the se netw orks, and in section 4 we introduce the family of Cone S chedu les and their geometry . Stability and performance implicatio ns are discus sed in sections 5 and 6 respecti vely . W e con clude in Section 7. 2 The Pr ocessing Structur e Let R t 0 A q ( z ) dz be the total wor kload that arriv es to qu eue q in the time interv al (0 , t ] ; that is, A q ( t ) ≥ 0 is the instanta neous worklo ad arriv al rate at time t ≥ 0 . The traffic trace A q = { A q ( t ) , t ≥ 0 } is a (determin istic) functio n, which may hav e disconti nuities and ev en δ -jumps for each q ∈ Q . The o verall (ve ctor) instantane ous traffic rat e is A ( t ) = ( A 1 ( t ) , A 2 ( t ) , ..., A q ( t ) , ..., A Q ( t )) at time t > 0 and the tr affic tra ce is A = { A ( t ) , t ≥ 0 } . W e as sume that the (long-ter m) traf fic load of the trace 4 A , lim t →∞ R t 0 A ( z ) dz t = ρ ( A ) ∈ R Q 0+ , (2.1) is well-defined on the traf fic trace A . C orresp ondingl y , we define the set of tra ffic traces of load ρ ∈ R Q 0+ , A ( ρ ) = ( A = { A ( t ) , t ≥ 0 } : lim t →∞ R t 0 A ( z ) dz t = ρ ) , (2.2) restric ting our attention in this paper to traffic traces of well-defined load. A vari ety of natural arriv al proces ses are include d. Fo r example, A q ( t ) = P ∞ j =1 σ j q 1 { t = t j q } models jobs of service requ irement σ j q arri ving at times t j q > 0 to queue q . In th is case, A q ( t ) is zero between conse cuti ve δ -jumps. In general, there could be positive instant aneous workload arriv al rate between consecu tiv e δ -jumps, which would represent a contin uous inflow of w ork. No further restriction s are placed on the arrivin g traf fic trace. It may be generated by an underlying stocha stic process, or ev en an adv ersary specifically designed to destabilize the system w hene ver possible . The arriv ing wo rkload is queue d up in the queues q ∈ Q , which are ass umed to be of infinite capacity . Let X q ( t ) be the work load (total workload or service requiremen t) in queue q at time t ≥ 0 and X ( t ) = ( X 1 ( t ) , X 2 ( t ) , ..., X q ( t ) , ..., X Q ( t )) ∈ R Q 0+ , the ov erall (vecto r) worklo ad. 4 Throughout this study we employ the notation Z + = { 1 , 2 , 3 , · · · } , Z 0+ = { 0 , 1 , 2 , · · · } , R = ( −∞ , ∞ ) , R + = (0 , ∞ ) , R 0+ = [0 , ∞ ) 4 The pro cessing system operates in a flu ctuating en vir onment , which can be in one of E distinc t states at an y poin t in time, inde xed by e ∈ E = { 1 , 2 , · · · , E } . Let e ( t ) ∈ E be the en vironment state at ti me t and E = { e ( t ) , t ∈ R } the ov erall en vironment trace over time. It is assumed that the propor tion of time the en vironment trace E spends in each state e ∈ E is well-defined , that is, lim t →∞ R t 0 1 { e ( z )= z } dz t = π e ( E ) with P e ∈E π e ( E ) = 1 , π e ( E ) > 0 , e ∈ E . Correspond ingly , we define the set o f en vironment traces E with time proporti ons π e , e ∈ E as E ( π e , e ∈ E ) = ( E = { e ( t ) , t ≥ 0 } : lim t →∞ R t 0 1 { e ( z )= z } dz t = π e , e ∈ E ) , (2.3) and restrict ou r attention in this paper to en vironment traces that hav e well-define d time proporti ons. Finally , E = 1 natural ly correspo nds to the degene rate case of a constan t (non-fluctuat ing) en vironmen t. When the en vironment is in state e ∈ E , a (nonempty) set of service vectors S e becomes a va ilable to the s ystem mana ger , who can select a s ervice v ector S ∈ S e at any point in time to operate the system. Each S ∈ S e is a Q -dimens ional vector S = ( S 1 , S 2 , ..., S q , ..., S Q ) ∈ R Q , where S q ∈ R is the dr ain ( or fill, see below) rate of queue q w hen the ser vice v ector S is use d. For example, in a simple system with two queues ( Q = 2 ), a service vector S 1 = (1 . 35 , 2 . 17) would serv e (drain) queue 1 at rate 1.35 a nd que ue 2 at rate 2.17 (work units per time un it). T his is the stand ard way of vie w ing s ervice vec tors. In this general model, howe ver , we also allo w for ne gative ‘service ’ rates, actually correspo nding to traf fic workload ‘feed’ rates, as expla ined below . In the previo us simple example of two queues , a service vec tor S 2 = (1 . 2 , − 0 . 8) would serve (drai n) the fi rst queu e at rate 1.2, bu t feed workload to the second queue at rate 0.8, filling it up. The motiv ation to allo w for negati ve components S q < 0 in the service vectors S ∈ S e comes from th e need to model en vir onmental (bac kgr ound) cr oss-traf fic sharing the queue b uf fers with t he primary (fore ground ) tra ffic { A ( t ) , t ≥ 0 } . This cross-tr affic depends exp licitly on the service vector S ∈ S e used, and implic itly on the en vironment state e ∈ E through the set S e where the servi ce vector S should be chosen from. When service vect or S is used w ith S q < 0 for some queue q ∈ Q , this corresp onds to cross- traf fic workloa d fed into queu e q at constan t rate − S q > 0 , in add ition to the pri mary traf fic worklo ad { A q ( t ) , t ≥ 0 } . It is easy t o see th at − S q > 0 can be int erpreted as the ‘ net’ cross-tr affic throu gh the queue; that is, workloa d could be fed into queue q at rate r 1 > 0 and removed (serv ed) at rate r 2 > 0 , with the net cross- traf fic load fed into the queue being − S q = r 1 − r 2 . One special case rel ated to this model is a feed-forwa rd networ k. A service vector representing the transfe r of worklo ad from one upstream queue q u to another do wnstream queue q d would be repr esented with S q u = − S q d and all other S q = 0 . The model here could handle the aggregat e of many transf ers, as w ell as gain and loss in the system at any queue. The concept of cross-traf fic considered here is more genera l, requiring no restrict ions on the physic al struc ture of the network. Feed-forward network s requi re some additional assumption s and are not the primary focus of this paper , but ha ve been studied extensi vely else w here, such as [Dai and Lin, 2005]. 5 Note that the abov e en vironmental cross -traf fic is far less ‘in nocuous ’ than simply allo wing the pri- mary traf fic to be modulated 5 by the en viron ment state. Indeed, the cross-tra ffic depends on the choice of servic e vec tor S , hence , the schedulin g decisio ns acti vel y influence it. The en vironment pla ys only a sec - ondary rol e by defining S e , hence , restrict ing the rang e of sche duling choices. Actual ly , the intro duction of cross-traf fi c is sho wn to hav e significan t implications on the stability beha vior of the scheduling policies studie d later . The sets S e , e ∈ E may be over lapping, that is, a service vector may be av ailable under one or more en vironment states. Let S = S e ∈E S e . It is assumed that each service vect or set S e , e ∈ E is complete , that is, for each e ∈ E and any q ∈ Q ( S 1 , S 2 , ..., S q − 1 , S q > 0 , S q +1 , ..., S Q ) ∈ S e ⇒ ( S 1 , S 2 , ..., S q − 1 , S q = 0 , S q +1 , ..., S Q ) ∈ S e . (2.4) Hence, any ‘sub-v ector’ of a service vec tor in S e (i.e. with one or more p ositive components reduced to zero) is als o 6 a service vect or in S e . The reas on for requi ring completeness of eac h S e is to accommodate the follo wing situatio n: when some queues become empty and ceases r eceiving service , the r esulti ng ef fective servic e vector is a feasible one. Under the latter perspec tiv e, the imposed assumpti on (2.4) is a natural one indeed . A s seen belo w , it allows us to naturally handle schedu les which provid e zero service rate to empty queue s. The ke y issue is choosing the service vector S ( t ) ∈ S e ( t ) at time t , when the en viron ment is in state e ( t ) and the v ectors S e ( t ) are av ailable to cho ose fro m. In general, the decisio n can be based on the observ able histories of the workload { X ( z ) , z ≤ t } , the en viron ment { e ( z ) , z ≤ t } , and prior service choice s { S ( z ) , z < t } . T he scheduling policy is the ov erall trace of service vector choices S = { S ( t ) , t ≥ 0 } . Our primar y obje cti ve is to de sign sche dules S which maxi mize the syste m through put (keep the system stable und er the maximum possib le load ρ ), while being r obust and util izing minimum information , like th e curren t workload and en vironment states, with no kno w ledge of the actual load ρ and the en vironment time propo rtions { π e , e ∈ E } . W e elab orate on such issues late r . W e are interested in natural schedu les S = { S ( t ) , t ≥ 0 } that nev er apply positi ve service to empty queue s. That is, whenev er X q ( t ) = 0 the scheduler chooses a service vector S ( t ) ∈ S e ( t ) with S q ( t ) ≤ 0 . This is possib le because we hav e assumed that the sets S e , e ∈ E are complete . Therefore, we can w rite X ( t ) = X (0) + Z t 0 A ( z ) dz − Z t 0 S ( z ) dz (2.5) without ha ving to explicitl y ‘compens ate’ for any idlin g time. 5 Actually , the en vironment could also modulate the primary tr af fic trace { A ( t ) , t ≥ 0 } in t he followin g sense. There is a collection of traf fic traces { A e ( t ) , t ≥ 0 } one for each en vironment state e ∈ E . W hen the env ironment is in state e , the traffic dri ven into the system is selected f rom { A e ( t ) , t ≥ 0 } . T herefore, the ov erall traffic trace is simply { A ( t ) = P e ∈E A e ( t ) 1 { e ( t )= e } , t ≥ 0 } . Hence, this basically re verts to the stand ard model (as long as the limit lim t →∞ R t 0 A ( z ) dz /t exists) and this is why we do not treat this case explicitly . 6 Note that if an y service v ector in S e has no n egati ve components, th en the zero vector (0 , 0 , ..., 0) must be in S e as a sub-vector of the former vector , due to completeness. But if each service vector in S e has at least one neg ativ e component, the zero vector does not necessarily hav e to be in S e unless it is by design. 6 3 The Stability Issue In th e interest o f rob ustness of the results, we emplo y the ‘light est’ possible (see belo w ) concep t of stabili ty , that is, rat e stability [Bambos and W alrand, 1993]. Specifically , we call the system stable iff lim t →∞ X ( t ) t = lim t →∞  X 1 ( t ) t , X 2 ( t ) t , ..., X q ( t ) t , ..., X Q ( t ) t  = 0 . (3.1) Note that from (2.5) and (2.1), rate-stabil ity implie s that ρ = lim t →∞ { R t 0 S ( z ) dz /t } . M oreo ver , when the traf fic trace in volv es pure ‘job-arr iv als’ ( δ -jumps) with zero workloa d arri va l rate between them, then rate- stabili ty (3.1) implies that the long-te rm job departu re rate from each que ue is equal to the long -term job arri v al rate [Armony and Bambos, 2003]. Therefore, there is flow conser vation throug h the system and the inflo w at each queue is equal to the outflo w . On the contrary , when the system is unstable there is a inflo w- to-out flow deficit, which accu mulates in the que ues. This is consis tent with engine ering intu ition and, in that sense, the concept of rate-stabili ty is quite natural. Of course, it can be further tighte ned by imposing progre ssi vely hea vier statistic al assumption s on the traffic and en viron ment traces. W e resist doin g that at thi s point, in ord er to pres erve the gen erality of the results and keep them as rob ust and ‘assumptio ns- agnos tic’ as possible. Definition 3.1 (Stability Region) W e define formally the stabil ity regio n R of the system as the set of traf fic loads ρ ∈ R Q 0+ for which there exists a schedulin g policy S = { S ( t ) , t ≥ 0 } under w hich the system is rate-s table (3.1) for all traffic traces A = { A ( t ) , t ≥ 0 } with ρ ( A ) = ρ and all en viron ment traces E = { e ( t ) , t ≥ 0 } w ith π e ( E ) = π e , e ∈ E . As sho wn belo w , the uni ver sal stability reg ion R can be characteriz ed as R ( S e , π e , e ∈ E ) = ( ρ ∈ R Q + : 0 ≤ ρ ≤ X e ∈E π e X S ∈S e φ e S S, for some φ e S ≥ 0 with X S ∈S e φ e S = 1 , e ∈ E ) (3.2) The intuition is that ρ is in the stability regi on R if it is domina ted (co vered ) by a con vex combina tion of the service vector s S ∈ S , induc ed under the va rious service vectors in S e , e ∈ E . Thus, R is the ‘weighte d sum’ of the vari ous ‘stabili ty regions’ generated by the indivi dual sets S e for each state e ∈ E of the en vironment. If ρ ∈ R and π e , e ∈ E were kno wn in adv ance and φ e S could be co mputed, then selectin g each mode S ∈ S e for a fraction φ e S of the time while the system is in en vironment state e ∈ E would kee p the system stable . This could be achiev ed through round-ro bin or rando mized algorithms. A scheduling algorithm which mainta ins stability (3.1) for any ρ ∈ R is referre d to as thr oughput maximizing . Howe ver , we are primarily interested in adapti ve schedu ling schemes which maintai n stabi lity (3.2) for all ρ ∈ R , without actual prior knowledg e of ρ or π e . The cone schedule s defined belo w are shown to provide such univ ersal stabili ty for any traf fic loa d in R , while being agn ostic to particulars of the traffic and en viron ment traces ρ ( A ) and π e ( E ) = π e , e ∈ E ; they respond only to current workload and en vironment state. In gene ral, the stability behav ior of scheduli ng rules could require the arri v al trace to satisfy stronger condit ions than tho se abov e. For example, re stricting the s tudy to Markov ian or st ationary arri v al pro cesses, 7 or disallo wing mixing, may provid e speci al cases of stability . Instead, we allow the arriv al tra ffic trace A and en vironment trace E to be desig ned by an adversar y to stress the system. Con sider for example an arri v al trac e w here arri v als to que ue q are deliberately cor related to the en viron ment states when q cannot be served at maximum capa city . Even further , an adv ersaria l trace may push arriv als to queues in a stat e- depen dent way w hich respon ds to the scheduling rules themselves. T hese are very difficult to capture by a natura l probabi listic framewo rk, but are simply treat ed as particular traf fic traces here. T o moti va te th e definition of the stability region for the process ing system under consideratio n, we e x- amine first the c ase where S q ≥ 0 for all q ∈ Q and th ere is on ly one en vironment state ( E = 1 , no en viron- ment fluctuation) ; that is, service is always non-ne gati ve and all s ervice vectors S are av ailable at e very point in time . U nder the trac e-based perspec tiv e employ ed in this paper , it is kno wn [Armony and Bambos, 200 3 ] that for any load ρ in the region ( ρ ∈ R Q 0+ : ρ ≤ X S ∈S φ S S for some φ S ≥ 0 , S ∈ S , such that X S ∈S φ S ≤ 1 ) the system can be m ade rate stable with an appropriat e scheduli ng rule. The non-ne gati ve parameters φ S , S ∈ S are essentially proportio nal weights, which are cho sen so that the load v ector ρ is component- wise dominated by the weighted linear combina tion P S ∈S φ S S of the service vecto rs. Extendin g this ‘geometric’ stability perspect iv e to allow cross traffic and varyin g en vironment states is not a trivi al task. Intuition may suggest that the stability region in networ ks in fl uctuat ing en vironments should be reduced according to ho w often each mode is av ailable. Consider the follo wing simple network to illustrate that the distrib ution of en vironment states { π e , e ∈ E } is criti cal to sta bility . T ake a 2-qu eue netwo rk with three servic e v ectors , S 1 = (1 , 0) , S 2 = (0 , 1) , S 3 = (1 , 1) . C learly , if all vectors are av ailabl e all the time, by employing alwa ys S 3 the system can accommodate any input vecto r ( ρ 1 , ρ 2 ) ∈ [0 , 1] 2 . On the other hand, if the re are two en vironment sta tes E = { e 1 , e 2 } with service vecto r set s S e 1 = { S 1 , S 2 } and S e 2 = { S 3 } with π e 1 = 0 . 5 , π e 2 = 0 . 5 , then the syst em can accommodate any input vect or ρ ≥ 0 satisfy ing the conditio ns ρ 1 + ρ 2 ≤ 1 . 5 , ρ 1 ≤ 1 , and ρ 2 ≤ 1 . Ho wev er , a differ ent configuration of the service vector sets, say S e 1 = { S 1 } and S e 2 = { S 2 , S 3 } with π e 1 = 0 . 5 , π e 2 = 0 . 5 , yields ρ 1 ∈ [0 , 1] and ρ 2 ∈ [0 , 0 . 5] for stability . Note that although the sets S e 1 and S e 2 ensure that each service vector is a va ilable for the same portio n o f time in both scenario s, the relati ve combina tions of the av ailable service vect ors change the stability region . W e illust rate (and generalize ) this perspe cti ve in Figure 1. W e establish first that if ρ / ∈ R , it is imposs ible to maintain stability and fl o w conserv ation in all queue s, no matter what scheduling policy one employs. A t leas t one queue will suf fer an outflo w deficit (compare d to its inflow), which will accumulat e in the queue and cause its workload to explode linearly it in time. Pro position 3.1 (Instability) For an y arbitrarily fixe d traffic traf fic trace A and en vironmen t trace E , w e ha ve ρ ( A ) / ∈ R = ⇒ lim sup t →∞ X q ( t ) t > 0 , (3.3) for at least one queue q ∈ Q under any sche duling polic y . Pro of: For con ve nience, we drop the fixed argumen t A from ρ ( A ) and write it traf fic load as simply ρ , 8 ρ 2 2 S 4 S 1 S 3 S 5 S ρ 1 1 ρ 2 7 S ρ S 6 ρ 1 ρ 2 ρ 2 0.8S 1 0.2S 6 0.2S 7 ρ 1 Figure 1: The stability reg ion. The set of allo wable arri v al rate vectors ρ is called the stability region R . T wo separate sets of service vectors are shown in the first two plots, with their respecti ve stability region s if they w ere the only en vironment state, and av ailable 100% of the time. The third plot shows the stabilit y reg ion when π 1 = 0 . 8 and π 2 = 0 . 2 . This corres ponds to the en vironment state fluctuating so that 80% of the time, the service vector s from the first group are av ailable , and 20% of the time the service vectors from the second grou p are a vailab le to be scheduled. For any ρ in the region R abo ve, there is a con vex combina tion of service modes within the resource sets which would apply a total servic e rate to each queue which is at least the a rri val rate to th at queue. For ρ outside R there is no such combin ation. Service modes S 2 and S 4 are strictly dominated by a con ve x combin ation of other service ve ctors and there fore do not contri bu te to the stability regi on (and in fa ct need not be utili zed to maintain stability ). Service vecto rs w ith neg ati ve components such as S 1 and S 5 abo ve may contrib ute to the stab ility region without being ins ide the stab ility regio n itself. The sta bility regi on for the combination of en vironments can be seen to be the weighted sum of the two original stabi lity reg ions, w ith care taken to the impact of ex treme points with neg ati ve components . 9 and proceed by contradict ion. If (3.3) does not hold, then from (2.5) we must hav e lim t →∞ R t 0 S ( z ) dz t = lim t →∞ R t 0 A ( z ) dz t + lim t →∞ 1 t X (0) = ρ . B ut then we ha ve ρ = lim t →∞ R t 0 S ( z ) dz t = lim t →∞ R t 0 P e ∈E P S ∈S e I e ( z )= e,S ( z )= S S dz t = X e ∈E π e X S ∈S e ˆ φ e S S where ˆ φ = lim t →∞ R t 0 I S ( z )= S S dz t satisfies ˆ φ e S ≥ 0 and P S ∈S e ˆ φ e S = 1 , which satis fies (3.2). W e then easily get (ar guing by contradictio n) that lim su p t →∞ X q ( t ) /t > 0 for at least one queue q ∈ Q . 4 Cone Sched ules a nd their Geometry W e focus in this pape r on sch edules that are workload-a ware and resource-a ware but not rate-a ware; that is, the system’ s operator can observ e and respon d to both the en vironment state e ( t ) and the worklo ad state X ( t ) , but has no kn owle dge of the long-t erm load vector ρ and state probabilit ies π e . In particul ar , we examine a family of resource allocation policie s that are called Cone Schedules and are parameter ized by a fixe d matrix B . These schedu les select the servic e vector ˆ S ∈ S e ( t ) that has the maximal projection on B X ( t ) , when the work load state is X ( t ) and the en vironment state is e ( t ) ∈ E . Specifically : Definition 4.1 (Cone Schedules) Giv en a fixed Q × Q real matr ix B , a cone sched ule is one th at, when the en vironment is in state e ∈ E and the the worklo ad is X ∈ R Q 0+ , it selects a service vec tor ˆ S e ( X ) in the set ˆ S e ( X ) = arg max S ∈S e h S, B X i = { ˆ S ∈ S e : D ˆ S , B X E = max S ∈S e h S, B X i} (4.1) which satisfies S q = 0 whene ver S q = 0 . W e sho w that such a vector must be c ontained in ˆ S e by propositi on 4.1 belo w . The set ˆ S e ( X ) ⊆ S e is nonempty , but may contain se vera l service vectors in S e , in which case one is arbitrari ly chosen by the cone schedule . Note that D ˆ S e ( X ) , B X E = max S ∈S e h S, B X i , (4.2) so the chosen ˆ S e ( X ) is one of maximal projectio n on B X amongst those in S e . Therefore, the service vec tor ˆ S ( t ) chosen by the cone schedul e at time t ≥ 0 is ˆ S ( t ) ∈ ˆ S e ( t ) ( X ( t )) = arg max S ∈S e ( t ) h S, B X ( t ) i , based on the obser ved current workload X ( t ) and en vironment state e ( t ) . 10 Notice that the maximization h S, B X i = P q S q ( B X ) q ensure s that cone sched ules follo w some importan t intuition for a schedu ling rule. W e see that ( B X ) q is increas ing in X q and decreasi ng in X p for p 6 = q . This will increase whene ver X q comes to dominate other queues. By maximizin g this sum, the cone sched ules all prefer lar ge positi ve servic e rates S q whene ve r ( B X ) q is lar ge and positi ve. T hus the schedu les will prefer remove the most worklo ad from the longer queues , and restrict the cross-tr affic added to those longe r queues. T he relati onship to perfo rmance and load balancin g is discussed in section 6. Pro position 4.1 (Matrices B with Negativ e or Z er o Off-Diagonal Elements) If the cone schedule ma- trix B = { B ij , i, j ∈ Q} has ne gative or zer o of f-dia gonal elements ( B ij ≤ 0 , i 6 = j ) and the service vec tor sets S e are complete for each en vironments state e ∈ E , then there m ust e xist some ˆ S e ( X ) ∈ ˆ S e ( X ) for which we ha ve X q = 0 = ⇒ ˆ S e q ( X ) ≤ 0 for each q ∈ Q . Thus, for such B matrices, the corresp onding cone sche dules can alwa ys select servi ce vec tors that provide no positive service rate to an empty qu eue . Pro of: Giv en a workload vecto r X such that X q = 0 for some (empty) queue q ∈ Q , let us examin e the inner product maximized by the cone schedul e (4.1) in selecting ˆ S e ( X ) ∈ ˆ S e ( X ) , that is, h S, B X i = X i ∈Q X j ∈Q S i B ij X j = X i ∈Q { S i B ii X i + S i ( X j ∈Q−{ i } B ij X j ) } . (4.3) with S ∈ S e . Consider the term correspo nding to the empty queue q in the abo ve sum, that is, S q B q q X q + S q ( X j ∈Q−{ q } B q j X j ) . (4.4) Since X q = 0 , the first term abo ve is automatically zero, irrespec tiv ely of S q and B q q . Howe ver , since B q j ≤ 0 and X j ≥ 0 for each j ∈ Q − { q } , we see that X j ∈Q−{ q } B q j X j ≤ 0 . (4.5) Arg uing by contradic tion, assume that ( S 1 , S 2 , ..., S q − 1 , S q > 0 , S q +1 , ..., S Q ) ∈ S e maximizes (4.3) with S q > 0 . But because S e is assumed to be complete (2.4), the ve ctor ( S 1 , S 2 , ..., S q − 1 , S q = 0 , S q +1 , ..., S Q ) also belon gs to S e and has S q = 0 , hence, leads to an equal or greater v alue of (4.3) becaus e of (4.4) and (4.5). This establish es a contradicti on and implies that the set of service vectors S that maximize (4.3) must alw ays inc lude one w here S q ≤ 0 (provide no positi ve servic e rate) for each empty queue q ∈ Q (that is, w ith wor kload X q = 0 ). T o justify the term ‘cone’ schedu le cons ider the follo wing perspecti ve. Defi ne first the set of work- loads X for which the cone schedule would choos e the service vector S when the en vironmen t is in state e , that is: C e S =  X ∈ R Q 0+ : h S, B X i = max S ′ ∈S e  S ′ , B X   11 for S ∈ S e , e ∈ E . This is simply the set of workloads X that ha ve m aximum projection on S ∈ S e amongst all other sets in S e . Note that C e S is a geometr ic cone because h S, B X i ≥ h S ′ , B X i impli es that h S, B αX i ≥ h S ′ , B αX i for any po siti ve scalar α ∈ R + and S, S ′ ∈ S e . Thus, if X belong s to C e S then an y up/do wn-scaling αX also belongs to it. For eac h en vironment state e ∈ E , the cones C e S , S ∈ S e form a partiti on of the workloa d space, that is, [ S ∈S e C e S = R Q 0+ . In gene ral, some cone s may actuall y be dege nerate (like those correspon ding to service vector s in S e that are fully domina ted compone nt-wise by other s in S e ) a nd se vera l cones ma y share common bound aries. Observ e that the cone schedule can now be geome trically defined as follo w s: When the en vironment state is e and the work load X ∈ C e S = ⇒ choose ˆ S e ( X ) = S ∈ S e . The cone structur e of the sets C e S moti v ates the name cone schedule s . 0 0.5 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 C S 1 C S 3 C S 4 X X 2 C S 2 X 1 X 3 Figure 2: The cone schedules assign a service vector from S e by identify ing the locatio n of X with respect to the cones formed by C e S . This figure shows the cone structu re for a system with Q = 3 queues and 4 servic e vectors for this particu lar en vironment. When X is in cone C e S , then service vector S corresp onding to that cone is used . The vector X will fluctuat e within ℜ 3 , switching between serv ice vecto rs when the arri v als and departu res cau se X ( t ) to cross a cone boundary , or when the en vironment state changes. The cone boundar ies are influenced by the en vironment state and the matrix B . When the en vironment is in state e ∈ E and the workload X is in the interio r of the non-d egene rate cone C e S , then the only service vect or that can be used by the cone schedu le is S ∈ S e . H o wev er , if X is on the b oundary of se ver al adja cent cones (for e xample, X ∈ C e S 1 T C e S 2 T C e S 3 ), th en an y of the serv ice vec tors corres ponding to these cones can be used ( S 1 , or S 2 , or S 3 ). Therefor e, gi ven a workloa d vector X , w e want to define the cone it belongs to, which conseque ntly specifies what service vector the cone schedule ought to use. W e pro ceed in this direc tion belo w . T o take another perspecti ve , recal l tha t when the en vironment is in state e ∈ E and the workload is 12 X , then the cone schedule choos es a service vecto r ˆ S e ( X ) in the set ˆ S e ( X ) =  ˆ S ∈ S e : D ˆ S , B X E = max S ∈S e h S, B X i  ⊆ S e ; any v ector is arbitra rily chosen, if there are more than one vector in the set ˆ S e ( X ) . When X is in the interio r of the (non-de genera te) cone C e S , then ˆ S e ( X ) = { S } is a singlet on and ˆ S e ( X ) = S . This follows since the in terior of a cone deno tes all worklo ad vector s X for which th e inne r product h S, B X i is un iquely maximized by S . T o cove r the general case of X being on a cone bounda ry (perh aps, a common bound ary of se vera l cones) , we define the ‘surrou nding’ cone of the workload vecto r X as C e ( X ) = [ S ∈ ˆ S e ( X ) C e S For example, if X is on the boundary of C e S 1 and C e S 2 only , then C e ( X ) = C e S 1 S C e S 2 . Note that the abov e definitio ns lead to the follo wing equi va lence C e ( X ) ⊆ C e ( Y ) ⇔ ˆ S e ( X ) ⊆ ˆ S e ( Y ) , as well as C e ( X ) ⊆ C e ( Y ) ⇒ X ∈ C e ( Y ) for any two w orkload vec tors X , Y ∈ R Q 0+ and en vironment state e ∈ E . This is illustrated in F ig. 3. 0 0.5 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 X on boundary C(X) X 2 X 1 X 3 Figure 3: Fo r work load vecto rs X w hich lie precise ly on the boundary of two or more cones , the cone C e ( X ) is the union of all of the cones in C e which include X . In contrast to F ig. 2, w here X was interior to a single cone, the abov e illustra tion sho ws X at the boundar y of 3 of the C S cones. In this case C ( X ) = C S 1 ∪ C S 2 ∪ C S 4 includ es all the elements of the thre e dif ferent cones. This definition is important in the proof because it captures the worklo ad vectors which share an optimal service vector with X . Note that if Y ∈ C e ( X ) then there m ust exist a service ve ctor ˆ S ∈ S e for which both h S, B X i is maximized at ˆ S and h S, B Y i is maximized at ˆ S , and if Y / ∈ C e ( X ) then no such vec tor can exist. W e observe that X cannot be on an interior boundar y of C e ( X ) (the only bounda ry it could be on is where the cone m eets an axis because of the non-neg ati vity constrain t). If X were on an interior bounda ry 13 1 C(X) C (X) C (X) e2 2 X X e1 X Figure 4: T he cone C ( X ) ov er en vironmen ts E is illustrate d. Here, X is in the cones C e 1 ( X ) and C e 2 ( X ) for the two en vironments e 1 and e 2 . The cone C ( X ) is the intersec tion of both of those cones. Since X is kno w n to be on the interior of each cone, X is also on the interior of C ( X ) . then there m ust e xist a direc tion vector δ 6 = 0 for which ( X + λδ ) ≥ 0 and ( X + λδ ) / ∈ C e ( X ) for an arbitra rily small positi ve scalar λ . This means that there exi sts some service vector S δ ∈ S e for which  S δ , B ( X + λδ )  > h S, B ( X + λδ ) i for all S ∈ ˆ S e . B ut since S δ / ∈ ˆ S e ( X ) we also hav e  S δ , B X  < h S, B X i for all S ∈ ˆ S e . This leads to the inequalit y λ ( D S δ , B δ E − h S, B δ i ) > h S , B X i − D S δ B X E > 0 . Since the left hand side can be made arb itrarily small this leads directly to a con tradictio n and we conclu de that X is indeed on the strict interior of C e ( X ) . This observ ation becomes critical in the proof of stabili ty . Finally , we define the cone around X with respect to all en vironment states e ∈ E as C ( X ) = \ e ∈E C e ( X ) . The cone C ( X ) is illustrated i n Fig 4 is of co urse non-empty because X belongs to each cones C e ( X ) , e ∈ E . This is the cone of worklo ads Y for which, at each en vironmen t state e ∈ E , the cone schedul e could hav e selecte d for Y the same service vector as for X (fixe d), that is, C ( X ) = { Y ∈ R Q 0+ : ˆ S e ( Y ) \ ˆ S e ( X ) 6 = ∅ , for each e ∈ E } ; hence, when Y ∈ C ( X ) , then for each e ∈ E we hav e ˆ S e ( Y ) ∈ ˆ S e ( X ) , besides ˆ S e ( X ) ∈ ˆ S e ( X ) of course . V iewed a nother way , 14 C ( X ) = { Y ∈ R Q 0+ : D ˆ S e ( Y ) , B X E = max S ∈S e h S, B X i for each e ∈ E } , that is, w hen Y ∈ C ( X ) , then for each e ∈ E we hav e that ˆ S e ( Y ) has maximal projection on B X , beside s also hav ing maximal projection on B Y (by definition). W e note that since X is strictly on the interior of each cone C e ( X ) and there are finitely many en vi- ronment states in E then X is stri ctly on the interi or of C ( X ) . The con e C ( X ) turns out to be of key importanc e in the stability proof belo w . This complet es the geometri c picture of cone schedules . 5 Univ ersal Stability of Cone Schedu les W e are primarily interested in the throughpu t maximizing properties of cone schedules for var ious famili es of matrices B , giv en the traffic load ρ . The follo wing theorem establishes that stability can be maintaine d for any ρ ∈ R by rich families of matrices B . Consider a cone schedu le generated by the matrix B and operating on any arbitrarily fi xed sys tem Σ chosen from the class S of process ing systems defined by: 1. some set of queue s Q and some set of en vironment states E , 2. some en vironment trace E ∈ E ( π e , e ∈ E ) , as per (2.3), 3. some (non-e mpty) service vector sets S e , e ∈ E that are complete , as per (2.4), 4. some traf fic trace A = { A ( t ) , t ≥ 0 } ∈ A ( ρ ) with load ρ ( A ) = ρ , as per (2.2). Theor em 5.1 (Universa l Stability of Cone Schedules) Giv en the abov e assumption s if B is positi ve-de finite, symmetric and has nega tiv e or zero off- diagona l elements ( B q p ≤ 0 , p 6 = q ∈ Q ), then ρ ( A ) ∈ R ( S e , π e , e ∈ E ) = ⇒ lim t →∞ X ( t ) t = 0 (5.1) univer sally on S . T hat is, each system in S is (rate) stable under such a cone sche dule, when ρ ( A ) ∈ R ( S e , π e , e ∈ E ) . It turns out that B being positi ve definite and hav ing nonpositi ve of f-diago nal elements are both necess ary for uni versa l stabilit y , which was sho w n in [Ross and Bambos, 2009]. T o see why nonpositi ve off diagonal ele ments are required, consider a simple netwo rk with Q = 2 queue s and E = 1 en vironmen t state, where B = [2 , 1; 1 , 2] is used. If S 1 = (1 , 0) and S 2 = (0 , 3) are the two a va ilable servic e vectors th en  S 1 , B X  = 2 X 1 + X 2 , an d  S 2 , B X  = 3 X 1 + 6 X 2 . Since  S 2 , B X  strictl y dominates  S 2 , B X  for an y nonzer o workload, S 1 would ne ver be s elected and an y arri val proces s with ρ 1 > 0 will be unstab le. 15 T o see why positi ve definiteness is required, consider a simple network w ith Q = 2 queues and E = 1 en vironment state, where B = [1 , − 2; − 2 , 1] is used. Let S 1 = (1 , 1) , S 2 = (0 , 0) , S 3 = (1 , 0) and S 4 = (0 , 1) be the a vailab le service vector s. Then w e ha ve  S 1 , B X  = − X 1 − X 2 < 0 =  S 2 , B X  , an d S 1 will ne ver be selected . The effe cti ve service rates applied to the queues ˆ S q = lim t →∞ 1 t R t o S q ( z ) must then satisfy ˆ S 1 + ˆ S 2 ≤ 1 . No w ρ = ( ρ 1 , ρ 2 ) with 0 . 5 < ρ q ≤ 1 is contain ed within R by (3.2), but rate stabili ty cann ot poss ibly be achie ved in (2.5). T he parameters of the non-pos itiv e-definite m atrix B cause the cone schedule to av oid utilizin g S 1 , which is critical for rate stability because it lies on the con ve x hull of R . 5.1 Pr oof o f Th eorem 5.1 W e prov e rate stability via a sequence of intermediat e steps. Consider any a rbitraril y fixed en vironment trace E = { e ( t ) , t ≥ 0 } , such that S e is complet e and lim t →∞ R t 0 1 { e ( z )= e } dz t = π e for each e ∈ E . Consider also any arbit rarily fi xed t raf fic trace A = { A ( t ) , t ≥ 0 } satisfy ing lim t →∞ R t 0 A ( z ) dz t = ρ ( A ) ∈ R ( S e , π e , e ∈ E ) . W e note that while A and E are fixed, they can be generated arbitrar ily , including by an underly ing stocha stic proce ss o r an advers ary . Recall that by Pro position (4.1) when B has negat iv e or z ero off-d iagonal elements the generat ed cone schedul e applies no positi ve rate to empty queues. Therefore, X ( t ) = X (0) + Z t 0 A ( z ) dz − Z t 0 S ( z ) dz (5.2) for the worklo ad X ( t ) at time t – as in (2.5) – without ha ving to compensate for any idle time. Pro position 5.1 U nder the condition s condit ions of Theorem 5.1, the service vector s ˆ S e ( X ) ∈ ˆ S e ( X ) selecte d by the cone schedule under va rious en vironment states e ∈ E satisfy h ρ, B X i ≤ X e ∈E π e D ˆ S e ( X ) , B X E = X e ∈E π e max S ∈S e h S, B X i . (5.3) for each workl oad X ∈ R Q 0+ . Pro of: First, cho ose an y workl oad X and fix it. Since ρ ∈ R , we ha ve ρ ≤ P e ∈E π e P S ∈S e φ e S S accordi ng to (3.2), or 0 ≤ ρ q ≤ X e ∈E π e X S ∈S e φ e S S q , for each q ∈ Q , (5.4) for some positi ve weights φ e S ≥ 0 such that P S ∈S e φ e S ≤ 1 . W e denot e v q = ( B X ) q and note that this may be neg ati ve for some q ∈ Q . W e examin e, the follo wing two cases : 16 1. If v q = ( B X ) q ≥ 0 , we get from (5.4) that ρ q v q ≤ X e ∈E π e X S ∈S e φ e S S q v q . (5.5) 2. If v q = ( B X ) q < 0 , we ha ve ρ q v q ≤ 0 , since ρ q ≥ 0 . Combining the two case s, w e get ρ q v q ≤ X e ∈E π e X S ∈S e φ e S S q 1 { v q S q ≥ 0 } v q ≤ X e ∈E π e X S ∈S e φ e S S q 1 { v q ≥ 0 } 1 { s q > 0 } v q for q ∈ Q . A dding the terms up ov er q ∈ Q , we get h ρ, v i ≤ X e ∈E π e X S ∈S e φ e S h V ( S ) , v i (5.6) where V ( S ) = ( S q 1 { v q ≥ 0 ,S q > 0 } ) , q ∈ Q ) is the vector generated by the service vector S ∈ S e by setting 0 the compone nts S q for which v q < 0 and S q > 0 . No w recall that for each e ∈ E and S ∈ S e , V ( S ) is a sub-v ector of S (dropping some positi ve compone nts to 0) and is also in S e becaus e the latter set is complete . But the serv ice vector ˆ S e ( X ) selected by the co ne schedule (4.2) has the maxi mal projecti on on v = B X amongst all those in S e , so h V ( S ) , v i ≤ D ˆ S e ( X ) , v E for e very S ∈ S e . Therefore, (5.6) becomes h ρ, v i ≤ X e ∈E π e X S ∈S e φ e S h V ( S ) , v i ≤ X e ∈E π e " X S ∈S e φ e S # D ˆ S e ( X ) , v E ≤ X e ∈E π e D ˆ S e ( X ) , v E , where the last inequal ity holds because P S ∈S e φ e S ≤ 1 for each e ∈ E . P utting back v = B X , we get h ρ, B X i ≤ X e ∈E π e D ˆ S e ( X ) , B X E , which complete s the proof. Lemma 5.1 W e ha ve that lim t →∞ X ( t ) t = 0 implies lim t →∞ R t 0 ˆ S ( z ) dz t = ρ . That is, the long-term applied servic e rate is equal to the long-term traf fic load, w hen the syst em is (rate) stabl e. Pro of: This is immediat ely obtained by di viding (5.2) by t and lettin g t →∞ . Lemma 5.2 Consider two arb itrarily fixed, increasi ng, unbounde d time sequ ences { t n } ∞ n =1 and { s n } ∞ n =1 with s n ≤ t n for each n ≥ 1 . If lim n →∞ t n − s n t n = 0 (or equi va lently lim n →∞ s n t n = 1 ), then lim n →∞ X ( t n ) − X ( s n ) t n = lim n →∞ X ( t n ) − X ( s n ) s n = 0 . 17 Pro of: Note that 0 ≤ R t n s n 1 { ˆ S ( z )= S } dz ≤ t n − s n for each S ∈ S = ∪ e ∈E S e . Dividi ng by t n and tak ing the limit as n →∞ , we get lim n →∞ R t n s n 1 { ˆ S ( z )= S } dz t n = 0 . Recalling that X ( t n ) − X ( s n ) = Z t n s n A ( z ) dz − X S ∈S S Z t n s n 1 { ˆ S ( z )= S } dz , di viding by t n and letting n →∞ , we get lim n →∞ X ( t n ) − X ( s n ) t n = lim n →∞ R t n s n A ( z ) dz t n − lim n →∞ P S ∈S S R t n s n 1 { ˆ S ( z )= S } dz t n = lim n →∞ R t n 0 A ( z ) dz t n − lim n →∞ R s n 0 A ( z ) dz s n s n t n − 0 = ρ q − ρ q . 1 = 0 (5.7) Moreo ver , lim n →∞ X ( t n ) − X ( s n ) s n = lim n →∞ X ( t n ) − X ( s n ) t n t n s n = 0 . T his compl etes the proof. Lemma 5.3 Consider an arbit rarily fixed, increasin g, unbounded time sequen ce { t n } ∞ n =1 . The follo w ing result then holds: lim n →∞ X ( t n ) − X ( t − n ) t n = 0 . Pro of: Clearly the result holds at times t when A ( t ) is finite. The issue arises at times t n when A ( t n ) has a δ -jump and the work load sudden ly shifts by a finite amoun t, which may actually be increa sing in consec uti ve jumps. Let t n be the time of a job arri v al to queue q ∈ Q , where j n the index of that job and σ j n q the worklo ad added by the job . It is then sufficien t to sho w that lim n →∞ σ j n q t n = 0 . Indeed, note that σ j n q = Z t n 0 A q ( t ) dt − Z t − n 0 A q ( t ) dt (5.8) Div iding by t n and letting n →∞ , we ha ve lim n →∞ σ j n q t n = ρ q − ρ q = 0 , which prov es the lemma. 5.1.1 Building a Contradictio n. The objecti ve of the proof is to show that lim t →∞ X ( t ) t = 0 , w hen ρ ∈ R . Since B is a pos itive-defi nite matrix, it is suf ficient to sho w that lim t →∞ D X ( t ) t , B X ( t ) t E = 0 . The proof proceeds by contradictio n. Assume that lim sup t →∞ D X ( t ) t , B X ( t ) t E > 0 , and let { t a } ∞ a =1 be an increa sing unbounde d time sequence on which the supremum limit is obtained; let lim a →∞ X ( t a ) t a = η 6 = 0 (5.9) be the correspond ing limit. Such a con ver gent subsequenc e must exis t by the compactnes s (since bounded) of the set of possible valu es 7 for X ( t ) t at lar ge times. W e will construct a related unboun ded time sequenc e 7 For an y arriv al trace, we hav e X ( t ) ≤ R t 0 A ( s ) ds , which implies that X ( t ) t ≤ R t 0 A ( s ) ds t → ρ 18 { s d } ∞ d =1 and sho w that it has the prope rty lim d →∞ D X ( s d ) s d , B X ( s d ) s d E > lim a →∞ D X ( t a ) t a , B X ( t a ) t a E > 0 . T he exi stence of such a sequence w ill cont radic t that the supremum limit is attained on { t a } ∞ a =1 . W e establis h the required contradi ction by finding an increasing unbou nded subsequ ence { t c } ∞ c =1 of { t a } ∞ a =1 , and a related sequen ce { s c } ∞ c =1 , which satisfy the follo wing two K e y Pr operties : I. lim c →∞ t c − s c t c = ǫ ∈ (0 , 1) and s c < t c for each c . This implies that lim c →∞ s c t c = 1 − ǫ . II. C ( X ( t )) ⊂ C ( η ) for all t ∈ ( s c , t c ] and each c . This implies that the workload X ( t ) drifts within the cone C ( η ) surroundi ng η = lim c →∞ X ( t c ) t c throug hout the time interv al ( s c , t c ] . The ass ociated intui tion is that s c marks t he last time before t c that the workload v ector X ( s c ) (re)ent ers the cone C ( η ) and reaches X ( t c ) ≈ η t c at time t c , drifti ng in C ( η ) throughout the time interv al ( s c , t c ] . Before constructing the abo ve sequence s w ith prope rties I and II we show their implicati ons for establ ishing the required contradic tion. Lemma 5.4 If the se quences { t c } ∞ c =1 and { s c } ∞ c =1 satisfy the Pr operties I a nd II abo ve, the n the supremum limit is not attained - as initially assumed - on the sequence { t c } ∞ c =1 (which is a subseque nce of { t a } ∞ a =1 ). This establis hes the tar geted contradictio n. Pro of: S ince B matrix has neg ati ve or zero off diagonal element s, the cone sched ule does not apply an y positi ve service rate to any empty queue (4.1). Therefore, by (5.2) we hav e X ( t c ) − X ( s c ) = Z t c s c A ( z ) dz − Z t c s c ˆ S ( z ) dz (5.10) and projec ting on B η (5.9 ) we get h X ( t c ) − X ( s c ) , B η i =  Z t c s c A ( z ) dz , B η  −  Z t c s c ˆ S ( z ) dz , B η  =  Z t c s c A ( z ) dz , B η  − X e ∈E Z t c s c D ˆ S e ( X ( z )) , B η E 1 { e ( z )= e } dz ( 5.11) where ˆ S e ( X ( z )) ∈ ˆ S e ( X ( z )) for z ≥ 0 . But becaus e of P ropert y II abov e, the workload X ( z ) drifts in the cone C ( η ) throug hout z ∈ ( s c , t c ] , which implies that D ˆ S e ( X ( z )) , B η E = max S ∈S e h S, B η i when e ( z ) = e ∈ E . Substitu ting into (5.11 ) we get h X ( t c ) − X ( s c ) , B η i =  Z t c s c A ( z ) dz , B η  − X e ∈E  Z t c s c 1 { e ( z )= e } dz  max S ∈S e h S, B η i (5.12) Observ e now tha t lim c →∞ R t c s c A ( t ) dt t c − s c = lim c →∞ R t c 0 A ( t ) dt t c lim c →∞ t c t c − s c − lim c →∞ R s c 0 A ( t ) dt s c lim c →∞ s c t c − s c = ρ 1 ǫ − ρ ( 1 ǫ − 1) = ρ 19 becaus e of Property I abo ve. Dividin g 5.12 by ( t c − s c ) and lettin g c →∞ , we get lim c →∞  X ( t c ) − X ( s c ) t c − s c , B η  = h ρ, B η i − X e ∈E π e max S ∈S e h S, B η i = − γ ( η ) ≤ 0 (5.13) for γ ( η ) ≥ 0 . The ineq uality − γ ( η ) = h ρ, B η i − P e ∈E π e max S ∈S e h S, B η i ≤ 0 is due to (5.3), since it is assumed that ρ ∈ R . Since { t c } ∞ c =1 is a subsequen ce of { t a } ∞ a =1 we hav e lim c →∞ X ( t c ) t c = η . Using Property I and (5.13) we get the follo wing inequality lim c →∞  X ( s c ) s c , B η  = lim c →∞  X ( s c ) − X ( t c ) s c , B η  +  X ( t c ) s c , B η  = lim c →∞  t c − s c s c  −  X ( t c ) − X ( s c ) t c − s c , B η  + t c s c  X ( t c ) t c , B η  = ǫ 1 − ǫ γ ( η ) + 1 1 − ǫ h η , B η i > h η , B η i (5.14) The last inequ ality is due to the facts that ǫ ∈ (0 , 1) and γ ( η ) ≥ 0 . By successi ve thi nnings of the components of the workload vector , we can obtain an increasin g un- bound ed subsequ ence { s d } ∞ d =1 of { s c } ∞ c =1 such that lim d →∞ X ( s d ) s d = ψ and from (5.14) h ψ , B η i > h η , B η i (5.15) Since B is posi tive-defin ite we ha ve h ψ − η , B ( ψ − η ) i ≥ 0 . This implies h ψ, B ψ i + h η, B η i ≥ h ψ , B η i + h η , B ψ i . Since B is symmetric (self-adj oint) we hav e h η , B ψ i = h ψ , B η i . There fore, h ψ , B ψ i + h η , B η i ≥ 2 h ψ , B η i > 2 h η , B η i , using (5.15) for the last inequal ity . Thus, h ψ , B ψ i > h η , B η i or lim d →∞  X ( s d ) s d , B X ( s d ) s d  = h ψ , B ψ i > h η , B η i = lim sup t →∞  X ( t ) t , B X ( t ) t  > 0 , gi ving a contradi ction to the definition of η . This completes the proof of L emma 5.4. 5.1.2 Constructing Sequences with Pro perties I and II It no w remains to constru ct sequen ces { t c } ∞ c =1 and { s c } ∞ c =1 satisfy ing properti es I and II . Their const ruction is based on the intuitio n mentioned abov e, which is made formal in the follo wing lemma. Lemma 5.5 Suppose lim k →∞ X ( t k ) t k = η 6 = 0 for some increasing unboun ded sequence { t k } ∞ k =1 and nonze ro η . Let s k = sup { t < t k : C ( X ( t )) * C ( η ) } (5.16) be the last time befor e t k that the cone C ( X ( t )) is not include d in C ( η ) . This is the last time that X ( t ) crosse s from outside C ( η ) to inside , hence, X ( t ) ∈ C ( η ) for e very t ∈ ( s k , t k ] and the workload drifts in 20 C ( η ) throughout that interv al. By con ventio n s k = 0 if the wo rkload has alway s been in C ( η ) before t k . W e then ha ve lim inf k →∞ t k − s k t k = ǫ 1 > 0 (5.17) for some ǫ 1 ∈ (0 , 1) . Pro of: Argui ng by contradi ction, supp ose that there exi sts an increasi ng unbound ed sub sequenc e { t n } ∞ n =1 of { t k } ∞ k =1 such that lim n →∞ t n − s n t n = 0 . From Lemma 5.2 we hav e that lim n →∞ X ( t n ) − X ( s n ) s n = 0 . Since lim n →∞ X ( t n ) t n = η , we then get lim n →∞ X ( s n ) s n = η . Further , to allo w for the possib ility of job arriv al that instan taneous ly shifts the workload from outsid e C ( η ) to inside, we note from Lemma 5.3 that we ha ve lim n →∞ X ( s − n ) s n = η . But according to the definition of s n the workload X ( s n ) must be outside C ( η ) , so lim n →∞ X ( s − n ) s n could not con ver ge to η . T his establishe s the necessary contr adiction , showing 5.17 and completi ng the proof of the Lemma 5.5. W e are no w ready to constru ct sequ ence { s c } ∞ c =1 satisfy ing prope rties I and II. W e rename the se- quenc e defined in (5.16) t o be { ˆ s c } and choose s c = max { ˆ s c , (1 − ǫ 2 ) t c } , f or s ome ǫ 2 ∈ (0 , 1) . (The se cond term (1 − ǫ 2 ) t c is used to guard agains t the degenera te case where ˆ s c is finite becaus e the worklo ad X ( t ) is alw ays in C ( η ) after some finite time.) Then w e ha ve the prop erties: 1. lim c →∞ t c − s c t c = ǫ ∈ (0 , 1) and s c < t c for each (lar ge) c . 2. C ( X ( t )) ⊂ C ( η ) for all t ∈ ( s c , t c ] and each (lar ge) c . This means that { s c } ∞ c =1 and { t c } ∞ c =1 satisfy both Properties I and II, and Lemma 5.4 completes the proof of rate stability in Theorem 5.1. 6 Pe rf ormance Issues Section 3 established the uni versa l sta bility of cone schedule s for an entire class of matrices B . A natural questi on t o consider is ho w the selectio n o f B from within this class of matrice s will af fect other performance measures such as a verag e worklo ad and waiting time. In [Stolyar , 2004] it was sho wn tha t for a simil ar queu eing syste m (with one en vironment state and nonne gati ve S q vec tors), the class of MaxW eight schedules, which are equiv alent to cone schedu les with a diagon al B m atrix, will minimize the total workload in the system as well as the holdin g cost rate asymp- totical ly in heavy traffic. While a formal proof of a correspon ding result is beyon d the scope of this paper , we conjecture that a similar result will hold for these generalized cone schedules . T his can be observ ed by consid ering the limiting beha vior of the cone schedules w hen X is large.  X ( t + ) , B , X ( t + )  = h X ( t ) , B , X ( t ) i + h A ( t ) − S ( t ) , B ( A ( t ) − S ( t )) i + 2 h A ( t ) , B X ( t ) i − 2 h S ( t ) , B X ( t ) i (6.1) for t + > t the workload immediate ly after time t . If X ( t ) is lar ge and fixed, then minimizing the exp ectation of the abo ve equ ation is equ iv alent to maximizing h X ( t ) , B S ( t ) i , beca use the first term is fix ed 21 by X ( t ) and no other terms gro w with X ( t ) . T his causes us to conjecture that the schedule that minimizes lim t →∞ 1 t R t 0 h X ( z ) , B X ( z ) i w ill be th e co ne sche dule with matrix B . A full proof of this o ptimality w ould requir e conside rably more restriction on the arriv al trace than has been prese nted in this paper . From a geometric point of vie w , the cones C e m shift (and ex pand or contra ct), as the weights assig ned to particu lar queues are adjusted. Figure 5 illustrate s a simple system where the matrix B transforms the cone space. The diag onal elements of B exp and and contract the cones in the dimension of the corre spondin g queue . The of f-diagon al elements mov e the boundar y between adjace nt cones where both cone s hav e a nonze ro service rate to the two corre spondin g queues. 0 0.5 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 C S 1 C S 3 C S 4 X 2 C S 2 X 1 X 3 0 0.5 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 B 11 C S 3 C S 1 X 2 C S 4 C S 2 X 1 X 3 0 0.5 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 C S 1 −B 12 C S 3 C S 4 X 2 C S 2 −B 21 X 1 X 3 Figure 5: The cone space with vary ing B matrices . The plots abov e illustra te the impact of matrix B on the cone space for a system with Q = 3 queues and M = 4 service vectors . The fi rst plot sho ws the cones for an iden tity matrix B . The middle plot has a diagonal B with larg e positi ve weight on the first queue, focusi ng more attentio n on service vecto rs w hich serve that queue. T he third plot sho ws the impact of off diagon al elements of B . T hese elements af fect the bounda ry between two queues . As highlight ed in the stability proof, as X increases in magnitud e in any gi ven direction, the service vec tors applied will rotate through the vectors w hich maximize h S, B X i for each en vironment. While in a particu lar en vironment , X will be drawn towa rd the boundary of its current cone. In particula r , if no ne w arri v als were allo wed and the en vironment stayed constant, the n X would follow a determinis tic path to a cone boun dary . Once at a boundary , the cone sch edule would fluctua te around the ser vice vecto rs which are optimal at that bou ndary and the workload wou ld be drawn down along that bo undary . T he bound ary planes act as attracto rs for the balanc e of queues in the system. Therefore one can vie w the matri x B as transfo rming the cone space in order to set the appropri ate attracting boundary planes giv en by the cone interse ctions. T his lead s to an understa nding of B as an importan t control on the relati ve importance of dif ferent workload dimensio ns. Cone schedule s perform constrained dyn amic load balanc ing of the queu e work loads (weigh ted by the elements of B ), observi ng the service constrain ts encoded in the service vecto rs S . As the workload of a queue increases excessi vely , the schedul e shifts attention to it and selects av ailable service configura tions S ∈ S e that provi de more service capaci ty to that queue, potenti ally at the expen se of others. T hat lowers the worklo ad at the queu e, trading it for increas ed workloa d in others and load balancin g them. A strictly diagon al matrix B ind uces a direct simple priority scheme . That is, as the w eight B q q of queue q is increase d (while those of others remain constant) , the queue attains higher servic e priority . T his results in the queue recei ving more service bandwidth ov er time and enjoy ing a lo wer workload. When B has neg ati ve of f-diagonal elements, those ha ve an in direct ef fect on se rvice prio rities, entan- gling the queues and inducin g a coupled priority sch eme . That is, when B pq < 0 with p 6 = q , the relati ve priorit y of queue p decreases as the wor kload of queu e q increa ses. As X q gro ws in size, more attention 22 needs to be paid in servicing queue q , while as X p gro ws in size, less attention is paid to queue q . It can be seen that the weight B pq < 0 induces a specific coupling between the correspo nding queues. W e also observ e that the proof of stability in section 5 is robus t to any sublinear perturba tions of informat ion or time. In particu lar , if there is a switching delay between configurat ions or an informa- tion lag in knowled ge of X ( t ) , or some error in the calcula tion of max S h S, B X i then as long as the corr espondin g perturb ation does not gr ow linear ly w ith t , then rate stability will still be assur ed . See [Ross and Bambos, 2009] for mor e detail on t his observ ation. For e xample, if a ny calc ulation error or de lay is bounded, then stability will hold. For clarity in the proofs we hav e not included additional terms, b ut the intuiti on is that any such sublin ear term will ha ve no impact on the limiting case as X ( t ) becomes lar ge. For some process ing systems, there can be computat ional issues in the requirement to calculate max S h S, B X i in real-time ov er ev ery possible service vector . The geometric structure of the cone sched- ules helps to ove rcome this by recognizin g that w hen the workload vector X is lar ge , any bounded chang e in workload will m ove the workload between adjacen t cones (adjacent cones ha ve a common bound ary when ℜ Q is di vided into the cone s C S ). Therefore, cone sch edules can be imple mented by e valuat ing h S, B X i ov er a much smaller subset of service vectors at each point in time. This was sho wn to be a special case of sublin ear perturba tions in the cone schedules , and discussed in detail in [Ross and B ambos, 200 9]. T o concl ude, the adjustment of the Q × Q entries of the matrix B allo ws for generating a rich family of stable servic e sched ules. The dynamic prio rities of the queues relate directly to the quality of service (QoS) they rec eiv e and the workloa d they see. W e are currently explor ing such perfor mance issues further . 7 Conclusions and Further Resear ch W e hav e establis hed that the family of Cone Schedule s maximizes the system throughp ut for very gen- eral process ing systems under very general condit ions. These schedu les natura lly sele ct the bes t av ailable vec tors, and this leads to the maximum possible system thro ughput. Arri val and service proc esses are as genera l as possible, and the stability proofs are presented w ith m inimal assumptions. Prev ious stability re- sults are generali zed here to a setting with generaliz ed service vectors , fluctuating resource av ailability and contin uous time schedul ing. By e xplor ing the ana lysis from a geometric standp oint we ha ve gleaned important intu ition for st abil- ity as well as performance and scalabilit y of the schedules. F urther research is necessary to deeper exp lore the ef fects of chan ges to the B matrix. 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