On he Sabiliy Region of Muli-Queue Muli-Server Queueing Sysems wih Saionary Channel Disribuion Hassan Halabian, Ioannis Lambadaris, Chung-Horng Lung Deparmen of Sysems and Compuer Engineering, Carleon Universiy, Oawa, ON, Canada Email: {hassanh, ioannis, chlung}@sce.carleon.ca arxiv:2.8v [cs.it] 6 Dec 20 Absrac In his paper, we characerize he sabiliy region of muli-queue muli-server MQMS queueing sysems wih saionary channel and packe arrival processes. Toward his, he necessary and sufficien condiions for he sabiliy of he sysem are derived under general arrival processes wih finie firs and second momens. We show ha when he arrival processes are saionary, he sabiliy region form is a polyope for which we explicily find he coefficiens of he linear inequaliies which characerize he sabiliy region polyope. I. INTRODUCTION Sabiliy region or nework capaciy region of a communicaion nework is he closure of he se of all arrival rae vecors for which here exiss a resource allocaion policy ha can sabilize he sysem [] [3]. Having he sabiliy region of a nework characerized by a se of convex inequaliies, we can obain a precise soluion for nework fairness opimizaion problems wih he general form of. Maximize: Subjec o: N f n r n n= r = r,r 2,...,r N Λ 0 r n λ n n =,2,...,N In, N is he number of raffic sources. By r = r,r 2,...,r N, we denoe he admied rae vecor, λ n is he inpu rae of each source n and Λ denoes he sabiliy region. Funcions f n r n are non-decreasing and concave uiliy funcions. The choice of f n r n deermine he desired fairness properies of he nework. The goal of his paper is o characerize he nework capaciy region of Muli-Queue Muli- Server MQMS queueing sysems wih saionary channel disribuion i.e., Λ for MQMS sysem. Such queueing sysems are used o model muliuser wireless neworks wih muliple orhogonal subchannels such as OFDMA access neworks. The resource allocaion in such neworks can be modelled as he server allocaion problem in muli-queue muli-server queueing sysems wih ime varying channel condiions [4] [9]. The sabiliy problem in wireless queueing neworks was mainly addressed in [] [3], [5], [9] []. In [3], he auhors inroduced he noion of capaciy region of a queueing nework. They considered a ime sloed sysem and assumed ha arrival processes are i.i.d. sequences and he queue lengh process is a Markov process. In [0], hey characerized he nework capaciy region of muli-queue single-server sysem wih ime varying ON-OFF conneciviies. They also proved ha for a symmeric sysem i.e., wih he same arrival and conneciviy saisics for all he queues, LCQ Longes Conneced Queue policy maximizes he capaciy region and also provides he opimal performance in erms of average queue occupancy or equivalenly average queueing delay. In [], [2] and [], he noion of nework capaciy region of a wireless nework was inroduced for more general arrival and queue lengh processes. The problem of server allocaion in muli-queue muli-server sysems wih ime varying conneciviies was mainly addressed in [4] [7], [9]. In [5], Maximum Weigh MW policy was proposed as a hroughpu opimal server allocaion policy for MQMS sysems wih saionary channel processes. However, in [5] he condiions on he arrival raffic o guaranee he sabiliy of MW were no explicily menioned. In our previous work [9], we characerized he nework capaciy region of muli-queue muli-server sysems wih ime varying ON-OFF channels. We also obained an upper bound for he average queueing delay of AS/LCQ Any Server/Longes Conneced Queue policy which is he hroughpu opimal server allocaion policy for such sysems. In his paper, we characerize he sabiliy region of muliqueue muli-server queueing sysem wih saionary channel and arrival processes. Toward his, he necessary and sufficien condiions for he sabiliy of he sysem are derived for general arrival processes wih finie firs and second momens. For saionary arrival processes, hese condiions esablish he nework sabiliy region of such sysems. Our conribuion in his work is o characerize he sabiliy region as a convex polyope specified by a finie se of linear inequaliies ha can be numerically abulaed and used o solve nework opimizaion problem. We furher inroduce an upper bound for he average queueing delay of MW policy which is a hroughpu opimal policy for MQMS sysems. The res of his paper is organized as follows. Secion II describes he model and noaion required hroughou he paper. In secion III, we discuss abou some preliminary definiions sabiliy definiion, polyope, ec. used in he paper. In secion IV, we will derive necessary and sufficien condiions for he sabiliy of our model and also find an upper bound for he average queue occupancy or average queueing delay. Secion V summarizes he conclusions of he paper. II. MODEL DESCRIPTION Before we proceed o describe he model, we inroduce basic noaions ofen used hroughou he paper. Oher no-
X A A 2 A N X 2 X N C K C C 2 C N C N 2 C NK Fig. : Muli-queue muli-server queueing sysem wih saionary channel disribuion aions are inroduced whenever i is necessary. Through his paper, all he vecors are considered o be row vecors. The ime average of a funcion f is denoed by f, i.e., f = lim τ=fτ. The operaor is used for enry-wise muliplicaion of wo marices. By K 0 K, we denoe a row vecor of size K whose elemens are all idenically equal o 0. The cardinaliy of a se is shown by. For any vecor α = α,α 2,...,α N and a nonempy index se U = {u,u 2,...,u U } {,2,...,N} and u < u 2 <... < u U, we define α U = α u,α u2,...,α u U. We consider a ime sloed queueing sysem wih equal lengh ime slos and equal lengh packes. The model consiss of a se of parallel queues N = {,2,...,N} and a se of idenical servers K = {,2,...,K} Figure. Each server can serve a mos one queue a each ime slo, i.e., we do no have server sharing in he sysem. A each ime slo, he capaciy of each link beween each queue n N and server k K is assumed o be C n,k packes/slo. We assume ha C n,k M where M = {m Z + m M} for given M. Therefore, a each ime slo, he channel sae may be expressed by an N K marix of C = C n,k,n N,k K,C n,k M. The channel process is defined as {C} = wih he sae space S. Noe ha S is a finie se wih S = M + NK. We label each elemen of S by a posiive ineger index s {,2,...,M + NK }. Suppose ha he channel sae marix associaed o he channel sae s is denoed by C s. The channel sae of he sysem is assumed o have saionary disribuion, wih saionary probabiliies π s = PrC = C s. Assume ha he arrival process o each queue n N a ime slo i.e., he number of packe arrivals during ime slo is represened by A n. Thus, he arrival vecor a ime slo is denoed by A = A,A 2,...,A N. Assume ha E[A 2 n ] < A2 max < for all. Each queue has an infinie buffer space. New arrivals are added o each queue a he end of each ime slo. Le X = X,...,X N be he queue lengh process vecor a he end of ime slo afer adding new arrivals o he queues. A server scheduling policy a each ime slo should decide on how o allocae servers from se K o he queues in se N. This mus be accomplished based on he available informaion abou he channel sae of he sysem a ime slo which is C and also he queue lengh process a he beginning of ime slo which is X. In oher words, a each ime slo, he scheduler has o deermine an allocaion 2 K marix I I where { I = I N K = I n,k,i n,k {0,} } N I n,k = k K. n= We can easily observe ha he queue lengh process evolves wih ime according o he following rule. X T = X T C I T K + +A T 2 where + is defined as follows: For an arbirary vecor v of size v, v + is a vecor of he same size whose i h elemen v + i = v i if v i 0 and zero oherwise. A. Srong Sabiliy III. BACKGROUND We begin wih inroducing he definiion of srong sabiliy in queueing sysems [], [2], []. Oher definiions can be found in [3], [0], [2]. Consider a discree ime single queue sysem wih an arrival process A and service process µ. Assume ha he arrivals are added o he sysem a he end of each ime slo. Hence, he queue lengh process X evolves wih ime according o he following rule. X = X µ + +A 3 Srong sabiliy is given by he following definiion []. Definiion : A queue saisfying he condiions above is called srongly sable if lim sup τ=0 E[Xτ] < 4 Naurally for a queueing sysem we have he following definiion []. Definiion 2: A queueing sysem is called o be srongly sable if all he queues in he sysem are srongly sable. In his paper, we use he srong sabiliy definiion and from now we use sabiliy and srong sabiliy inerchangeably. B. Some Fundamenal Conceps of Polyopes In his par, we will have a brief review on he noion of convex polyopes and some fundamenal properies of hem. These conceps are needed when describing he sabiliy region of MQMS sysem in secion IV. Definiion 3: A convex polyope is defined as he convex hull of a finie se of poins [3], [4]. According o he Weyl s Theorem [3], a polyope in R N always can be expressed by a se { x R N α l x T β l, l =,2,...,L } for some posiive ineger L and α l R N and β l R. Dimension Theorem [3]: For a polyope P R N, dimension of P is equal o N minus he maximum number of linearly independen equaions saisfied by all he poins in P. Definiion 4: An equaliy ax T b is called valid for polyope P if for every poin x 0 P, ax T 0 b. Definiion 5: A face of polyope P is defined as F = {x P ax T = b} where inequaliy ax T b is a valid inequaliy for P.
We call he hyperplane ax T 0 = b associaed o he valid inequaliy ax T 0 b a face defining hyperplane for P if is associaed face is no empy. In oher words, ax T 0 = b is a face defining hyperplane for P if i inersecs wih P on a leas one poin. Noe ha P has finiely many faces. However, he face defining hyperplanes of a polyope can be infinie. Definiion 6: A face of polyope P is a maximal face disinc from P [4]. In oher words, all faces of P wih dimension dimp are called faces of P. For polyope P = { x R N α l x T β l, l =,2,...,L } an inequaliy is redundan if polyope P is unchanged by removing he inequaliy. Redundancy Theorem in Polyopes: Face defining hyperplanes inequaliies describing faces of dimension less han dimp are redundan [3]. Redundancy heorem saes ha o describe a polyope compleely, only face defining hyperplanes inequaliies are sufficien. IV. NECESSARY AND SUFFICIENT CONDITIONS FOR THE STABILITY OF MQMS SYSTEM A. Necessary Condiions for he Sabiliy of MQMS Sysem We inroduce he deparure marix H N K = H n,k,n N,k K in which H n,k represens he oal number of packes served by server k from queue n a ime slo. Obviously H n,k C n,k. Thus, he deparure process from queue n a ime slo would be K k= H n,k. The following equaliy illusraes he evoluion of queue lengh process wih ime. X n = X n K H n,k +A n n N 5 k= Le G be he class of deerminisic policies in which a each sae of he sysem, each policy g G allocaes he servers according o a predeermined allocaion marix. Therefore, each deerminisic policy g is specified by S allocaion marices I s g, s S. Noe ha se I is a finie se and since he channel sae space is also finie, se G is finie. A deerminisic policy g will resul ino an average ransmission rae for each queue n. Le R n g denoe he ime averaged ransmission rae provided o queue n and R g := R g,rg 2,...,Rg N. I follows ha, T R g = π s C s I g K T 6 s S Each rae vecorr g deermines a single poin inr N +. Now, consider he convex hull of all he poins R g, g G in R N +, i.e., R = conv.hullr g. According o he above discussion and definiion of convex polyope, we can show ha he se of achievable ransmission rae vecors R is specified by a polyope. Henceforh, we denoe he achievable ransmission rae polyope by P. Noe ha convex hull represenaion of he sabiliy region may no be suiable o use in nework opimizaion problems like. Since i does no provide mahemaical saemens in a form o be used as he consrains in. In he following, s we will characerize he sabiliy region by a se of linear inequaliies. To inroduce he necessary condiions for he sabiliy of MQMS sysem we need o prove a series of Lemmas Lemmas o 5. In his paper, he proofs were skipped due o space limiaion. Lemma : If he MQMS sysem is srongly sable under a server allocaion policy, hen for any vecor α = α,α 2,...,α N R N + we have lim τ= αe[aτ] T = lim αe[hτ] T K. 7 τ= Lemma 2: If he MQMS sysem is srongly sable under a server allocaion policy, hen E[A] P. Lemma 3: If he MQMS sysem is srongly sable under a server allocaion policy, hen for all α R N + αe[a] T s Sπ s max αcs I T K As we can see, he number of inequaliies defined in 8 is infinie. Noe ha each inequaliy in 8 deermines a valid inequaliy for polyope P. I is no hard o show ha he hyperplanes associaed o he valid inequaliies of 8 are all face defining hyperplanes of polyope P. To see his, le I α denoe a se of allocaion marices of {I α s,s S} ha maximizes he righ hand side of 8, i.e., 8 I α s = argmax αc s I T K. 9 Noe ha I α is no unique and here may be more han one se of allocaion marices of I α whose elemens maximize αc s I T K. According o 6 and definiion of polyope P, s S π sc s Is α T K T P. On he oher hand, s S π sαc s Is αt K = α s S π sc s Is αt K. Thus, s S π sc s Is α T K T is locaed on he hyperplane associaed o 8. In oher words, he hyperplane associaed o 8 ouches polyope P. So, he se of inequaliies of 8 are all face defining hyperplanes of polyope P. According o redundancy heorem in polyopes [3], no all he face defining hyperplanes of a polyope are required o deermine a polyope compleely. The inequaliies corresponding o he faces of a polyope are jus sufficien o characerize a polyope. In he following, we will characerize he faces of polyope P. Le V denoe he se of vecors α R N + wih he following propery, i.e., V := { α R N + U N,U,α U 0 U,α U c 0 U c i U,j U c, m,n M, α i,α j,m,n 0 : α i m = α j n} 0 In oher words, a vecor α belongs o se V if for any pariioning of elemens of vecor α ino wo non-empy disjoin subvecors in which all he elemens of each vecor are no idenically equal o zero, here exiss a leas one nonzero elemen in each vecor, he raio of which is equal o raio of wo non-zero elemens of se M. To clarify he definiion of se V consider he following example. Example: Le M = {0,,2} and N = 4. Consider vecor α =,2,5,0. We can pariion α o α {,2} =,2 and
α {3,4} = 5,0. Noe ha here exis no elemens in α {,2} and α {3,4} whose raio is or 2 and herefore, α / V. Anoher example is he vecor α 2 = 0,2.5,0,0 in which for any pariioning of he vecor ino wo non-empy disjoin pars, one par will always have all he elemens idenically equal o zero. Therefore, α 2 V. In he following we will show ha any vecor in R N + V canno form a face defining hyperplane for polyope P and herefore he inequaliies derived by he vecors of se R N + V are redundan o characerize polyope P. Lemma 4: The hyperplane associaed o he valid inequaliy of 8 is no a face defining hyperplane of P if α R N + V. Se V is an infinie se. However, we can show ha se V is finie up o muliplicaion of vecors by { posiive scalars. To show his, we define se W = z Z + z = } N j= m j, m j M. Then, we consider he se W N = {α,α 2,...,α N α n W}. We can prove he following Lemma. Lemma 5: There exiss V W N such ha any vecor α V can be wrien as α = qβ for some β V and q > 0. M+N 2 N We can show ha W N = +. Since V N W N, herefore V W N excluding he vecor 0 N. We now inroduce he necessary condiions for he sabiliy of MQMS sysem wih saionary channel disribuions by he following heorem. Theorem : If here exiss a server allocaion policy under which he sysem is sable, hen αe[a] T s Sπ s max αcs I T K, α V. The proof follows from Lemmas o 5. Noe ha according o Theorem, in order o characerize polyope P, we have o specify all he elemens of se V which can be obained afer considering all possible vecors α W N 0 N and removing redundancies following 0. This can be done using numerical compuaions. We may also useα W N 0 N insead ofα V in. Alhough by using W N 0 N we may obain some redundan inequaliies, since W N 0 N < we sill have a finie number of inequaliies o describe polyope P. Table I depics V and W N for some simple cases of N = 2,3 and M =,2,3,4. TABLE I: V and W N for N = 2,3 and M =,2,3,4 V, W N M = M = 2 M = 3 M = 4 N = 2 3,3 5,8 9,5 2,24 N = 3 7,7 25,63 09,342 253,330 Corollary : For an MQMS sysem wih saionary arrival processes, we have E[A] = λ = λ,λ 2,...,λ N and he necessary condiions for he sabiliy of he sysem would be αλ T s Sπ s max αcs I T K, α V, 2 Corollary 2: For an MQMS sysem wih ON-OFF channels, we have W = {0, } and herefore he necessary condiions for he sabiliy of he sysem become αe[a] T s Sπ s max αcs I T K, αn {0,} 3 In a special case considered in [9] where he channels are modelled by independen Bernoulli random variables wih E[C n,k ] = p n,k, he necessary condiions for he sabiliy of he sysem are given by K E[A n τ] K p n,k Q N. 4 n Q k= n Q In his case, he oal number of inequaliies needed o describe he polyope P is equal o 2 N. B. Sufficien Condiions for he Sabiliy of MQMS Sysem Consider a server allocaion policy which deermines he allocaion marix a each ime slo by solving he following maximizaion problem. I = argmax X C IT K 5 This policy is called Maximum Weigh MW and was firs inroduced in [3], [5]. According o he consrains on allocaion marix I, we can easily conclude ha MW policy allocaes he servers by he following rule: A each ime slo, MW policy allocaes each server k o queue n who has he maximum X n C n,k. In he special case where he channels are ON-OFF, he policy is equivalen o AS/LCQ Any Server/Longes Conneced Queue inroduced in [9]. In he following, we derive sufficien condiions for he sabiliy of our model and prove ha MW sabilizes he sysem as long as condiion 6 is saisfied. An upper bound is also derived for he ime averaged expeced number of packes in he sysem. Theorem 2: The MQMS sysem is sable under MW if for all αe[a] T < s Sπ s max αcs I T K, α V 6 Furhermore, he following bound for he average expeced aggregae occupancy holds. lim sup N τ=0n= E[X n τ] NA2 max +MK2 2δ 7 where δ is he maximum posiive number such ha for all we have E[A]+δ N P. Corollary 3: For an MQMS sysem wih saionary arrival processes, we have E[A] = λ and he sufficien condiions for he sabiliy of he sysem under MW policy become αλ T < s S π smax αcs I T K, α V. According o Corollaries and 3 and he definiion of nework capaciy region, we can conclude ha for an MQMS sysem wih saionary channel disribuion and saionary arrival processes, he sabiliy region is characerized by 2. Consider an MQMS sysem wih ON-OFF channels. For such a sysem, he MW policy is equivalen o AS/LCQ policy inroduced in [9]. In AS/LCQ he policy akes an arbirary ordering of servers o be allocaed o he queues and hen for each server, AS/LCQ allocaes i o is longes conneced queue LCQ. Noe ha in such a sysem, W = {0,} and herefore we will have he following corollary.
Corollary 4: The MQMS sysem wih ON-OFF channels is sable under AS/LCQ if for all αe[a] T < s max αcs I s Sπ T K, αn {0,} 8 In a special case where he channels are modelled by independen Bernoulli random variables wih E[C n,k ] = p n,k, AS/LCQ sabilizes he sysem if K E[A n τ] < K p n,k Q N 9 n Q k= n Q C. Sabiliy Region for Fluid Model MQMS Sysems wih Saionary Coninuous Channel Disribuions We will consider a ime sloed fluid model MQMS sysem wih saionary channel disribuions. In his case, he amoun of work ha arrives ino and depars from he queues is considered o be a coninuous process. We also assume ha he channel sae of he link from each queue o each server is modelled by a coninuous random variable. The channel sae marix in he fluid model MQMS sysem is defined as C = C n,k,n N,k K,C n,k R +. We assume ha he channel sae follows a saionary disribuion f C c. f C is he join disribuions of all C n,k variables. We can easily check ha in he fluid model MQMS sysem, se V is equivalen o R N + as in he fluid model MQMS sysem, se M is replaced wih R +. Therefore, sabiliy region for he fluid model sysem is characerized by he following se of inequaliies. αλ T c n,k =0 c n,k =0 c,=0 max αc I T K f C c dc, dc n,k dc n,k α R N + 20 Noe ha in order o characerize he sabiliy region of he fluid model sysem we need o compue an infinie number of inegrals in 20. In fac, he sabiliy region of fluid model sysem is characerized by infinie number of half spaces which means ha he sabiliy region is a convex surface. In his case, depending on he channel disribuion and dimension of he sysem, we could characerize he sabiliy region by a limied number of non-linear inequaliies insead of infinie number of linear inequaliies as we show in he following example. Example: Consider a fluid model MQMS sysem wih wo queues and one server. Suppose ha he channel sae variables C, and C 2, are independen and follow exponenial disribuion wih means µ and µ 2, respecively. Such a modelling is used o model slow Rayleigh fading channels wih low SNR where he approximaionlog+x x is valid for small posiive x. We can show ha he sabiliy region for his example is described by λ 2 µ 2 λ 2 λ µ µ 2 for λ 0 and λ 2 0. The derivaion of 2 is skipped because of space limiaion. As we can see from equaion 2, he sabiliy region in his example is characerized by jus one non-linear inequaliy 2 and wo linear inequaliies λ 0 and λ 2 0. The sabiliy region for his example for µ = 2 and µ 2 = is illusraed in Figure 2. λ2 0.8 0.6 0.4 0.2 λ 2 µ 2 λ 2 λ µ µ = 2 and µ 2 = 0 0 0.2 0.4 0.6 0.8.2.4.6.8 2 Fig. 2: Sabiliy region for µ = 2 and µ 2 = V. CONCLUSIONS In his paper, we characerized he sabiliy region polyope of muli-queue muli-server MQMS queueing sysems wih saionary channel and packe arrival processes by a finie se of linear inequaliies given by 2. We also argued ha in general i may be compuaionally hard o quanify se V from se W N. In his case, we can use se W N iself insead of V in 2 as V W N, alhough we may have some redundan inequaliies. We also considered he sabiliy region for a fluid model MQMS. In his case, we deermine he sabiliy region by an infinie se of linear inequaliies given by 20. Finally, by use of an example we showed ha depending on he channel disribuion and number of queues, we may characerize he sabiliy region by a finie se of non-linear inequaliies insead of infinie se of linear inequaliies. REFERENCES [] L. Georgiadis, M. J. Neely, and L. Tassiulas, Resource Allocaion and Cross Layer Conrol in Wireless Neworks. Now Publisher, 2006. [2] M. J. Neely, Dynamic power allocaion and rouing for saellie and wireless neworks wih ime varying channels, Ph.D. disseraion, Massachuses Insiue of Technology, LIDS, 2003. [3] L. Tassiulas and A. 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