Optimal Server Assignment in Multi-Server

Transcription

1 Opimal Server Assignmen in Muli-Server 1 Queueing Sysems wih Random Conneciviies Hassan Halabian, Suden Member, IEEE, Ioannis Lambadaris, Member, IEEE, arxiv: v2 [mah.oc] 21 Jun 2013 Yannis Viniois, and Chung-Horng Lung, Member, IEEE, Absrac We sudy he problem of assigning K idenical servers o a se of N parallel queues in a imesloed queueing sysem. The conneciviy of each queue o each server is randomly changing wih ime; each server can serve a mos one queue and each queue can be served by a mos one server during each ime slo. Such a queueing model has been used in addressing resource allocaion problems in wireless neworks. I has been previously proven ha Maximum Weighed Maching (MWM) is a hroughpu-opimal server assignmen policy for such a queueing sysem. In his paper, we prove ha for a sysem wih i.i.d. Bernoulli packe arrivals and conneciviies, MWM minimizes, in sochasic ordering sense, a broad range of cos funcions of he queue lenghs such as oal queue occupancy (which implies minimizaion of average queueing delays). Then, we exend he model by considering imperfec services where i is assumed ha he service of a scheduled packe fails randomly wih a cerain probabiliy. We prove ha he same policy is sill opimal for he exended model. We finally show ha he resuls are sill valid for more general conneciviy and arrival processes which follow condiional permuaion invarian disribuions. I. INTRODUCTION Opimal sochasic conrol is one of he main objecives in he design of emerging wireless neworks. One of he primary goals in sochasic conrol and opimizaion of wireless neworks is o disribue he shared resources in he physical (e.g., power) and MAC layers (e.g., radio H. Halabian, I. Lambadaris, C-H Lung are wih he Deparmen of Sysems and Compuer Engineering, Carleon Universiy, Oawa, ON, K1S 5B6 Canada ( hassanh@sce.carleon.ca; ioannis@sce.carleon.ca; chlung@sce.carleon.ca). Y. Viniois is wih he Deparmen of Elecrical and Compuer Engineering, Norh Carolina Sae Universiy, Raleigh, Norh Carolina ( candice@ncsu.edu).

2 2 inerfaces, relay saions and orhogonal sub-channels) among muliple users such ha cerain sochasic performance aribues are opimized. While various performance crieria including he sable hroughpu region, power consumpion and uiliy funcions of he admied raffic raes have been sudied in several papers [1] [19], average queueing delay has received less aenion. The inheren randomness in wireless channels makes delay-opimal resource allocaion a challenging problem in wireless neworks. In his paper, we focus on delay-opimal server assignmen in a ime-sloed, muli-queue, muli-server sysem wih random conneciviies. Random conneciviies can model unreliable and randomly varying wireless channels. Our queueing model can be applied o sudy resource allocaion in wireless access neworks where he wireless users are modeled by he queues; he shared resources are modeled by he servers and he wireless channels are modeled by he random conneciviies beween he queues and he servers. Alhough his model is a simplified represenaion of a real wireless sysem, neverheless i does provide valuable inuiion for he performance opimizaion of real sysems. Similar modeling approaches have already appeared in [2], [3], [10], [15] [17], [20] [23]. A. Relaed Work and Our Conribuions The problem of hroughpu-opimal server allocaion in muli-queue, single-server sysems wih random conneciviies was addressed in [2], [10], [20], [21]. In [2], he auhors considered a ime-sloed, muli-queue single-server sysem wih Bernoulli packe arrivals and conneciviies from each of he queues o a single server. They inroduced LCQ (Longes Conneced Queue) policy as a hroughpu-opimal policy and also characerized he sabiliy region by a se of linear inequaliies. The auhors in [20] considered a coninuous-ime version of he model sudied in [2] wih finie buffer space and showed ha under saionary ergodic inpu job flow and modulaion processes, LCQ policy maximizes he sable hroughpu region of his sysem. In [10], C-FES (Conneced queue wih he Fewes Empy Spaces) policy, a policy ha allocaes he server o he conneced queue wih he fewes empy spaces, was inroduced for his sysem. I was shown ha C-FES sochasically minimizes he loss flow and maximizes he hroughpu of he sysem. In [21], a model similar o he model of [2], [10] was sudied and i was shown ha he Bes User (BU) policy maximizes he expeced discouned number of successful ransmissions. While in hroughpu-opimal server allocaion he objecive is o find a policy ha maximizes

3 3 he hroughpu region of he sysem and keeps he queues sable [1], [4], in delay-opimal server allocaion he goal is o deermine a policy ha minimizes he average queueing delay. Thus, he objecive in delay opimaliy is more sringen han he objecive in hroughpu opimaliy. A server allocaion policy may be hroughpu-opimal bu no delay-opimal; however, a delayopimal policy (for all he arrival raes) is always hroughpu-opimal. In [2], he auhors (oher han proving he hroughpu opimaliy of LCQ as menioned earlier) proved ha for a muliqueue, single-server sysem wih i.i.d. Bernoulli arrival and conneciviy processes, he LCQ policy is also delay-opimal. The exension of his resul for non-i.i.d. case is sill an open problem. In generalizing he resuls o muli-queue, muli-server (MQMS) sysems, various muli-server sysems have been sudied [3], [15], [16], [22] [25]. In [22], Maximum Weigh (MW) policy was proposed as a hroughpu-opimal server allocaion policy for an MQMS queueing sysem wih general, saionary channel processes. MW policy can be considered as a special case of backpressure algorihm which was proven in [1], [4] o be a hroughpu-opimal resource allocaion algorihm in a general queueing sysem. In [15], he auhors characerized he nework sabiliy region of muli-queue, muli-server sysems wih ime-varying, independen conneciviies. The resuls were furher exended in [16] for more general, saionary channel disribuions (and no jus independen Bernoulli channels). In all he models sudied in [15], [16], [22], here is no resricion on he number of servers ha can be allocaed o a queue. For ease of reference, we will call such an MQMS sysem as MQMS-Type1. In [3], i was shown ha for an MQMS sysem in which he queues are resriced o ge service from a mos one server during each ime slo, Maximum Weighed Maching (MWM) policy is hroughpu-opimal. For ease of reference, we will call such an MQMS sysem wih his exra assumpion as MQMS-ype2. The auhors also considered he effec of infrequen channel sae measuremens on he nework sabiliy region of MQMS sysems. Similar o MQMS-Type1, for MQMS-Type2 he hroughpu-opimal policy (MWM) can be considered as a special case of he back-pressure algorihm. In conras o he single-server sysem (where LCQ was boh hroughpu-opimal and delayopimal), in MQMS-Type1 sysem he MW policy is no necessarily delay-opimal. More specifically, in [15] i was also shown ha alhough MW policy is hroughpu-opimal, even for a sysem wih i.i.d. Bernoulli arrivals and conneciviy processes, MW policy in is general form, is no delay-opimal.

4 4 The delay-opimal server allocaion problem in muli-server sysems was addressed in [23] [25]. The auhors in [23] considered a queueing model wih a se of parallel queues and i.i.d. Bernoulli packe arrivals ha are compeing o arac service from K idenical servers forming a server-bank. The conneciviies of he queues o he enire server-bank are assumed o be i.i.d. Bernoulli processes. Each queue is resriced o receive service from a mos one server during each ime slo. The auhors proposed LCQ policy in which he servers of he server-bank are allocaed o he K longes conneced queues a each ime slo. Using dynamic coupling and sochasic ordering, hey proved he delay opimaliy of LCQ policy for such a sysem. In our work, he focus would be on delay opimaliy of MWM policy in MQMS-Type2 sysem in which he servers are no resriced o form a server-bank. Insead, we assume ha each queue has an independen conneciviy o each individual server (as seen in Figure 1). The work in [24], [25] focuses on delay opimal server allocaion problem in he MQMS-Type1 sysem. In [24], he auhors inroduced MTLB (Maximum-Throughpu Load-Balancing) policy and using dynamic programming showed ha his policy minimizes a class of cos funcions including oal average delay for he case of wo queues wih i.i.d., Bernoulli-disribued arrivals and conneciviies. In [24], no general argumen was provided for he opimaliy of MTLB for more han wo queues. The work in [25] considers his problem for a general number of queues and servers. In [25], a class of Mos Balancing (MB) policies was characerized among all work-conserving policies which minimize, in sochasic ordering sense, a class of cos funcions including oal queue occupancy (and hus are delay-opimal). However, his class of proposed MB policies is jus characerized by a propery of his class; he auhors did no inroduce an explici implemenaion for he opimal policy. In his paper, we focus on MWM policy and prove ha his hroughpu-opimal policy is also delay-opimal for an MQMS-Type2 sysem wih i.i.d. arrival and conneciviy processes. Our work exends he resuls derived in [2], [23]. In paricular, he researchers in [2], [23] have considered queueing models where a single server or a server-bank is randomly conneced o a se of parallel queues. In his paper, we consider a more general model where each individual server is randomly conneced o he queues (as seen in Figure 1). Alhough he wo models bear cerain similariies, exending he resuls from single-server (server-bank) sysem o muli-server sysem is no a sraighforward procedure. Our work is differen from he work in [2], [23] from boh he modeling power and he difficuly in proof poins of view.

5 ag replacemens 5 Single server or server-bank λ λ X 1 () X 2 () p p 1 X 1 () p λ p p X 2 () 1 2 λ 2 λ X N () p K λ X N () p p p K a) The model sudied in [2], [23] b) MQMS-Type2 sysem in his paper. Fig. 1: Previous models vs. our model. (λ is he arrival probabiliy and p is he conneciviy probabiliy) For more informaion on opimal scheduling and resource allocaion problems in wireless neworks he reader is encouraged o also consul wih [4], [26] [30]. Our conribuions in his paper are summarized as follows: Firs, for an MQMS-Type2 sysem we prove ha during each ime slo, Maximum Weighed Maching (MWM) policy will resul in he mos balanced queue vecor in he sysem, i.e., maximizaion of he maching weigh and balancing of he queues are equivalen. Graph heoreic argumens were applied o prove his resul ha is formally inroduced in Lemma 1 and Lemma 2 laer in he paper. Noe ha our approach o prove his resul is only applicable o he MQMS-Type2 model (due o he srucure of he model and he MWM policy) and canno be easily exended o MQMS-Type1 sysem. Second, using his resul in conjuncion wih he noions of sochasic ordering and dynamic coupling, we prove he delay opimaliy of MWM policy for an MQMS-Type2 sysem wih i.i.d. Bernoulli arrivals and conneciviies. More specifically, we prove ha MWM minimizes, in sochasic ordering sense, a range of cos funcions of queue lenghs including oal queue occupancy 1. Third, we hen exend our model by considering imperfec services where i is assumed ha he service of a scheduled packe fails randomly wih a cerain probabiliy. We prove ha MWM is sill opimal for he exended model. We finally show ha he resuls are 1 The opimaliy of MWM is proven among all causal server assignmen policies.

6 6 sill valid for some more general conneciviy and arrival processes which follow condiional permuaion invarian disribuions. The res of his paper is organized as follows. In Secion II, we inroduce he queueing model and he required noaion. In Secion III, we describe he Maximum Weighed Maching (MWM) server assignmen policy. In Secion IV, we prove he delay opimaliy of MWM server assignmen policy. In Secion V, we presen simulaion resuls where we compare he performance of MWM policy wih he performance of wo oher server assignmen policies in erms of average oal queue occupancy (or equivalenly average queueing delay). Finally, we summarize our conclusions in Secion VI. II. MODEL DESCRIPTION Throughou he paper, random variables are represened by CAPITAL leers and lower case leers are used o represen sample values of he random variables. Moreover, we use boldface fon o represen marices and vecors. We consider a ime-sloed, MQMS-Type2 sysem consising of a se of parallel queues N = {1,2,...,N} wih infinie buffer space for each queue (see Figure 2). Packes in his sysem are assumed o have consan lengh and require one ime slo o complee service. The service o his se of queues is provided by a se of idenical servers K = {1,2,...,K}. The conneciviy of each queue n N o each server k K a each ime slo is random and varying across ime slos. We denoe he conneciviy of queue n o server k a ime slo by C n,k () {0,1}. When C n,k () = 1 (C n,k () = 0), queue n is conneced o (disconneced from) server k a ime slo. The conneciviy variables C n,k () are assumed o be i.i.d. Bernoulli random variables wih a fixed parameer p 2. A any ime slo, each server can serve a mos one packe from a conneced, non-empy queue. We do no allow server sharing in he sysem, i.e., a server can serve a mos one queue per ime slo. We also assume ha a mos one server can be assigned o any conneced queue during a ime slo. Le A n () denoe he number of packe arrivals o queue n a ime slo. We assume ha new arrivals a each ime slo are added o he queues a he end of he ime slo. The arrival 2 The acual value of p does no involve in our analysis. We only rely on he fac ha he conneciviies are i.i.d. Bernoulli processes.

7 7 A 1 () A 2 () X 1 () X 2 () C 1,1 () C 1,2 () C 1,K () 1 2 C N,1 () A N () X N () C N,2 () C N,K () K Fig. 2: Discree-ime MQMS-Type2 sysem wih N parallel queues and K servers. variables A n () are assumed o be i.i.d. Bernoulli random variables wih he same parameer λ for all n and 3. We denoe he lengh of queue n a he end of ime slo (i.e., afer adding he new arrivals) by X n (). Hence, X n () represens he number of packes in he nh queue a he end of ime slo (or beginning of ime slo +1). A. Server Assignmen Policy A each ime slo he server assignmen policy has o decide abou a biparie (graph) maching 4 beween ses N and K. We assume ha his decision is made in a causal fashion, i.e., based on he available hisory of arrival processes, service processes, queue saes and he conneciviy saes unil ime. A policyπ is fully deermined by is indicaor variables M (π) n,k () n N, k K, = 1,2,... which are defined as M (π) n,k () = 1, if server k is assigned o queue n by policy π a ime slo, 0, oherwise. We define hen K marixm (π) () = (M (π) n,k ()), n N, k K as he employed maching by policy π a ime slo. Hence, a server assignmen policy π can be defined as he se of all (1) 3 The acual value of λ does no involve in our analysis. We only rely on he fac ha he arrivals are i.i.d. Bernoulli processes. 4 A maching in a biparie graph is a sub-graph of he original graph in which no wo edges share a common verex.

8 8 he employed machings by policy π a ime slos = 1,2,..., i.e., π = {M (π) ()} =1. We denoe he maching space conaining all he possible assignmens of he servers o he queues by M. The se M is equivalen o he se of all he possible machings in an N K complee biparie graph 5. We can observe ha X n (), he queue lengh random variable, evolves in ime as follows: ( ) + K X n () = X n ( 1) C n,k ()M (π) n,k () +A n () n N (2) k=1 The operaor ( ) + reurns he erm inside he parenheses if i is non-negaive and zero oherwise. The queueing model inroduced in his secion is useful in providing inuiion for modeling resource assignmen problems in various sysems wih shared resources [3], [17]. In wireless communicaion sysems, resources such as communicaion sub-channels, relay saions, ec. are shared among users. As an example, we can consider a relaying access nework wih N users and K shared relays. By modeling he cooperaive wireless channel beween each user, each relay and he base saion as an erasure channel, he performance of such a sysem can be sudied following our model in Figure 2. III. MAXIMUM WEIGHTED MATCHING (MWM) SERVER ASSIGNMENT POLICY A. MWM Opimizaion Problem In [1], [4], i was shown ha back-pressure algorihm maximizes he sable hroughpu region of a general daa nework, i.e., i is hroughpu-opimal. The reader may refer o [1], [4] for more informaion abou back-pressure algorihm. For he model inroduced in Secion II, he backpressure algorihm reduces o he following opimizaion problem a each ime slo [3]. In his ineger programming problem, M n,k () variables are he opimizaion variables and X n ( 1) 5 A complee biparie graph is a biparie graph in which each verex in each par is conneced o all he verices in he oher par. An N K biparie graph has NK edges.

9 9 and C n,k () are known parameers. Maximize: M n,k (), n,k Subjec o: N K X n ( 1) M n,k ()C n,k () n=1 k=1 K M n,k () 1 (n = 1,2,...,N), k=1 N M n,k () 1 (k = 1,2,...,K), n=1 M n,k () {0,1} (k = 1,2,...,K),(n = 1,2,...,N) (3) Finding he soluion of problem (3) is equivalen o finding a maximum weighed maching in he N K biparie graph G = (N,K,E) shown in Figure 3. Hence, he back-pressure algorihm for he queueing model of Figure 2 is also known as Maximum Weighed Maching (MWM) algorihm. In G, N and K are he wo ses of verices in each par of he graph and E = {e n,k, n N, k K} is he se of edges beween hese wo pars. In G, he associaed weigh o each edgee n,k isx n ( 1)C n,k (). A maching in graphg is a sub-graph ofg in which no wo edges share a common verex. Any maching M (π) () a any ime slo is corresponding o a sub-graph of G namely G (π) = (N,K,E (π) ) in which e n,k E (π) if and only if M (π) n,k () = 1. There are several algorihms o find he maximum weighed maching in biparie graphs. The mos well-known one is he Hungarian algorihm wih O((min{N,K})(max{N,K}) 2 ) complexiy [31]. B. MWM Policy Assume ha M (MWM) () = (M (MWM) n,k ()) n N, k K is he maching whose indicaor variables are he soluion of he opimizaion problem (3). M (MWM) () has he following properies: (a) M (MWM) () always exiss a all ime slos. (b) The maximum weighed maching in a biparie graph may no be unique, i.e., here may be more han one maching M (MWM) () for he graph of Figure 3 a each ime slo. Definiion 1: A Maximum Weighed Maching (MWM) server assignmen policy is defined as a policy ha employs maximum weighed maching M (MWM) () a all ime slos, i.e., π (MWM) = {M (MWM) ()} =1.

10 10 1 X 1 ( 1)C 1,1 () X 1 ( 1)C 1,2 () 1 2 X 2 ( 1)C 2,K () X 2 ( 1)C 2,2 () 2 N X N ( 1)C N,1 () X N ( 1)C N,2 () X N ( 1)C N,K () K Fig. 3: Biparie graph for he Maximum Weighed Maching (MWM) policy. An MWM policy a each ime slo observes he queue lenghs X n ( 1) and he conneciviy variables C n,k () and deermines a maximum weighed maching (he maching indicaor variables) in he biparie graph of Figure 3. Noe ha, by consrucion, he MWM policy is causal. Definiion 2: We denoe he se of all policies ha employ maximum weighed maching a all ime slos by Π MWM. According o propery (a) above, he se Π MWM is no empy. Moreover, according o propery (b), we conclude ha Π MWM may conain an infinie number of policies. IV. DELAY OPTIMALITY OF MWM POLICY In his secion, we prove he delay opimaliy of an MWM policy π Π MWM. This resul is formally presened in Theorem 2. More specifically, we show ha in an MQMS-Type2 sysem wih i.i.d. Bernoulli arrival and conneciviy processes, any MWM policy is opimal in minimizing, in sochasic ordering sense, a class of cos funcions of queue lengh processes including average queueing delay. For breviy we will use he erm delay opimaliy o refer o he opimaliy of MWM in his sense. A. Delay Opimaliy of MWM Policy-Ouline of he Proof The proof of Theorem 2 proceeds along he following seps: Firs, We will inroduce he noion of balanced queue vecors and he corresponding balancing server reallocaion a ime slo in Definiion 4. For any given policy π and a fixed ime slo

11 11 ny π / Π MWM Queue Lemma 1 MW index Balancing Lemma 2 Maximizaion Theorem 1 Delay-opimal policy Queue Policy Improvemen belongs o Π MWM Any MWM policy Balancing Lemma 3 (Delay Reducion) All MWM policies resul is delay opimal in he same average delay Lemmas 4 and 5 Fig. 4: Ouline of he proof, we will also define he Maching Weigh index MW π () in Definiion 5. Noe ha his index is no direcly relaed o (average) delay; i is, however, a crucial link in comparing arbirary policies o he MWM ones in he se Π MWM defined in he previous secion. We show hen in Lemmas 1 and 2 ha he noions of maximizing he Maching Weigh index and producing balanced queue vecors via balancing server reallocaions are equivalen. This propery allows us o characerize MWM policies as ones ha produce he mos balanced queue size vecors possible. Second, we use he balanced queue size propery o show in Lemma 3 ha for any arbirary policy π ouside he se Π MWM, we may consruc a policy in he se Π MWM ha improves π in erms of delay. In he words of Theorem 1, we prove ha he delay-opimal policy belongs o he se of MWM policies Π MWM. Third, in Lemmas 4 and 5 we will show ha all policies in he se Π MWM resul in he same cos (and hence average delay). Finally, by using Theorem 1 and Lemma 4 we conclude Theorem 2 where we show ha he policies in he se Π MWM are all delay-opimal. Graph heoreic analysis is applied in he proof of Lemmas 2 and 5 and sochasic ordering and dynamic coupling argumens are used o prove Lemmas 3 and 4. B. Equivalence of Queue Lengh Balancing and Maximum Weighed Maching We sar his secion by inroducing he inermediae queue sae in he following definiion. Definiion 3: Le X () = (X 1 (),X 2 (),...,X N ()) denoe he queue lengh vecor a ime slo exacly afer serving he queues according o a server assignmen policy π and before

12 12 adding he new arrivals of ime slo, i.e., X n () = (X n ( 1) K k=1 C n,k ()M (π) n,k () ) +. (4) We call his vecor as a he inermediae queue sae. Recall ha he final sae of queue n a ime slo is deermined afer adding he new arrivals. Given x () as a sample value of random vecor X (), we define a balancing server reallocaion a ime slo as follows. Definiion 4: Assume ha he employed maching a ime slo (assignmen of servers o he queues a ime slo ) will resul in he inermediae queue vecor x (). A balancing server reallocaion a his ime slo is a new maching resuling in inermediae vecor x () such ha one of he following condiions is saisfied. (C1) x n() x n() for all n = 1,2,...,N and here exiss an m {1,2,...,N} such ha x m () < x m (). (C2) x () and x () are differen in only wo elemens n and m such ha x n () < x n () x m() < x m() and he following consrains are saisfied: x n() = x n() + 1 and x m() = x m () 1. A balancing server reallocaion is a crucial ool in defining new policies ha improve he delay performance of an arbirary policy as we will see in he proof laer. Example: Consider a sysem wih hree queues and hree servers. Assume ha x( 1) = (3,2,5) is he queue lengh vecor righ a he end of ime slo 1 (or a he beginning of ime slo ). We consider wo disinc examples o show he definiion of balancing server reallocaions corresponding o each of he cases C1 and C2 in Definiion 4. Figures 5a and 5b show hese examples of balancing server reallocaions. In each case, we also show he weigh of each edge (n,k) which is equal o c n,k ()x n ( 1). In hese figures, since none of he queues is empy, he edges wih weigh 0 are he ones which are disconneced. We have specified he original allocaions by solid lines and he balancing ones by dashed lines. For he sysem in Figure 5a, he original allocaion will resul in he inermediae vecor x () = (3,1,4) while he balancing server reallocaion will resul in he inermediae vecor x () = (2,1,4). The vecors x () and x () saisfy Condiion C1. For he sysem in Figure 5b, he original allocaion will resul in he inermediae vecor x () = (2,1,5) while he balancing server reallocaion will resul in x () = (3,1,4). The vecors x () and x () saisfy Condiion C2.

13 x 3 ( 1) = x 1 ( 1) = x 2 ( 1) = x 3 ( 1) = (a) Saisfying condiion C1 x 1 ( 1) = x 2 ( 1) = 2 x 3 ( 1) = (b) Saisfying condiion C Fig. 5: Examples of balancing server reallocaions (he weigh c n,k ()x n ( 1) of each edge (n,k) is also shown) Definiion 5: For a server assignmen policy π wih he allocaion variables {M (π) n,k ()} =1, k K and n N, we define Maching Weigh (MW) index a ime slo by N K MW π () = X n ( 1) C n,k ()M (π) n,k (). (5) n=1 MW index is exacly he objecive of he opimizaion problem (3). MW π () is an index associaed wih policy π a ime slo whose value is dependen on he sae of he sysem (queue lenghs and conneciviies) as well as he maching employed by policy π a ime slo. In he following lemmas (Lemmas 1 and 2), we relae he noions of balancing server reallocaion and Maching Weigh index and we prove ha maximizaion of MW π () index and balancing of he queues are equivalen. More specifically, we show ha if he MW π () index for policy π is no maximized a ime slo (π is no using a maximum weighed maching), hen here exiss a balancing server reallocaion (i.e., a new maching ha saisfies eiher C1 or C2) ha resuls in a larger MW index. Furhermore, if π is using a maximum weighed maching, hen here exiss no balancing server reallocaion a ha ime slo, i.e., no maching can be found ha saisfies eiher C1 or C2. These facs are formally saed in he following wo lemmas. Lemma 1: For a given policy π employing maching M (π) () a ime slo, by applying a balancing server reallocaion a ime slo (if here exiss any), we can creae a new policy π (differing from π only a ime slo ) such ha MW π () < MW π (). The deailed proof of he lemma is given in Appendix I-A. Based on Lemma 1, we can sae he following corollary. k=1

14 14 Corollary 1: For a given policy π a ime slo, if MW π () is maximized, i.e., policy π employs a maximum weighed maching a ime slo, hen here exiss no balancing server reallocaion a ha ime slo. Lemma 1 saes ha any balancing server reallocaion sricly increases he maching weigh index. However, i does no imply he exisence of a balancing server reallocaion when MW π () is no maximized. In he following, we prove he exisence resul i.e., he inverse of Lemma 1. Lemma 2: For a given policy π a ime slo, if MW π () is no maximized, i.e., if MW π () < MW MWM (), hen here exiss a balancing server reallocaion a ha ime slo. For he deailed proof, please refer o Appendix I-B. Using Lemmas 1 and 2, we can conclude ha maximizing he maching weigh is equivalen o balancing he queues (in a sense ha here is no furher maching ha can saisfy C1 or C2 in Definiion 4). Hence, an MWM maching will resul in he mos balanced inermediae queue sae where no balancing server reallocaion is possible. This propery of an MWM maching will be crucial in he proof of Lemma 3. C. Background on Sochasic Ordering and Dynamic Coupling In his secion, we briefly review he conceps of sochasic ordering (sochasic dominance) and dynamic coupling echniques. These conceps are needed in he proof of delay opimaliy of MWM policy in he res of our discussion. The reader is encouraged o consul [32] [34] for more deails abou sochasic ordering and dynamic coupling. Definiion 6: Consider wo real-valued, discree-ime sochasic processes A = {A()} =1 and B = {B()} =1 in R. We say A is sochasically smaller han B and we wrie A s B if Pr(A() > r) Pr(B() > r) for all = 1,2,... and all r R [32], [33]. The following wo properies of sochasic ordering are useful: if A s B, hen (a) E[A()] E[B()] (b) f(a) s f(b) for all non-decreasing funcions f. Process A is sochasically smaller han B, if here exiss a process à = {Ã()} =1 defined on he same probabiliy space as B, has he same probabiliy disribuion as A and saisfies Ã() B() almos surely (a.s.) for every = 1,2,... [23]. The las saemen is known as coupling of A and Ã. When applying coupling echnique, given he process A, we consruc a coupled process à wih he same disribuion as A and Ã() B() a.s. for all. This gives

15 15 us a ool for comparing he processes A and B sochasically when i is infeasible o derive he disribuions of A and B (e.g., in our queueing model when comparing he oal occupancy process for differen server assignmen policies). D. Delay Opimaliy of MWM In his subsecion, we will elaborae on proving he delay opimaliy of any MWM policy. We firs inroduce some definiions. We denoe by Z + he se of non-negaive inegers and by Z N + Z N + he N dimensional Caresian space of non-negaive inegers. We define he relaion on as follows. Definiion 7: For wo vecors x, x Z N +, we wrie x x if one of he following relaions holds: D1: x n x n for all n = 1,2,...,N. D2: x is obained by permuaion of wo disinc elemens of x, i.e., x and x are differen in only wo elemens n and m such ha x n = x m and x m = x n. In his case, we say x and x are equal in permuaion and we wrie x = p x. D3: x and x are differen in only wo elemens n and m such ha x n < x n x m < x m and he following consrains are saisfied: x n = x n +1 and x m = x m 1. The hree relaions D1, D2 and D3 are muually exclusive. In D3, we say ha x is more balanced han x and can be obained by decreasing a larger elemen of x (i.e., m) by one and increasing a smaller elemen (i.e., n) by one. We call such an inerchange as a balancing inerchange on vecor x. Thus, he resul of a balancing inerchange on a vecor x would be a vecor x such ha x x. According o Definiion 4, a balancing server reallocaion saisfying Condiion C2, will resul in a balancing inerchange beween x () and x (). We define he parial order p on Z N + as he ransiive closure of relaion [35]. In oher words, x p x if and only if x is obained from x by performing a sequence of reducions (i.e., reducing an elemen of he vecor x such ha x and x saisfy D1), permuaions of wo elemens (permuaion of wo elemens of he vecor x such ha x and x saisfy D2) and/or balancing inerchanges (such ha x and x saisfy D3). When x and x are wo queue lengh vecors, we wrie x p x if and only if queue lengh vecor x is obained from x by applying a sequence of packe removals, wo-queue permuaions and balancing inerchanges.

16 16 Definiion 8: We define F as he class of real-valued funcions on Z N + ha are monoone and non-decreasing wih respec o he parial order p, i.e., f F x p x f( x) f(x). (6) We can easily check ha funcion f(x) = N n=1 x n belongs o F. This funcion represens he oal queue occupancy of he sysem. Definiion 9: We define Π, = 1,2,..., as he se of all policies ha employ maximum weighed maching in every ime slo τ = 1,...,. We observe ha Π 1 Π and Π MWM = =1 Π. Consider a policy π Π 1 which is using an arbirary maching M (π) () a ime slo. If M (π) () is no a maximum weighed maching, hen from Lemmas 1 and 2 we conclude ha by applying a sequence of balancing server reallocaions 6 we can creae a policy π Π. Le h π denoe he number of balancing server reallocaions required o conver he employed maching in policy π a ime slo o a maximum weighed maching. Definiion 10: We define he disance of policy π Π 1 from he se Π o be h π balancing server reallocaions. According o Lemmas 1 and 2, since by applying each server reallocaion, he maching weigh index sricly increases, he number of balancing server reallocaions needed o conver π o a maximum weighed maching is bounded, i.e., h π H < for all,π. Hence, afer applying he firs balancing server reallocaion a ime slo we reach a policy π 1 whose disance from Π is h π 1 balancing server reallocaions. By repeaing his procedure we finally idenify a policy whose disance o Π is zero, i.e., i belongs o Π. Figure 6 illusraes he definiion of he disance h π and how balancing server reallocaions resul in idenifying a policy ha employs a maximum weighed maching a ime slo. In his figure, x π() is he inermediae queue sae due o he employed maching a ime slo and x π 1 (),x π 2 (),...,x π h π () are he inermediae queue saes afer applying he balancing server reallocaions. Definiion 11: By Π h (0 h H) we denoe he se of all server assignmen policies in Π 1 whose disance from Π is h balancing server reallocaions. Recall ha Π 0 = Π. 6 According o Lemma 1, each balancing server reallocaion sricly increases he maching weigh index.

17 17 x( 1) ime slo x() x( 1) employed maching π x π () = MW π () < MW MWM () x( 1) firs balancing reallocaion π 1 x π 1 () = MW π () < MW π1 () < MW MWM () h π x( 1) second balancing reallocaion π 2 x π 2 () = MW π1 () < MW π2 () < MW MWM () x( 1) h π h balancing reallocaion π h π x π h π () = This balancing reallocaion creaes maximum weigh maching MW πh π () = MW MWM () Fig. 6: h π balancing server reallocaions are required o creae a policy in Π from policy π Π 1. MW indices given he sae of he sysem a ime slo are also compared. Definiion 12: For any wo policies π and π wih queue lengh processes X = {X()} =1 and X = { X()} =1, respecively, we say π dominaes π, if f( X) s f(x), f F, i.e., he queue lengh cos (delay) of policy π is sochasically less han ha of policy π. If π dominaesπ we havee[f( X)] E[f(X)] 7. In he following lemma, we will inerconnec he noions of maximizing he maching weigh index and delay opimaliy and show ha maximizaion of he maching weigh index (a any given ime ) will improve he delay performance (will decrease he queue lengh cos funcion f(x) sochasically). The key elemen in he inerconnecion is he noion of balancing server reallocaion. In paricular, we show ha, for any given policy π Π h, h = hπ ha does no employ a maximum weighed maching a ime slo (i.e., h > 0), here exiss a balancing server reallocaion a ime slo. In he following lemma, we show ha by using such a balancing server reallocaion a ime slo we can consruc a new policy π ha dominaes he original policy π. For he deailed proof, please refer o Appendix II-A. We used sochasic ordering and dynamic coupling o prove his lemma. Lemma 3: For any policy π Π h where h = h π > 0, we can consruc a policy π Π h 1 such ha π dominaes π. Thus, π ouperforms π in erms of average queueing delay. 7 Choosing f(x) = N n=1xn, we conclude ha he expeced oal queue occupancy (or equivalenly average queueing delay) of policy π is smaller han ha of policy π in every ime slo.

18 18 Using Lemma 3, we can prove he following heorem which saes ha any MWM policy ouperforms any non-mwm policy in erms of average queueing delay. Theorem 1: For any server assignmen policy π / Π MWM, here exiss an MWM policy π Π MWM such ha π dominaes π. Proof: Leπbe any arbirary non-mwm policy. Thenπ Π H 1 1 whereh 1 = h π 1. By applying Lemma 3 repeaedly, we can consruc a sequence of policies such ha each policy dominaes he previous one. Thus, we obain policies ha belong o Π H 1 1,Π H 1 1 1,Π H 1 2 1,...,Π 0 1 = Π 1. The las policy is called π 1 for which we have π 1 Π H 2 2 where H 2 = h π 1 2. By coninuing such an argumen, we obain a sequence of policies π Π, = 1,2,... such ha π j dominaes π i for j > i. This sequence of policies defines a limiing policy π ha agrees wih MWM a all ime slos. Thus, π is an MWM policy ha dominaes all he previous policies, including he saring policy π. This proves ha he delay-opimal policy is an MWM policy in Π MWM. As we menioned before, he se Π MWM may conain an infinie number of policies. In he following, we show ha any MWM policy is delay-opimal. To achieve his, we need o prove he following lemma 8. Lemma 4: The queue lengh coss of all he maximum weighed maching policies in Π MWM are equal in disribuion, i.e., for any wo MWM policies π 1,π 2 Π MWM, we have f(x (π 1) ) D = f(x (π 2) ) where X (π 1) and X (π 2) are he queue lengh processes under π 1 and π 2, respecively. The proof of his lemma is provided in Appendix II-D. Using Theorem 1 and Lemma 4, we can conclude he main resul of his secion in he following heorem. Theorem 2: Any Maximum Weighed Maching policy dominaes any server assignmen policy, i.e., any MWM policy is delay-opimal. E. Exensions 1) Imperfec Services: We can exend Theorems 1 and 2 for he case where he service of a scheduled packe by a conneced server fails randomly wih a cerain probabiliy. This can model he operaion of realisic wireless neworks where service failures usually occur due o unexpeced and unpredicable effecs of noise, inerference, ec. In he case of a packe service 8 As par of he proof for his lemma, we need preliminary Lemma 5 presened and proven in Appendix II-C

19 19 failure, he packe will be kep in he queue and will be rescheduled and reransmied in fuure ime slos. By he random variable Q n,k () {0,1}, we denoe he successful/unsuccessful service of queue n provided by server k a ime slo ; a value of 1 (resp. 0) denoes ha he service is successful (resp. unsuccessful). We assume ha Q n,k (), n N, k K are i.i.d. Bernoulli random variables wih he same success probabiliy q. The parameer q (similar o parameers λ and p) is no explicily involved in our analysis oher han he fac ha E[Q n,k ()] = q, n,k,. The queue lenghs are hen updaed a he end of each ime slo by he following rule. ( + K X n () = X n ( 1) C n,k ()M (π) n,k n,k()) ()Q +A n () n N (7) k=1 The nework scheduler (ha performs server assignmen process) canno observe he variables Q n,k () and from is perspecive hey are assumed o be random. The random vecor X () is defined similar o equaion (4). Hence, X () represens he queue lenghs before adding he new arrivals of ime slo as if all he services a ha ime slo are successful. For such a sysem, we can verify ha Lemmas 1 and 2 are valid. We can exend Lemma 3 for he sysem wih service failures by considering he random variables Q n,k () in our dynamic coupling argumen. The proof is followed by using he same approach as in Lemma 3. The deailed analysis is brough in Appendix II-B. By applying he same approach as in he proof of Theorem 1 and Lemma 4, we can similarly prove he delay opimaliy of MWM policy for he sysem wih imperfec services. 2) Exensions for Conneciviy and Arrival Processes: The argumens in Lemmas 3 and 4 and Theorem 1 remain valid if he i.i.d. assumpion for conneciviy and arrival processes is relaxed as follows; we will consider conneciviy and arrival processes which follow condiional permuaion invarian disribuions. Given even H (which is used o denoe he hisory of he sysem), we define a condiional mulivariae probabiliy disribuion f(y 1,y 2,...,y n H) o be permuaion invarian if for any permuaion of he variables y 1,y 2,...,y n namely y 1,y 2,...,y n we have f(y 1,y 2,...,y n H) = f(y 1,y 2,...,y n H). We can readily see ha for all he conneciviy and arrival processes whose join disribuions a each ime slo given he hisory of he sysem 9 9 By hisory of he sysem we mean all he channel saes, arrivals and machings of he previous ime slos up o ime slo.

20 20 (i.e., f A() (a 1,a 2,..,a N H) and f C() (c 1,1,c 1,2,...,c N,K 1,c N,K H)) are permuaion invarian, Lemmas 3, 4 and Theorem 1 are sill valid and herefore MWM is delay-opimal. We also consider he generalizaion of Theorems 1 and 2 for non-bernoulli arrival processes. Suppose ha he number of arrivals o each queue can be represened by he summaion of some i.i.d. Bernoulli random variables, i.e., has Binomial disribuion. Also suppose ha A n () A max for all n N and all. In his case, we can creae a new (virual) sysem in which afer each ime slo we append A max 1 virual ime slos and pu he conneciviies all equal o zero, i.e., for each virual ime slo, C n,k () = 0, n N, k K. We hen disribue he arrivals of he acual ime slo among hese A max ime slos (one acual ime slo and A max 1 virual ime slos) randomly such ha a each ime slo a mos one packe arrival occurs. Since he conneciviies and he arrivals in boh sysems are permuaion invarian, we can sill prove Theorems 1 and 2 for he virual sysem. We observe ha he operaion of he wo sysems (he original sysem and he virual sysem) are he same. Therefore, we can conclude ha Theorem 1 is also valid for a muli-server sysem wih Binomial arrival processes. V. SIMULATION RESULTS We have compared he delay performance of MWM policy wih wo alernaive server assignmen policies described in he following. Maximum Maching (MM) policy applies he maximum maching on marix C(). The maximum maching policy a each ime slo employs a server assignmen (or maching) M (MM) () which is obained by solving he following problem (equivalen o finding he maximum maching in he conneciviy marix). N K Maximize: M n,k ()C n,k () n=1 k=1 K Subjec o: M n,k () 1, (n = 1,2,...,N), k=1 N M n,k () 1, (k = 1,2,...,K). (8) n=1 The MM maximizes he insananeous hroughpu a each ime slo wihou considering he queue lengh informaion in is server assignmen decisions.

21 21 A heurisic policy ha assigns he servers o he queues a each ime slo according o he following rule: I selecs a server randomly and assigns i o is longes conneced queue. Then, updaes he se of servers by removing he seleced server from K and he se of queues (i.e., N ) by removing he queue o which he seleced server was assigned. This procedure is repeaed K imes. For some servers he updaed se N may be empy (e.g., when K > N) and herefore hose servers are no assigned o any queue. Algorihm 1 Heurisic Policy Pseudocode inpu: N, K, c() and x( 1) iniialize: M (H) () = (0) N K for i = 1 o K do end Choose a server k K randomly if N endif n argmax n N c n,k ()x n ( 1) M (H) n,k () 1 N N {n } K K {k } Reurn M (H) () The moivaion for he heurisic policy is coming from Longes Conneced Queue (LCQ) policy which was proven in [2] o be opimal for a single-server sysem. For muli-server sysem, we will use he same principle for each server. However, he order in which servers are seleced for assignmen is random. We have preformed a comprehensive se of simulaions in which we invesigae he effecs of he number of servers K, he probabiliy of conneciviy p and he probabiliy of service success q on he performance of he aforemenioned policies. In all he simulaions, we se N = 8 and he arrivals are i.i.d. Binomial disribued which is he summaion of 10 Bernoulli random variables. We use log-scale for he y-axis in he figures so ha we can easily compare he performance of differen policies in low arrival raes (where he average queue lenghs are very close for

22 22 Average Queue Occupancy (Packes) Maximum Weighed Maching Policy Maximum Maching Policy Heurisic Policy Arrival Rae per Queue (Packes/ime slo) (a) p = 0.2 Average Queue Occupancy (Packes) Maximum Weighed Maching Policy Maximum Maching Policy Heurisic Policy Arrival Rae per Queue (Packes/ime slo) (b) p = 0.5 Fig. 7: Average oal queue occupancy, N = 8, K = 4, q = 0.8 differen policies). Figures 7-9 illusrae he simulaion resuls. In all he cases, he confidence inerval is very small and is no visible in he graphs. As we can see in all cases, MWM exhibis improved performance wih respec o he oher policies in erms of average queue occupancy or average queueing delay. Figure 7 shows he simulaion resuls for K = 4, q = 0.8 and p = 0.2, 0.5. In hese cases, since he number of servers is relaively low, server assignmen will be more compeiive. As shown earlier, he MWM minimizes he queue imbalance. The heurisic policy

23 Average Queue Occupancy (Packes) Maximum Weighed Maching Policy Maximum Maching Policy Heurisic Policy Arrival Rae per Queue (Packes/ime slo) (a) p = 0.2 Average Queue Occupancy (Packes) Maximum Weighed Maching Policy Maximum Maching Policy Heurisic Policy Arrival Rae per Queue (Packes/ime slo) (b) p = 0.5 Fig. 8: Average oal queue occupancy, N = 8, K = 8, q = 0.8 follows he same principle. However, since he selecion of servers for assignmen is random, in cerain cases i may happen ha wo or more servers have he same longes conneced queue. In such cases, he order of selecing he servers for assignmen does have an effec on he sysem performance. Maximum Maching policy however, does no ry o balance he queues since by consrucion i does no consider he queue lenghs in is assignmens and ha is why i performs worse han he oher wo policies. We observe ha as he conneciviy probabiliy ges larger, he performances of MWM and he heurisic policy ge closer. I is worh menioning ha he

24 Average Queue Occupancy (Packes) Maximum Weighed Maching Policy Maximum Maching Policy Heurisic Policy Arrival Rae per Queue (Packes/ime slo) (a) q = Average Queue Occupancy (Packes) Maximum Weighed Maching Policy Maximum Maching Policy Heurisic Policy Arrival Rae per Queue (Packes/ime slo) (b) q = 0.2 Fig. 9: Average oal queue occupancy, N = 8, K = 6, p = 0.5 heurisic policy inroduced here performs he same as MWM for K = 1 (which is equivalen o LCQ whose opimaliy has been previously shown in [2]). Figure 8 shows he resuls for 8 servers. In his case, since he number of servers is relaively large and comparable o he number of queues, in MWM and MM policies each queue ges service wih high probabiliy when he probabiliy of server conneciviies increases. As he conneciviy probabiliy ges smaller, he difference in performance of MWM and MM becomes more apparen. In his case, he heurisic policy performs worse han he oher wo policies since

25 25 i is more probable o lead o cases where wo or more servers have he same longes conneced queue. As he number of servers increases, we expec MM o perform he same as MWM as in his case he probabiliy of serving all he queues increases. Therefore, in he limiing case where K becomes very large, MM and MWM resul in very close performance. In Figure 9 we have invesigaed he effec of service success probabiliy. As we can see in he figures, he only effec of his parameer is o change he sabiliy poin (he arrival rae a which queue occupancy ends o infiniy). In his case, again we can see ha for boh q = 0.2, 0.8, MWM policy ouperforms he oher policies. VI. CONCLUSIONS In his paper, we considered he problem of assigning K idenical servers o a se of N parallel queues in a ime-sloed, muli-server queueing sysem wih random conneciviies. For such sysems, i has been previously shown ha MWM is hroughpu-opimal, i.e., has he maximum sabiliy region. In his paper, we showed ha for a sysem wih i.i.d. Bernoulli arrival and conneciviy processes, MWM is also opimal for minimizing a class of cos funcions of queue lenghs including he average queueing delay. We firs proved ha MWM and queue lengh balancing are equivalen. Then, using his resul and by applying he noions of sochasic ordering and dynamic coupling echniques, we proved he delay opimaliy of MWM. Finally, we considered exensions of he model in which we have imperfec packe services or more general packe arrival and server conneciviy processes. We have shown he opimaliy of MWM in hese cases as well. A. Proof of Lemma 1 APPENDIX I PROOF OF LEMMA 1 AND LEMMA 2 Proof: Le M ( π) () denoe he employed maching afer applying he balancing server reallocaion. According o he definiion of balancing server reallocaion, a server reallocaion a ime slo resuls in an inermediae queue lengh vecor x () ha saisfies eiher condiion C1 or C2. Therefore, we consider he following wo cases: Case 1: Condiion C1 is saisfied a ime slo. Thus, x i() x i() for all i = 1,2,...,N and here exiss a leas one m {1,2,...,N}, such ha 0 x m () < x m (). We denoe

26 26 he (sub)se of queues for which we have 0 x i() < x i() by Q. Therefore, here exiss no queue ha was served by policy π bu no by policy π. Also he queues in subse Q which were no receiving service by policy π a ime slo, are now receiving service afer applying he balancing server reallocaion. Therefore, for all i Q, x i ( 1) K k=1 c i,k()m (π) i,k () = 0, x i ( 1) K k=1 c i,k()m ( π) i,k () = x i( 1) > 0. Thus, MW π () = i/ Qx i ( 1) < i/ Qx i ( 1) and he resul follows. K k=1 K k=1 c i,k ()M (π) i,k ()+ i Qx i ( 1) c i,k ()M ( π) i,k ()+ i Qx i ( 1) K k=1 K k=1 c i,k ()M (π) i,k () c i,k ()M ( π) i,k () = MW π(), (9) Case 2: Condiion C2 is saisfied a ime. In his case, by using policy π a ime slo queue n is receiving service bu queue m is no. In conras, by using policy π, a ime slo queue m is receiving service bu queue n is no. The service of oher queues is no disurbed, i.e., he oher queues which were receiving service by policy π sill receive a service by policy π a ime slo and he ones ha were no receiving service under policy π sill do no ge service under policy π. Therefore, MW π () MW π () = N i=1 i m,n x i ( 1) N i=1 i m,n K k=1 Therefore, MW π () < MW π (). B. Proof of Lemma 2 x i ( 1) K k=1 c i,k ()M ( π) i,k ()+x m( 1) c i,k ()M (π) i,k () x n( 1) = x m ( 1) x n ( 1) > 0 (10) Proof: Wihou loss of generaliy, we may conver he biparie graph G o a complee weighed biparie graph G wih max{n,k} verices in each par. This is done by adding some verices and edges of zero weigh as necessary. In paricular, if N > K, we will add N K servers on he righ hand side wih edges of weigh zero o each queue (each verex on he lef hand side). If N < K, we will add K N queues on he lef hand side wih edges of weigh zero o each server (each verex on he righ hand side). This will no change he operaion of he sysem since he added queues and servers are disconneced from he whole

27 27 sysem. We denoe he ses of verices on each par of G by N and K, respecively and he se of edges by E. Consequenly, a policy π is defined as π = {M (π) ()} =1 where M(π) () is a perfec maching 10 in he complee biparie graph G. We can easily verify ha a maximum weighed perfec maching M (MWM) () in he complee biparie graph G is he same as he maximum weighed maching in graph G if we remove he added edges of weigh zero from maching M (MWM) (). Consider a policy π which is employing perfec maching M (π) () a ime slo. Suppose ha M (π) () is no a maximum weighed perfec maching on graph G, i.e., MW π () < MW MWM (). Also, consider a maximum weighed perfec maching M (MWM) () a ime slo. Now, consider hese wo machings on G = (N,K,E ). Each of M (π) () and M (MWM) () corresponds o a disinc sub-graph of G namely G (π) respecively. We now build wo direced, weighed sub-graphs D (π) = (N,K,E (π) ) and G (MWM) = (N,K,E (MWM) ), and D (MWM) as follows: D (π) is he same asg (π) wih all he edges direced fromn ok wih he same edge weighs asg (π). D (MWM) is he same as G (MWM) equal o he negaive of edge weighs of G (MWM) i.e., he union of he sub-graphs D (π) wih all he edges direced from K o N wih edge weighs. Now, consider graph U = D (π) (MWM) D, and D (MWM). The graph U can be seen as he union of a number of even cycles 11 denoed by L. This is direcly concluded from he fac ha D (π) D (MWM) are each perfec machings of G and and hus each verex is inciden o an incoming edge and an ougoing edge. Furhermore, for he weigh of U shown by w(u), we have w(u) = l L w(l) = MW π () MW MWM () < 0. (11) In (11), w(l) is he weigh of edge l in L. Therefore, here mus exis a negaive cycle 12 in U. We denoe his negaive cycle by l. The cycle l is an even cycle and conains an even number of nodes and edges. We assume ha l conains 2W nodes (W nodes from N and W nodes from K ) and also 2W edges. Le us denoe he nodes of ses N and K ha form l by n 1,n 2,...,n W and k 1,k 2,...,k W, respecively. Thus, he cycle l can be represened by he sequence of is edges as l = e n1,k 1,e k1,n 2,e n2,k 2,e k2,n 3,...,e kw 1,n W, e nw,k W,e kw,n 1 (see Figure 10). 10 A perfec maching is a maching ha maches all verices of he graph. 11 A cycle wih even number of verices. 12 A cycle whose oal edge weigh is negaive.