Control of parallel non-observable queues: asymptotic equivalence and optimality of periodic policies

Transcription

1 Contol of paallel non-obsevable queues: asymptotic equivalence and optimality of peiodic policies Jonatha Anselmi, Buno Gaujal, Tommaso Nesti To cite this vesion: Jonatha Anselmi, Buno Gaujal, Tommaso Nesti. Contol of paallel non-obsevable queues: asymptotic equivalence and optimality of peiodic policies. stochastic systems, INFORMS Applied Pobability Society, 2015, 5 1), < /14-SSY146>. <hal > HAL Id: hal Submitted on 13 Jan 2015 HAL is a multi-disciplinay open access achive fo the deposit and dissemination of scientific eseach documents, whethe they ae published o not. The documents may come fom teaching and eseach institutions in Fance o aboad, o fom public o pivate eseach centes. L achive ouvete pluidisciplinaie HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau echeche, publiés ou non, émanant des établissements d enseignement et de echeche fançais ou étanges, des laboatoies publics ou pivés.

2 Contol of paallel non-obsevable queues: asymptotic equivalence and optimality of peiodic policies Jonatha Anselmi, Buno Gaujal and Tommaso Nesti Abstact We conside a queueing system composed of a dispatche that outes deteministically jobs to a set of non-obsevable queues woking in paallel. In this setting, the fundamental poblem is which policy should the dispatche implement to minimize the stationay mean waiting time of the incoming jobs. We pesent a stuctual popety that holds in the classic scaling of the system whee the netwok demand aival ate of jobs) gows popotionally with the numbe of queues. Assuming that each queue of type is eplicated k times, we conside a set of policies that ae peiodic with peiod k p and such that exactly jobs ae sent in a peiod to each queue of type. When k, ou main esult shows that all the policies in this set ae equivalent, in the sense that they yield the same mean stationay waiting time, and optimal, in the sense that no othe policy having the same aggegate aival ate to all queues of a given type can do bette in minimizing the stationay mean waiting time. This popety holds in a stong pobabilistic sense. Futhemoe, the limiting mean waiting time achieved by ou policies is a convex function of the aival ate in each queue, which facilitates the development of a futhe optimization aimed at solving the fundamental poblem above fo lage systems. 1 Intoduction In compute and communication netwoks, the access of jobs to esouces web seves, netwok links, etc.) is usually egulated by a dispatche. A fundamental poblem is which algoithm should the dispatche implement to minimize the mean delay expeienced by jobs. Thee is a vast liteatue on this subject and the stuctue of the optimal algoithm stongly depends on i) the infomation available to the dispatche, ii) the topology of the netwok and iii) how jobs ae pocessed by esouces. We ae inteested in a scenaio whee: The dispatche has static infomation of the system; The netwok topology is paallel; Resouces pocess jobs accoding to the fist-come-fist-seved discipline. Static infomation means that the dispatche knows the pobability distibutions of job sizes and inte-aival times but cannot obseve the dynamic state of esouces e.g., the cuent numbe of jobs in thei queues). This is motivated by the fact that eal netwoks ae composed of hundeds o thousands of esouces, and allowing fo dynamic infomation implies a non-negligible communication ovehead and the poblem of delayed infomation [29]. On the othe hand, infeing statistics of inte-aival times and job sizes in advance is a much easie task. The scenaio above is of inteest in voluntee computing, cloud computing, web seve fams, etc.; see, e.g., [23, 24, 19] espectively. Reseach patially suppoted by gant SA-2012/00331 of the Depatm ent of Industy, Innovation, Tade and Touism Basque Govenment) and gant MTM Ministeio de Ciencia e Innovación, Spain) which sponsoed an intenship of Tommaso Nesti at BCAM. Basque Cente fo Applied Mathematics BCAM), Al. de Mazaedo 14, Bilbao, Basque County, Spain; INRIA Bodeaux Sud Ouest, 200 av. de la Vieille Tou, Talence, Fance. jonatha.anselmi@inia.f INRIA and LIG, Zist 51, Av. J. Kuntzmann, MontBonnot Saint-Matin, 38330, Fance. buno.gaujal@inia.f Basque Cente fo Applied Mathematics BCAM), Al. de Mazaedo 14, Bilbao, Basque County, Spain; Dipatimento di Matematica, Univesity of Pisa, Lago B. Pontecovo 5, Pisa, Italy. nesti@mail.dm.unipi.it 1

3 In this famewok, the poblems of finding a policy that minimizes the mean stationay delay and of detemining the minimum mean stationay delay ae both consideed difficult; see, e.g., [2, 1] fo an oveview. A policy can be defined as a function that maps a natual numbe n, coesponding to the n-th job aiving to the dispatche, to a pobability mass function P n ove the set of esouces. When the n-th job aives, the dispatche sends it to esouce i with pobability P n i) independently of any othe event in the past, pesent o futue. Unfotunately, the poblem of finding an optimal policy is intactable and fo this eason two exteme families of policies have eceived paticula attention in the liteatue: pobabilistic policies, obtained when P n is constant in n), and deteministic policies, obtained when P n puts the whole mass on a single esouce. When dealing with pobabilistic policies, the difficulty of the poblem is simplified by the fact that the aival pocess at each esouce is a enewal pocess, povided that the same holds fo the aival pocess at the dispatche. This allows one to decompose the poblem and, using the theoy of the mean waiting time of the single GI/GI/1 queue, to immediately educe it to a elatively simple optimization poblem. This poblem is usually convex and thee exist efficient numeical pocedues fo thei solution; e.g., [30, 34, 32, 12, 13, 8]. Contaiwise, when dealing with deteministic policies, one of the main difficulties is that the aival pocess at each esouce is hadly eve a enewal pocess. This pevents one fom decomposing the poblem and diectly using the classic theoy of the single queue as it has been done fo pobabilistic policies. Given this difficulty, eseaches divided this poblem in two subpoblems: i) In the fist subpoblem, the optimal deteministic policy is seached among all the deteministic policies ensuing that the long-tem factions of jobs to be sent to each esouce is kept fixed denote such factions by vecto p); ii) In the second subpoblem, the output of the fist subpoblem is employed to develop a futhe optimization ove p. In this pape, we focus on the fist subpoblem and, unde some system scaling, we identify a set of policies that ae optimal. This esult is used to educe the second subpoblem to the solution of a convex optimization. One of the folk theoem of queueing theoy says that deteminism in the inte-aival times minimizes the waiting time of the single queue [17, 22]. In view of this classic insight and fixing factions p, it is not supising that an optimal policy ties to make the aival pocess at each esouce as egula o less vaiable) as possible. Thus, ou stochastic scheduling poblem can be conveted into a poblem in wod combinatoics. If the dispatche must ensue factions p, the main esult known in the liteatue is that balanced sequences ae optimal admission sequences [18, 1]. Howeve, only vey few vectos of ates p ae balanceable and to decide whethe o not p is balanceable is itself a had combinatoial poblem; see, e.g., [2, Chapte 2], which also contains an oveview of which vectos ae balanceable, and [36]. Matte of fact, the poblem of finding an optimal deteministic policy is still consideed difficult [10, 3, 37, 21, 2, 7, 20]. The only exceptions ae when esouces ae stochastically equivalent, whee ound-obin 1 is known to be optimal in a stong sense [26], o when the dispatche outes jobs to two esouces, whee balanced sequences can be always constucted no matte the value of ates p [18]. In pesence of moe than two queues, we stess that balanced sequences with given ates p do not exist in geneal. This non-existence makes the poblem difficult and one still wondes which stuctue should an optimal policy have when p is not balanceable. When the outing is pefomed to two esouces, jobs join the dispatche following a Poisson pocess and sevice times have an exponential distibution, the optimal ates p as function of the inte-aival and sevice times have a factal stuctue, see [15, Figue 8]. This puts futhe light on the complexity of the poblem. While deteministic policies ae believed to be moe difficult to study than pobabilistic policies deteministic and pobabilistic in the sense descibed above), they can achieve a significantly bette pefomance [3]. This holds also fo the vaiance of the waiting time because, as discussed above, the aival pocess at each esouce is much moe egula in the deteministic case, especially if thee ae seveal esouces as we show in this pape. 1.1 Contibution In the famewok descibed above, we ae inteested in deiving stuctual popeties of deteministic policies when the system size is lage. We study a scaling of the system whee the aival ate of jobs, λk, gows to 1 Round-obin sends the n-th job to esouce n mod R) + 1, whee R is the total numbe of esouces. 2

4 infinity popotionally with the numbe of esouces queues in the following), Rk, while keeping the netwok load o utilization) fixed. This scaling is often used in the queueing community. Specifically, thee ae R types of queues, and k is the numbe of queues in each type, i.e., the paamete that we will let gow to infinity. Beyond issues elated to the tactability of the poblem, this type of scaling is motivated by the fact that the size of eal systems is lage and that eplication of esouces is commonly used to incease system eliability. Fist, with espect to a class of peiodic policies, we define the andom vaiable of the waiting time of each incoming job. This is done using Lindley s equation [25] and a suitable initial andomization. Using such andomization, we can adapt the famewok developed by Loynes in [27] to ou setting whee jobs ae sent to a set of paallel queues, and in ou fist esult, Theoem 1, we show the monotone convegence in distibution of the waiting time of each incoming job. To the best of ou knowledge, this is the fist pape that chaacteizes the stationay distibution of the waiting time of incoming jobs in paallel FCFS queues when policies ae deteministic. Then, with espect to a given vecto p N R, we define a cetain subset of policies that ae peiodic with peiod k and such that exactly jobs ae sent in a peiod to each queue of type. While futhe details will be developed in Section 3.1, this set is meant to imply that queues of a given type ae visited in a ound-obin manne and that aivals ae well distibuted among the diffeent queue types. When k, ou main esult states that all the policies in this set ae equivalent, in the sense that they yield the same mean stationay waiting time, and optimal, in the sense that no othe policy having the same aggegate aival ate to all queues of a given type can do bette in minimizing the mean stationay waiting time. In paticula, we show that the stationay waiting time conveges both in distibution and in expectation to the stationay waiting time of a system of independent D/GI/1 queues whose paametes only depend on p, λ and the distibution of the sevice times. This is shown in Theoems 2 and 3, espectively. The main idea undelying ou poof stands in analyzing the sequence of stationay waiting times along appopiate subsequences. Along these subsequences, it is possible to extact a poweful patten fo the aival pocess of each queue that is common to all membes of the subsequence. Such patten is exploited to establish monotonicity popeties in the language of stochastic odeings. These ae popeties holding fo the consideed subsequences only: they do not hold tue along an abitay subsequence and counteexamples can be given. These popeties will imply the unifom integability of the sequence of stationay waiting times and will allow us to wok on expected values. Summaizing, fixing the popotions p of jobs to send to each queue type and given k lage, ou esults state that all the peiodic policies have almost the same pefomance and this lets us conclude that ou limit is obust fo pedicting pefomance. Using known popeties of the D/GI/1 queue, we also obseve that the stationay mean waiting time obtained in ou limit is a convex function of p. This facilitates the solution of subpoblem ii) above, which is thus captued by the solution of a convex optimization poblem. This pape is oganized as follows. Section 2 intoduces the model unde investigation and povides a chaacteization of the stationay waiting time Theoem 1); Section 3 intoduces a class of policies and pesents ou main esults Theoems 2 and 3); Section 4 is devoted to poofs; finally, Section 5 daws the conclusions of this pape. 2 Paallel queueing model We conside a queueing system composed of R types of queues o esouces, seves) woking in paallel. Each queue of type is eplicated k times, fo all = 1,..., R, so thee ae kr queues in total. Paamete k is a scaling facto and we will let it gow to infinity. The sevice discipline of each queue is fist-comefist-seved FCFS) and the buffe size of each queue is infinite. A steam of jobs o customes) joins the queues though a dispatche. The dispatche outes each incoming job to a queue accoding to some policy and instantaneously. Figue 1 illustates the stuctue of the queueing model unde investigation. In the following, indices, κ, n will be implicitely assumed to ange fom 1 to R, fom 1 to k, in N, espectively. All the andom vaiables that follow will be consideed belonging to a fixed undelying pobability tiple Ω, F, P). Let T n ) n N and S n,κ,) n N be given sequences of i.i.d. andom vaiables in R + 2. These sequences ae 2 Fo any E R, we let E + def = {x E : x > 0}. 3

5 Type 1 μ 1 Type 1 μ 1 k queues λ k jobs/time π Dispatche Type 1... μ 1 Type R μ R Type R Type R μ R μ R k queues Figue 1: Stuctue of the paallel queueing model unde investigation. all assumed to be independent each othe. Quantity T n is intepeted as the inte-aival time between the n-th and the n + 1)-th jobs aiving to the dispatche. Quantity S the n-th job aiving at the κ-th queue of type. We assume that S 1,κ, = st S 1,1, Fo the aival pocess at the dispatche, we will efe to the following cases. n,κ, is intepeted as the sevice times of and that ES n,κ, = µ 1. Case 1. The pocess T n ) n N is a enewal pocess with ate λk and such that Va T n = o1/k). Case 2. The pocess T n ) n N is a Poisson pocess with ate λk. Case 3. The pocess T n ) n N is constant with ate λk, i.e., T n = λk) 1. It is clea that Cases 2 and 3 ae both moe estictive than Case 1. Let denote the L 1 -nom. Let q def = q,κ ),κ Q R + Q k + be such that q = 1. Quantity q,κ will be intepeted as the popotion of jobs sent to queue, κ) in a peiod, o equivalently in the long tem. Let n def = n q) def = min{n Z + : n q Z R + Z k +}. Since q is a vecto of ational numbes, n <. Let V be a discete andom vaiable with values in {1,..., n } such that PV = i) = 1/n, fo all i = 1,..., n. We assume that V is independent of any othe andom vaiable. Let A q be the set of all functions π : N {1,..., R} {1,..., k} such that fo all and κ n q,κ = 1 n 1 {πn)=,κ)}, πn) = πn + n ) 1) fo all n, whee 1 E denotes the indicato function of event E. Thus, these functions ae peiodic with peiod n and n q,κ is the numbe of jobs sent in a peiod to queue, κ). We efe to each element π A q as a policy o a q-policy) opeated by the dispatche, and it is intepeted as follows: πn) =, κ) means that the n-th job aiving to the dispatche is sent to the k-th queue of type if n V, othewise it means that the n-th job is discaded. Thus, the outcome of andom vaiable V gives the index of the fist job that is actually seved by some queue. In othe wods, πv ) is the fist queue that seves some job. Let T n,κ,π)) n N be the sequence of inte-aival times that ae induced by policy π at the κ-th queue of type unde any of the cases above). By constuction, T n,κ,π) is the sum of a deteministic numbe of inte-aival times seen at the dispatche. The aival pocess T n,κ,π)) n N can be made stationay if it is allowed a shift in time and a suitable andomization fo the inte-aival time of the fist aival of each queue. We assume fo now that this has been done details will be given at the beginning of Section 4). As done in 4

6 [27] and accoding to [14, p. 456], this implies that we can extend the stationay pocess T n,κ,π)) n N to fom a stationay pocess T n,κ,π)) n Z clealy, the same holds fo the pocess S n,κ,) n N ). The waiting time of the n-th job aiving to the κ-th queue of type induced by a policy π A is denoted by W n,,κπ). It is the time between its aival at the dispatche o equivalently at the queue) and the stat of its sevice, and it is defined as follows: fo n = 0, W n,,κπ) = 0 and fo n > 0, W n,,κπ) def + = W n 1,,κ π) + S n,κ, T n,κ,π)), 2) whee x + def = max{x, 0}. Equation 2) is known as Lindley s ecusion [25]. The assumption that W 0,,κ π) = 0 seves to avoid technicalities 3. It is known that the sequence of andom vaiables W n,,κπ)) n N conveges in distibution to the andom vaiable W,κ π) def = sup n n 0 n =0 S n,κ, T n,κ, π)) + 3) and that W n,,κπ) st W n+1,,κ π), whee st denote the usual stochastic ode; see [27]. W We efe to,κ π) as the stationay waiting time of jobs at the κ-th queue of type. Given π, let f : N N {1,..., R} {1,..., k} be a mapping with the following meaning: fn) = n,, κ) means that the n-th job aiving to the dispatche is the n -th custome joining queue, κ). If n < V, no job is sent to any queue and thus we assume fn) = 0. Note that fn) is a deteministic function of andom vaiable V. Since V is unifom ove {1,..., n }, fo all n n we have Pf 2 n) =, f 3 n) = κ) = q,κ, 4) whee f j efes to the j-th component of f. With this notation, quantity W fn) π) is the waiting time of the n-th job aiving to the dispatche induced by a policy π A q, fo all n n. Now, let Q,κ ),κ denote a patition of set {1,..., n }. The intevals Q,κ, fo all and k, ae thus disjoint and we futhe equie that the numbe of points in Q,κ is n q,κ. Let W π) def = R k =1 κ=1 1 {V Q,κ}W,κ π). 5) Since V is independent of the W,κ π) s, the distibution of W π) is the finite mixtue of the W,κ π) s with weights q. Next theoem says that W π) can be intepeted as the ight andom vaiable descibing the stationay waiting time of jobs achieved with policy π A q. It is poven by adapting the famewok developed by Loynes in [27]. In the emainde of the pape, convegence in distibution in pobability) is denoted by d P espectively, ). Theoem 1. Let Case 1 hold. Let q be such that q,κ λ < µ fo all, κ and π A q. Then, and W fn) π) st W fn+1) π) 6) W fn) π) d n W π). 7) By using the monotone convegence theoem, Theoem 1 implies that also the moments of W fn) π) convege to the moments of W π), povided that they ae finite. Finally, fo a given p R R +, we define the auxiliay andom vaiable W p), which coesponds to the stationay waiting time of a D/GI/1 queue with inte-aival times T n, p)) n N whee T n, p) = p / λ) 3 Using a standad coupling agument and that each queue will empty in finite time almost suely, what follows can be genealized easily to the case whee W 0,,κ π) 0. 5

7 and sevice times S n,1, ) n N, fo all. Let also V def = V p) be a andom vaiable with values in {1,..., R} independent of any othe andom vaiable and such that PV = ) = / p, fo all, and let W p) def = R 1 {V =} W p). 8) =1 Note that EW p) = p EW p) can be intepeted as the waiting time of R independent D/GI/1 queues aveaged ove weights p/ p. We will be inteested in establishing convegence esults fo W when k. With espect to some policies to be defined, we will show foms of convegence to W p). 2.1 Discussion Some emaks about the model above and Theoem 1 follow: To the best of ou knowledge, the chaacteization of the distibution of the stationay waiting time W given in Theoem 1 is new in the liteatue, though its poof is based on classical aguments. While we believe that such chaacteization has its own inteest, in ou pape we need to fomally define the distibution of the stationay waiting time athe than only its expected value as existing woks do) because we can only pove ou main esult, Theoem 3, though a distibutional convegence agument, Theoem 2. Though seveal woks focused on finding policies that minimize the expected value EW see the Intoduction), the analysis of EW in ou scaling whee k seems new. We assume that q is a vecto of ational numbes. Fom a pactical standpoint, this is not a loss of geneality fo obvious easons. Ou appoach needs this stuctue to pove 7). In the case whee πn) 1 n is not peiodic, a case that we do not conside hee, and the limit lim n n n =1 1 {πn )=,κ)} does not exist, it would be inteesting to know whethe some convegence in distibution of W fn) π) occus. The fact that in Case 1 we equie that Va T n = o is not a loss of geneality fo Theoem 1, as we do not let k thee. Case 1 coves the case whee T n has a Gk, α)-phase-type distibution whee both G and α ae not functions of k. We may have assumed that each queue of type was eplicated kz times, z N, instead of just k times. This is equivalent to ou setting, as we can assume that thee ae z queues of type when k = 1. 3 Main esults Fo p Z R +, let and P p def = { q Q R + Q k + : A p def = q P p k κ=1 q,κ k = p p, } 9) A q. 10) The set A p is intepeted as the set of all peiodic policies fo which λ p p is the mean aival ate of jobs to each queue of type. We conside the following poblem. Poblem 1. Let p Z R + be given. Detemine the optimizes and the optimal objective function value of min EW π). 11) π A p 6

8 As discussed in the Intoduction, this is consideed as a difficult poblem. We ae inteested in establishing stuctual popeties of Poblem 1 when k is lage. In the following, we define a cetain class of policies and then pesent ou main esults. 3.1 A class of peiodic policies Fo p Z R +, we define C p as the subset of all policies π A p that satisfy the following popeties: Among evey k p consecutively-aiving jobs, exactly jobs ae sent to each queue of type, fo each. That is, k p +n n =n+1 1 {πn )=,κ)} =, n. Among evey p consecutively-aiving jobs, exactly jobs of type aive. That is, k p +n κ=1 n =n+1 1 {πn )=,κ)} =, n. Let n 1, n 2,... be the subsequence of all jobs that ae sent to queues of type. Then, π 2 n 1 ) = 1, and πn j+1 ) = πn j ) + 0, 1) if π 2 n j ) < k and πn j+1, ) =, 1) othewise. The fist popety says that π is peiodic with peiod k p and that in a peiod exactly jobs ae sent to each queue of type. The second popety says that aivals emain well distibuted among the diffeent types when k inceases. Finally, the thid popety says that queues of type ae accessed by jobs in a ound-obin o cyclic) ode stating fom queue 1, and we need it to ensue that the cadinality of C p, i.e., C p, does not vay with k. In fact, one can veify that C p = p!!). 12) These policies can be implemented in a distibuted manne. Moe pecisely, one can think that thee ae two ties of dispatches: the dispatche in the fist tie schedules jobs inte-goup, while the dispatches in the second tie, R in total, schedule jobs inta-goup and implement ound-obin. With espect to a sequence of policies π ) k N, whee π C p, we will show Lemma 2) T n,κ,π ) P p λ, n. 13) This means that the finite dimensional distibutions of the aival pocess at each queue will be close to the deteministic pocess, when k is lage, which implies that some fom of convegence to W p) should occu in view of the continuity of the stationay waiting time [11]. 3.2 Asymptotic equivalence and optimality Next theoem poves a fist fom of convegence of W π ) to W p). Theoem 2. Let Case 1 hold. Let p Z R + be such that λ p p < µ fo all. Let also an abitay sequence π ) k N be given whee π C p fo all k. Then W π ) d W p). 14) Unfotunately, Theoem 2 is not enough to claim that EW π ) conveges as well. Convegence of the fist moment is impotant fom an opeational standpoint, as in pactice one desies to optimize ove EW o Va W. Unde some additional assumptions, next theoem states that also the expected value and the 7

9 vaiance of W convege. Futhemoe, it states that all the policies in set C p ae both asymptotically equivalent and asymptotically optimal, with espect to the citeion in Poblem 1. Theoem 3. Let Case 2 o 3 hold. Let p Z R + be such that λ p p < µ fo all. Let also an abitay sequence π ) k N be given whee π C p fo all k. If E[S 1,κ, )3 ] <, then lim inf Futhemoe, if E[S 1,κ, )5 ] <, then min π A p lim Va W π ) = EW π) = lim EW π ) 15) lim EW π ) = EW p). 16) p Va W p) + EW p) EW p)) 2). 17) Povided that k is lage, thus, no othe policy in A p \ C p can do bette than any π C p to minimize EW. It also explicits the limiting value of EW π ), which is EW p). It is known that EW p) is a convex function in p e.g., [30]). To pove Theoem 3, we use the well-known fact that [5] W p) icx W,κ π) 18) fo any π A p. Using this lowe bound fist and then that the waiting time of the D/GI/1 queue is convex inceasing in its aival ate, it is not difficult to show that EW p) EW π). 19) Then, we pove that the sequence EW π ) is uppe bounded by a sequence that conveges to the lowe bound in 19). An obsevation hee is that the lowe bound 18) holds unde conditions that ae weake than those assumed in this pape; see [22]. Fo instance, it is possible to extend 19) and thus Theoem 3) to the case whee i) policies ae not peiodic, ii) the factions of jobs to send in each queue ae not necessaily ational numbes, and iii) policies ae andomized [31], that is the case whee πn) is any pobability mass function ove the set of queues. We do not investigate these extensions in futhe detail. 4 Poofs In this section, we develop poofs fo Theoems 1, 2 and 3. Befoe doing this, we fix some additional notation and show how it is possible to make the aival pocess at each queue stationay as assumed in Section 2). Let us conside the κ-th queue of type and its aival pocess T n,κ,) n N. Each inte-aival time depends on the policy π implemented by the dispatche, i.e., T n,κ, = T n,κ,π ), though in the following we dop such dependency fo notational simplicity. Since π is peiodic by constuction with peiod k p and jobs have to be sent within a cycle to each type- queue, the sequence T n,κ,) n N is composed of a epeated patten of inte-aival times, that we can wite as A 1,κ,, A 2,κ,,..., A,κ,, 20) whee each quantity A j,κ,, j = 1,...,, is the sum of a deteministic numbe that depends on π ) of inte-aival times to the dispatche. We denote such numbe by a and notice that it does not vay with κ because by symmety the aival pocesses of all queues of a given type ae equal, in distibution, up to a shift in time. Thus, A j,κ, = st 8 a T n, 21)

10 whee = st denotes equality in distibution. It is also clea that a j=1 = k p. 22) We want to make T n,κ,) n N stationay. This can be done as follows by andomizing ove the fist inteaival time of queue, κ). Now, let us conside the auxiliay andom vaiables U,κ, fo all and κ, which we assume independent each othe and of any othe andom vaiable and having a unifom distibution in [0, 1]. Then, we take T def 1,,κ = A,κ 1 {U,κ [ j 1 p j=1, j p ]} 23) Theefoe, if T 1,,κ = A,κ, fo some j <, then T 2,,κ = A j+1,,κ and so foth accoding to the patten 20). Defined in this manne, one can see that T n,,κ) n N is stationay. Futhemoe, using 22) ET n,κ, = 1 j=1 a ET n = 1 j=1 a kλ = p λ. 24) At this point, the issue is the following. Conside two queues, say, κ) and, κ ), and suppose that T 1,,κ = A,κ and T 1,,κ = A j,,κ. We should check whethe policy π is actually able to induce an aival pocess equal samplepath-wise) to the one built above. One can easily see that this can be done with a possible shift of time fo the aival pocess at the queues and possibly discading a finite numbe of jobs. This is allowed because these opeations do not change the stationay behavio. In the emainde, we will use stochastic odeings. We will denote by st, cx and icx, the usual stochastic ode, the convex ode and the inceasing convex ode, espectively; we point to, e.g., [33, 35] fo thei definition. We will also efe to the following lemma, which can be easily poven by using a diagonalization agument. Lemma 1. Let D k ) k N be a sequence of odeed finite sets having N > 1 elements each. Fo n D k, let #n N denote the position of n in D k e.g., if D k = {c, a, b} then #a = 2). Let n k ) k N be a sequence such that n k D k, fo all k. Let f k ) k N be a sequence of functions whee f k : D k R. If lim f k n k ) = c fo all sequences n k ) k N such that #n k = #n 1, fo all n 1 D 1, then lim f k n k ) = c fo all sequences n k ) k N. 4.1 Poof of Theoem 1 Let andom vaiables V 1 and V 2 be given such that V 1 = st V 2 = st V. We pove 6) though Stassen s theoem building a coupling W fn) π), W fn+1) π)) of W fn) π) and W fn+1) π) though V 1 and V 2 ensuing that W fn) π) W fn+1) π). This is done as follows. Fist, let W fn) π) and W fn+1) π) be the Loynes waiting times see [27, p. 501]) obtained when the fist queue to seve a job is given by the outcome of V 1 and V 2, espectively; thus, W fn) π) is the waiting time at time 0 with n jobs in the past at queue, κ), povided that fn) = n,, κ). Then, we take V 2 = V 1 1 if V 1 > 1 othewise V 2 = n, and let the Loynes) waiting times be diven by the same ealizations of the andom inte-aival and sevice times. We now pove 7). Since π is peiodic with peiod n, we fist obseve that π etuns the same queue along subsequence n n + i) n N, fo all i = 1,..., n. Similaly, also the second and thid component of fn n + i) do not change along these subsequences, though they ae not known in advance because they depend on the outcome of andom vaiable V, see 4). Thus, fo all i = n + 1,..., 2n, by constuction we have Pf 2 n n + i) =, f 3 n n + i) = κ) = Pf 2 i) =, f 3 i) = κ) = q,κ, 25) and we get lim PW n fnn +i) π) t) 26a) 9

11 = lim n =,κ,κ q,κ PW n,,κπ) t f 2 i), f 3 i)) =, κ)) q,κ PW,κ π) t) 26b) 26c) = PW π) t). 26d) In 26b), we have conditioned on fi). In 26c), we have used that W n,,κπ) conveges in distibution to W,κ π); see [27]. In 26d), we have used the definition of W π). Now, since the limit in 26d) does not depend on i, the poof is concluded by applying Lemma Poof of Theoem 2 We fist obseve that we can pove this theoem unde some assumption on the sequence π ) k N. Given π 1) C p 1), we equie that fo all k: π 1 n) = π1) 1 n), n. 27) These sequences will be assumed along this poof. If this theoem holds fo these sequences, then it also holds fo all the sequences in view of Lemma 1 and of the fact that the cadinality of C p does not vay with k. Fo m N, let def k m = m lcmp), 28) whee lcmp) denotes the least common multiple of p 1,..., p R. The subsequences k m + i) m N, fo all i = 1,..., lcmp), play a key ole in ou poof and will be especially used also in the poof of Theoem 3. Along these subsequences, next fact holds tue and follows by constuction of the policies in set C p : it is a diect consequence of the fact that queues of the same type ae visited in a ound-obin manne. Fact 1. Fo m > 1, j = 1,..., and i = 1,..., lcmp), a pm+i) = a pm 1)+i) Unfolding the ecusion in Fact 1, fo m N, i = 1,..., lcmp), we get and theefoe a km+i) = a km 1+i) i) + p. + p lcmp) = a i) + m p lcmp) 29) a km+i) k m + i = a + m p lcmp) m lcmp) + i m Since 30) holds fo all i = 1,..., lcmp), by using Lemma 1 we obtain lim a k p. 30) = p. 31) Lemma 2. Unde the hypotheses of Theoem 2, T n,,κ p λ in pobability, as k. Poof. Fo all ɛ > 0, P ) ) T n,,κ p λ ɛ = P T n,,κ ET n,,κ ɛ 1 Va T ɛ2 n,,κ = 1 ɛ 2 EVa T n,,κ U,κ ) + Va ET = 1 1 ɛ 2 j=1 n,,κ U,κ ) Va A,κ + EA,κ ET ) n,,κ) 2 32a) 32b) 32c) 32d) 10

12 = 1 1 ɛ 2 j=1 a Va T 1 + a λk ) 2 p λ 32e) 0. 32f) In 32b), 32c) and 32e), we have used Chebyshev s inequality, the law of total vaiance and that the T n s ae i.i.d., espectively. In 32f), we have used 31) and that a Va T 1 0 because Va T 1 = o and a k p. Since T n,,κ convege in pobability fo each n, also the finite dimensional distibutions of the pocess n,,κ) n N convege to the one of the constant pocess with ate λp. Togethe with the fact that ET n,,κ = T lim ET n,,κ = p λ, we can use the continuity of the stationay waiting time see [11, Theoem 22]) to establish that W,κ π ) p d W p) 33) holds. Using 33) and that Cesao sums convege if each addend conveges, we obtain lim PW π ) t) = lim = R =1 R =1 p k k κ=1 PW,κ π ) t) p PW p) t) = PW p) t) 34a) 34b) as desied. 4.3 Poof of Theoem 3 Poof of 15) and 16). Given that C p fo all π A p and A p, 15) and 16) hold tue if EW p) EW π) 35) lim EW π ) EW p). 36) Let EW,κ x) be the mean waiting time of a D/GI/1 queue with aival ate λx and i.i.d. sevice times having the same distibution of S 1,,κ. Inequality 35) is a faily diect application of known esults: fo all π A p, EW π) =,κ,κ,κ q,κ EW,κ π) q,κ EW,κ kq,κ ) k p EW,κ p p ) 37a) 37b) 37c) = p EW p) = EW p). 37d) In 37b), we have used the lowe bound in [22]. In 37c), we have used Kaamata s inequality once noticing that i) EW,κ x) = EW,1 x), ii) the majoization p p,..., p p ) kq,1,..., kq,k ) holds, and iii) the mean waiting time of a D/GI/1 queue is convex inceasing in the aival ate see, e.g., [30, Theoem 5], [16]), which means that q,κ EW,κ kq,κ ) is convex in q,κ. 11

13 We now pove 36). As in the poof of Theoem 2, this can be done assuming that 27) holds. Thus, the sequences 27) will be assumed along this poof. Associated to each queue of type, we define an auxiliay andom vaiable, T, such that T = st Next lemma povides popeties satisfied by T see 28). min j=1,...,p Lemma 3. Unde the hypotheses of Theoem 3, the following popeties hold: i) We have ii) T p λ in pobability, as k. iii) Fo all i = 1,..., lcmp), lim ET = lim T km+1+i) a T n. 38) that will be used late. We ecall that k m = m lcmp), min a j=1,..., Poof. Poof of i). This is an immediate consequence of 31). λk = p λ. 39) icx T km+i). 40) Poof of ii). Fo all i = 1,..., lcmp), let j i ag min j=1,..., a i) i. Then, fom 30), we get ji a km+i) ag min, m > 1. 41) j=1,..., k m + i In view of Lemma 1, the convegence in ii) holds if we can show that T km+i) i = 1,..., lcmp). Given 41), this amounts to show that a km+i) j i, T km+i) n P m p λ, fo all P p m λ, i = 1,..., lcmp). 42) We pove the fome by showing the stonge statement) that a T n P p λ, fo all j. Now, using that { X c > 2ɛ} { X EX > ɛ} { EX c > ɛ} fo a andom vaiable X, we have P a T n ) p p ɛ λ P a T n a P p p E λ a E ) T n ɛ + ) T n ɛ. The second tem in the ight-hand side of fome inequality tends to zeo as k by 31). The following shows that also the fist tem goes to zeo: P a T n a E ) T n ɛ 1 ɛ 2 12 a Va T n 44a) 1 ɛ 2 a o1/k) 0. 44b)

14 In 44a), we have used Chebyshev s inequality and that the T n s ae independent. In 44b), we have used that a k p. Poof of iii). We use that X icx Y if and only if thee exists an othe andom vaiable Z such that X st Z and Z cx Y [28]. Using 29) and that akm+i) j i, k p m+i) fo all m by 30)), the fist obsevation is that a km+1+i) j i, a km+i) j i, + lcmp) akm+i) ji, k m + i, 45) whee j i is defined above in point i). Thus, we have T km+1+i) = st st a k m+1 +i) j i, a km+i) j i, T km+1+i) n a km+i) j +lcmp) i, km+i T km+1+i) n Now, it emains to show that Z cx T km+i), which is equivalent to show that see [33, Theoem 3.A.12]. Since EZ = 1 λ a km+i) j i, 46a) def = Z. 46b) Z cx T km+i), 47) + lcmp) a km+i) j i, k m+i k m + i + lcmp) = 1 λ a km+i) j i, k m + i = ET km+i), 48) 47) holds tivially unde Case 3. Now, let Case 2 hold. Noticing that both T km+i) and Z have Elang distibutions with the same mean, to pove 47) is enough to show that Va Z Va T km+i) ; see [35, p. 14]. We have as desied. m+i) λ 2 Va Z = ak ji, + lcmp) a km+i) j i, k m+i k m+1 + i) 2 49a) m+i) = ak ji, k m + i)k m + i + lcmp)) k m + i) 2 k m + i + lcmp)) 2 49b) m+i) ak ji, k m + i) 2 = λ2 Va T km+i) We now pesent an agument ) that allows us to unifomly bound the second moment of W def Let δ = 1 p 2 p 1 λ µ and Lemma 4. k <. { k def = min k > 0 : 0 p p min a λ j=1,...,,κ. 49c) } k ) k λ δ, k k. 50) 13

15 Poof. This is immediate because δ > 0 by hypothesis, a min j=1,..., kλ fo all k. k lim min a ) k j=1,...,p k λ = p λ p see 39)), and λ Let W whee T n, = st T denote the stationay waiting time of a GI/GI/1 queue with i.i.d.) inteaival times T n,) n N see 38)) and sevice times S n,κ,) n N. By coupling the T n, s and the T n,κ, s in the obvious manne, one can easily see that T 1,,..., T n,) T 1,κ,,..., T n,κ,) 4 and theefoe we have T 1,,..., T n,) st T 1,κ,,..., T n,κ,). Using, e.g., [6, pp. 217, 220], this implies Futhemoe, given that T km+1+i) n, icx T km+i) n, T n,) n N ae independent, we can use [5, p. 337] to establish that fo all i = 1,..., lcmp). Theefoe, given E W,κ st W. 51) W km+1+i) the second moment of W,κ as follows sup E W,κ ) 2) sup E W k k k k fo all i = 1,..., lcmp) by Lemma 3) and that the icx W km+i), 52) W,κ ) 2 ) < fo all k and Lemma 4, we can unifomly bound = max ) 2) 53a) W km+i) ) 2) 53b) sup E i=1,...,lcmp) m:k m+i k = max E i=1,...,lcmp) W k m i +i) ) 2) 53c) <, 53d) whee m def i = min{m : k m + i k }. In 53a) and 53c), we have used 51) and 52), espectively. In 53d), we have used that ET n, = a min j=1,..., kλ p λ δ = 1 2 p p λ 2 µ > 1 µ, k k, 54) i.e. the egodicity condition, and that the thid moment of sevice times is finite, which imply that the second moment of W is finite [5, pg. 270]. Now, using the continuity of the stationay waiting time of GI/GI/1 queues [5, Coollay X.6.4] and pat i) and ii) of Lemma 3, we have W d W p), and given the unifom integability 53) we have that also the expected values convege, i.e., lim EW = EW p). 55) With the above elations, we can conclude the poof of 36) lim EW π ) = lim lim κ EW,κ π ) p k EW p 56a) 56b) 4 Given x, y R d, hee x y means x i y i fo all i = 1,..., d. 14

16 = p EW p). 56c) Poof of 17). Let Q be a discete andom vaiable with values in {1,..., R} {1,..., k} such that PQ =, κ)) =. We assume that this andom vaiable is independent of any othe andom vaiable. By p p k definition of W π ) and using the law of total vaiance, we obtain Va W π ) = EVa W π ) Q) + Va EW π ) Q) Va W,κ π ) + =,κ p k EW,κ π ) EW π ) 57a) ) ) 2. 57b) When k, we have aleady established that EW π ) EW p) and that EW p) EW,κ π ) EW,κ EW p). Theefoe, it only emains to show that the second moment of W,κ π ) convege to E[W p) 2 ]. This is done by using the same agument above fo the convegence of the fist moment. Hence, using the continuity of the waiting time and of the squae function [5, Coollay X.6.4] and pat i) and ii) of Lemma 3, we obtain W ) 2 d W p) 2, as k. Futhemoe, the second moment of W ) 2 is finite because the fifth moment of the sevice times is finite [5, pg. 270] and 52) ensues that the sequence W ) 2 is unifomly integable because it is non-inceasing along subsequences k m + i) m N, fo all i = 1,..., lcmp). Thus, E[W ) 2 ] E[W p) 2 ]. Togethe with 18), as desied we obtain 5 Conclusions lim E[W,κ π )) 2 ] = E[W p) 2 ]. 58) We have deived stuctual popeties concening Poblem 1. Fixing the popotion of jobs to send on each queue, p, we have shown that peiodic policies ae asymptotically equivalent and optimal. The limiting mean waiting time, EW p) see 8)), is expessed in tems of independent D/GI/1 queues and has the convenient popety of being convex in p. We believe that these stuctual popeties povide eseaches and pactitiones with new means about the consideed poblem. Fo instance, one consequence of these esults is that the poblem of computing the optimal popotions of jobs to send to each queue, which is consideed a difficult poblem see the Intoduction), can be captued by the solution of an optimization poblem of the fom: min EW p) s.t.: p S, fo S compact and convex, and we stess that EW p) is a convex function of p. In the case whee sevice times have an exponential distibution, EW p) admits a simple chaacteization because it is the weighted mean waiting time of R D/M/1 queues, which can be computed efficiently [9]; see also [4]. As discussed in Section 3.1, the policies in C p can be implemented in a distibuted manne in a two-tie hieachy. Hee, ou esults imply that the dispatche in the fist tie has a negligible influence on jobs waiting times, which means that it does not need to have full infomation about the statistics of netwok esouces. Refeences [1] E. Altman, B. Gaujal, and A. Hodijk. Balanced sequences and optimal outing. J. ACM, 474): , July [2] E. Altman, B. Gaujal, and A. Hodijk. Multimodulaity, convexity, and optimization popeties. Math. Ope. Res., 252): , [3] J. Anselmi and B. Gaujal. Optimal outing in paallel, non-obsevable queues and the pice of anachy evisited. In Intenational Teletaffic Congess, pages 1 8,

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