State Dependent Control of Closed Queueing Networks

Transcription

1 Submitte to Stochastic Systems manuscript MS Authors are encourage to submit new papers to INFORMS journals by means of a style file template, which inclues the journal title. However, use of a template oes not certify that the paper has been accepte for publication in the name journal. INFORMS journal templates are for the exclusive purpose of submitting to an IN- FORMS journal an shoul not be use to istribute the papers in print or online or to submit the papers to another publication. State Depenent Control of Close Queueing Networks Authors names bline for peer review) We stuy the esign of state-epenent control for a close queueing network moel inspire by share transportation systems such as those for riesharing an bike-sharing. We focus on the assignment policy, where the platform can choose which supply unit to assign when a customer request comes in, an assume that this is the exclusive control lever available. The supply unit once again becomes available at the estination after ropping the customer. We consier the proportion of roppe eman in steay state as the performance measure. We propose a family of simple an explicit state-epenent policies calle Scale MaxWeight SMW) policies an prove that uner the complete resource pooling CRP) conition analogous to the conition in Hall s marriage theorem), each SMW policy leas to exponential ecay of eman-ropping probability as the number of supply units scales to infinity. We further show that there is an SMW policy that achieves the optimal exponent among all assignment policies, an analytically specify this policy in terms of the customer arrival rates for all source-estination pairs. The optimal SMW policy protects structurally uner-supplie locations. Key wors : maximum weight policy, close queueing network, control, Lyapunov function, share transportation systems, large eviations

2 Article submitte to Stochastic Systems; manuscript no. MS Introuction Recently there is an increasing interest in the control of share transportation systems such as Uber an Lyft. These platforms are ynamic two-sie markets where customers emans) arrive at ifferent physical locations stochastically over time, an vehicles supplies) circulate in the system as a result of riving emans to their estinations. The platform s goal is to maximize throughput proportion of emans fulfille), revenue or other objectives by employing various types of controls. The main inefficiency in such systems comes from the geographic mismatch of supply units an customers: when a customer arrives, he has to be matche immeiately with a nearby supply unit, otherwise the customer will abanon the request ue to impatience. There are two sources of spatial supply-eman asymmetry: structural imbalance an stochasticity. In riesharing, the former is ominant uring rush hours in the city when most of the eman pickups concentrate in a particular region of the city while ropoffs concentrate on others, otherwise the latter source often ominates, see Hall et al. 205). We propose an assignment scheuling) policy that eals with both sources simultaneously if this is possible): the choice of policy parameters accounts for the structural imbalance, while its state epenence nature manages stochasticity optimally. Many control mechanisms have been propose an analyze in the literature. Pricing, for example, enjoys great popularity in both acaemia an inustry, see, e.g., Waserhole an Jost 206), Banerjee et al. 206). By ajusting prices of ries, the system can inirectly re-balance supply an eman. Empty-vehicle routing Braverman et al. 206) focus on sening available supply units to uner-supplie locations in orer to meet more eman. In this paper, we stuy another important form of control, assignment. When a customer request comes in, the platform can ecie from where to assign a supply unit, which will in turn influence the platform s future ability to fulfill emans. Previous work has stuie assignment ecisions base on flui limits by optimizing the system on flui scale, the platform can calculate the probability of assigning from any compatible location when a eman arises, an realize it by ranomization Ozkan an War 206) hereafter referre to as flui-base policy). However, this approach requires exact knowlege of customer arrival rates which is infeasible in practice), fails to react to the stochastic variation in the system an creates aitional variance ue to ranomization. Although this control guarantees asymptotic optimality in the Law of Large Numbers sense, it converges only slowly to the flui limit Banerjee et al. 206). One might expect that by implementing flui-base policy in close loop i.e. allow resolving once in a while, hereafter referre to as resolving flui-base policy) it will perform better, but in our setting where the

3 2 Article submitte to Stochastic Systems; manuscript no. MS objective is minimizing roppe eman) this is not the case. To counter these issues, we stuy the state-epenent assignment control of close queueing networks 2. We moel the system as a close queueing network with n servers representing physical locations, an K jobs that stan for supply units. This is a common moel for riesharing systems, see for example Banerjee et al. 206), Braverman et al. 206), Waserhole an Jost 206). For each location i there are some compatible supply locations that are close enough, from where the platform can assign supply units to serve customer at i. Customers arrive at the system stochastically, each has a estination in min. Each time a customer arrives, the platform makes an assignment ecision from a compatible supply location base on the current spatial istribution of available supplies. After a supply unit picks up a customer, it rops her at the estination an becomes available again. Supplies o not enter or leave the system.) The platform s goal is to maintain aequate supply in all neighborhoos an hence meet as much customer emans as possible, therefore we aopt the global) proportion of roppe emans as a measure of efficiency. For the formal escription of our moel, see section 2. To stuy state-epenent spatial rebalancing of supply while keeping the size of the state space manageable, we make a key simplification we assume that pickup an service of customers are both instantaneous. This allows us to get away from the complexity of tracking the positions of in transit supply units, while retaining the essence of our focal challenge, that of ensuring that all neighborhoos have supplies at almost) all times. To obtain tight characterizations, we further consier the asymptotic regime where the number of supply units in the system K goes to infinity. It s worth noting that the large supply regime eman arrival rates stay fixe while K ) an the large market regime eman arrival rate scales with K as K ) are equivalent in our moel. The reason is that we assume instantaneous relocation of assigne supply units, hence the large market regime is simply a spee-up of the infinite supply regime. A main assumption in our moel is the complete resource pooling CRP) conition. CRP is a stanar assumption in the heavy traffic analysis of queueing systems see e.g. Harrison an López 999, Dai an Lin 2008, Shi et al. 205). It can be interprete as requiring enough overlapping in the processing ability of servers so that they form a poole server. For the moel consiere in this paper, the CRP conition is closely relate to the conition in Hall s marriage theorem in bipartite matching theory. We show that the CRP conition is necessary for any assignment policy to have eman ropping probability that converges to zero. In riesharing the eman arrival rates are enogenously etermine by system manager s pricing/amission control ecisions In fact, any static flui-base policy that keeps the number of emans at each location unchange in the flui limit is also an optimal resolving flui-base policy. 2 Note that {flui-base policy} {resolving flui-base policy} {state-epenent policy}.

4 Article submitte to Stochastic Systems; manuscript no. MS though we on t focus on that here). If CRP conition oesn t hol for base prices, the platform may use spatially varying prices as in Bimpikis et al. 206), Banerjee et al. 206) to make it true, an then use our assignment control to manage the unpreictable fluctuations in the system. One key ifficulty in the analysis is the necessity to eal with a multi-imensional system in the limit. In many existing works that seek to minimize the workloa process or holing costs of a queueing system, asymptotic optimality of a certain policy relies on collapse of the system state to a lower imensional space in the heavy traffic limit uner the CRP conition. This is not the case for the objective we are consiering, i.e., minimizing eman ropping probability, or maximizing throughput. Since the exponent of eman ropping probability epens on the most likely event that leas to eman ropping event, we nee to protect all the subsets of locations simultaneously. An interesting observation rawn from our analysis is a critical subset property: given the current state, the most likely way a eman may be roppe is via raining a certain critical subset of locations. The critical subset changes with the state of the system. Formulate as classic close queueing network scheuling problem. Using terminology of classic queueing theory, the K supply units are jobs, each eman location is a server, each supply location is a buffer, inter-arrival times of customers with origin i are service times at server i. The istribution of customers estinations given origin captures routing probabilities. Servers are flexible i.e. can serve multiple queues), an assignment is equivalent to scheuling. Our moel eviates a little from classic moel in that a scheule job relocates to estination queue at the start of its service time, instea of the en... Main Contributions As a function of system primitives, we erive a large eviation rate-optimal assignment policy that minimizes eman ropping maximizes throughput). Our optimal policy is strikingly simple an its parameters epen in a natural way on eman arrival rates. Our contribution is threefol:. Achievability: We propose a family of state-epenent assignment policies calle Scale MaxWeight SMW) policies, an prove that all of them guarantee exponential ecay of emanropping probability uner CRP conition. The proof is base on a family of novel Lyapunov functions a ifferent one for each SMW policy) which are use to analyze a multi-imensional variational problem. An SMW policy is parameterize by an n-imensional vector consisting of a scaling factor for each location; each eman is serve by assigning a supply from the compatible location with the largest scale number of cars. We obtain an explicit specification for the optimal scaling factors base on location compatibilities an eman arrival rates. Further, we obtain the surprising fining that the optimal SMW policy is, in fact, exponent-optimal among all state-epenent policies Theorem ). SMW policies are simple, explicit an appear promising for practical applications.

5 4 Article submitte to Stochastic Systems; manuscript no. MS Converse bouns: We provie lower boun of eman ropping probability for any assignment policy using ranom-walk relate inequalities. We first show that no flui-base assignment policy can achieve exponential ecay rate Proposition 4), which emonstrates the value of stateepenent control even a naive state-epenent assignment policy with no knowlege of eman arrival rates beats the best flui-base assignment policy asymptotically. Then we justify the CRP assumption by showing that it is a necessary conition for exponential convergence Proposition ; in fact if any of the inequalities is reverse, a positive fraction of emans must be roppe for that instance even as K ). Finally, we obtain an upper boun on the eman ropping probability exponent for any state-epenent policy that matches the achievable exponent of the optimal SMW policy, thus proving that the best SMW policy is, in fact, exponent-optimal. 3. Qualitative insights: We characterize the system behavior uner SMW policies as K, which is technically challenging since the problem remains n imensional even in the limit. We establish the critical subset property of the problem: given a system state in the limit), there exists a state-epenent) subset J of eman locations that are most likely to be eplete of supply in compatible locations, hence leaing to eman ropping. The optimal SMW policy avois using supply from locations compatible with J for eman arising outsie of J..2. Literature Review MaxWeight scheuling. MaxWeight policy is a simple scheuling policy in constraine queueing networks that exhibits many goo properties. It attaches a weight to each scheule activity u, which is a function of current queue-lengths {f u X)}; in many cases f u X) is simply the queue length at the scheule server source). At each time perio it activates the amissible activities with the largest total weight. MaxWeight scheuling has been stuie intensively since the seminal work Tassiulas an Ephremies 992), which showe MaxWeight with weight efine as a constant multiple of source-estination queue-length ifference) achieves the entire stability region for an open one-pass system. Dai an Lin 2005) showe that MaxWeight with the same choice of weight as Tassiulas an Ephremies 992) achieves throughput optimality in open stochastic processing networks in flui limit uner mil conitions. Stolyar 2004) showe that for one-hop open network, any MaxWeight policy with weight f u X) chosen as a positive multiple of the β-th β > 0) power of source queue length, minimizes workloa weighte sum of queue lengths) uner Resource Pooling RP) conition in iffusion limit. A similar result was obtaine in Dai an Lin 2008), where they showe that for a more general family of open networks, MaxWeight policy with weight efine as in Dai an Lin 2005) minimizes workloa in iffusion limit uner CRP conition.

6 Article submitte to Stochastic Systems; manuscript no. MS Eryilmaz an Srikant 202) showe that for one-hop network, MaxWeight policy with weight f u X) = c sourceu) X sourceu) minimizes the expectation of c i ix i in heavy traffic limit with oneimensional state-space collapse. Maguluri an Srikant 206) showe that for cross-bar switch MaxWeight policy with f u X) = X sourceu) achieves the optimal scaling of total queue-length in heavy traffic with multi-imensional state space collapse. Shi et al. 205) consiere a moel similar to Eryilmaz an Srikant 202) with focus on esign of network topology. In contrast of the many of the above works where asymptotic optimality is guarantee for any choice of constant c sourceu) > 0 in weight f u X), we show that in our moel the coefficients nee to be chosen carefully in orer to achieve optimal exponent, though any choice of positive constants will result in exponential ecay uner CRP conition. For example, vanilla MaxWeight f u X) = X sourceu) ) can result in non-trivial sub-optimality of system performance. This is because for open queueing networks uner CRP conition, though the queue lengths can collapse to ifferent subspaces uner ifferent MaxWeight policies, the workloa process always converges to the same weak limit which is uniquely efine by the moel primitives. In our case, however, ifferent state space collapse usually leas to ifferent system performance exponent); an evaluation of each MaxWeight policy requires analysis of policy-specific most-likely sample paths. There are other variants of MaxWeight policy that are less relate to our work, e.g. Meyn 2009). Large eviations in queueing systems. There is a large literature on characterizing the ecay rate of probability of builing up large queue lengths in open queueing networks. To the best of our knowlege, we are the first to consier large eviation of controlle close queueing networks. Stolyar an Ramanan 200) showe the exponent optimality of Largest Weighte Delay First scheuling in minimizing the probability of waiting times exceeing large values for multi-class single server queueing system, an Stolyar 2003) extene the result to multi-class multi-server queueing networks the choice of weights correspons to the objective at han, in contrast with our work, where the optimal choice of weights will be etermine by the moel primitives). Boas et al. 204) consiere the large eviation optimal scheuling of parallel servers, but the asymptotic regime is ifferent in that they are scaling up the size of network while keeping buffer size fixe. Compare with these works, the ifficulty of analyzing our moel comes from its complex ynamics: supply units circulate in the system enlessly in contrast to one-hop system in Boas et al. 204), an each supply unit can be matche to emans at ifferent noes in contrast to multi-class moel in Stolyar an Ramanan 200, Stolyar 2003). As a result, the techniques use in these works cannot be irectly applie here. Our result is also qualitatively ifferent: in Stolyar 2003) the analysis of queueing networks with arbitrary topology reuces to stuying one noe in isolation, but in our work the optimal exponent inclue terms for each subset of noes.

7 6 Article submitte to Stochastic Systems; manuscript no. MS The closest work to ours is that of Venkataramanan an Lin 203), who establishe the relationship between Lyapunov function an buffer overflow probability for open queueing networks. Our Lyapunov function approach is inspire by their work, an we evise a family of Lyapunov functions such that the ecay rate of eman ropping probability is the same as that of Lyapunov function exceeing certain threshol. The key ifficulty of extening the Lyapunov approach to close queueing networks is the lack of natural reference state where the Lyapunov function equals to 0 in open queueing network it s simply 0). It turns out when optimizing the MaxWeight parameters we are also solving for the best reference state. There are also works that stuy the large eviation behavior of queueing networks without control aspect, see e.g. Majewski an Ramanan 2008), Blanchet 203). Share Transportation Systems. Optimization of share transportation systems such as those for riesharing an bike-sharing has rawn attention in recent years. Ozkan an War 206) stuie revenue-maximizing flui-base assignment control by solving a minimum cost flow problem in the flui limit. Braverman et al. 206) moele the system by a close queueing network an erive the optimal static routing policy that sens empty vehicles to uner-supplie locations. Banerjee et al. 206) aopte Goron-Newell close queueing network moel an consiere static pricing policy that maximize throughput, welfare or revenue. In contrast to our work, which stuies state-epenent control, these works consier static control that completely relies on system parameters. In terms of convergence rate to the flui-base solution, Ozkan an War 206) i not show the orer of convergence rate of their policy, Braverman et al. 206) prove an O/ K) rate as number of supply units in the close system K goes to infinity, an Banerjee et al. 206) showe an O/K) convergence rate as K. When CRP conition hols, we show that no fluibase policy can o better than O/K 2 ), while our state-epenent policy achieves an Oe γk ) convergence rate with optimal γ > 0. We remark briefly that our CRP conition has some similarity with the notion of balanceness in Bimpikis et al. 206), although balanceness requires an exact balance between inflows an outflows for each location 3, whereas the CRP conition oes not, ue to the flexibility from having multiple compatible supply locations for a eman location. There are many other works that also stuie riesharing platforms but focus on pricing aspects, see, e.g., Aelman 2007), Bimpikis et al. 206), Waserhole an Jost 206), Cachon et al. 207), Hall et al. 205). Online Stochastic Bipartite Matching. There is a stream of research on online stochastic bipartite matching OSBM) that are relate to our work, see e.g. Calentey et al. 2009), Aan 3 To compare with our close network moel, we consier here the special case of the moel in Bimpikis et al. 206) where rivers never exit or enter, i.e., their parameter β =.

8 Article submitte to Stochastic Systems; manuscript no. MS an Weiss 202), Bušić an Meyn 205), Mairesse an Moyal 206). In OSBM, ifferent types of supplies an emans arrive over time, an the system manager matches supplies with emans of compatible types using some matching policy then ischarge the matche pairs from the system. Many matching policies consiere in the literature have the flavor of MaxWeight, if not exactly the same: in Calentey et al. 2009) an Aan an Weiss 202) emans are matche to the olest compatible supplies available, Mairesse an Moyal 206) consiere match the longest policy. Our moel is ifferent from OSBM in the following ways: )our moel is close hence supply units never leave enter or leave the system, while in OSBM the network is open; 2)many OSBM works focus on the stability an other properties uner a given policy instea of looking for the optimal control, e.g. Calentey et al. 2009), Aan an Weiss 202), Mairesse an Moyal 206); 3)in terms of ynamics, in Calentey et al. 2009), Bušić an Meyn 205) exactly one supply an one eman enter the system each time, which is ifferent from our moel..3. Organization The remainer of our paper is organize as follows. In section 2 we introuce the basic notation use throughout the paper an formally escribe our moel together with performance measure. Some backgroun on sample path large eviation principle will also be provie. In section 3 we introuce the family of Scale MaxWeight policies. In section 4 we present our main theoretical result, i.e., exponent optimality of SMW policy. In section 5 we prove the achievability bouns of of SMW policies. In section 6 we prove the converse bouns an show the exponent optimality of SMW policy. In section 7 we conclue the paper with further iscussion of results an future irections. We also present there the converse bouns for flui-base policies an cases where CRP conition is violate. 2. Moel an Preliminaries 2.. Notation Wherever possible, we reserve capital letters for ranom quantities an small letters for their realizations; we also use bolface letters to inicate column vectors. We use e i to enote the i-th unit vector, an the all- vector. If vector b is strictly larger than a component-wise, we write b > a. For inex set A, efine A i A e i. For a set Ω in Eucliean space R n, enote its relative interior by relintω). We use Bx, ɛ) to enote a ball centere at x R n with raius ɛ > 0. For event C, we efine the inicator ranom variable {C} to equal when C is true, else Basic Setting Unerlying Moel an Simplifications: We moel the share transportation system as a finitestate Markov process, comprising of a fixe number of ientical supply units circulating among n

9 8 Article submitte to Stochastic Systems; manuscript no. MS noes i.e., a given partition of a city into neighborhoos). Customers i.e., prospective passengers) arrive at each noe i with esire estination j accoring to inepenent Poisson processes with rate ˆφ ij. To serve an arriving customer, the platform immeiately assigns a supply unit from a neighboring station of i i.e., one among a set of nearby stations, efine formally below), an subsequently, after serving the customer, the supply unit becomes available at the estination noe j. If however there are no supplies available in the neighboring noes of i, then we experience a eman rop, wherein the customer leaves the system without being serve. Customers o not wait. The aim of the platform is to assign supplies so as to minimize the fraction of emans roppe. Intuitively, to achieve this objective, the platform shoul ensure that it maintains aequate supply in or near) all neighborhoos, i.e., it nees to manage the spatial istribution of supply. To stuy the esign of assignment rules in the above moel, we make two simplifications. First, we assume that pickup an service are instantaneous. This allows us to reformulate the above moel as a iscrete-time Markov chain the so-calle jump chain of the continuous-time process), where in each time-slot t N, with probability proportional to ˆφ ij, exactly one customer arrives to the system at noe i with esire estination j. The customer is then serve by an assigne supply unit from a neighboring station of i, which then becomes available at noe j at the beginning of time-slot t +. This simplification removes the high-imensionality require for tracking the positions of all in-transit supply units, while still retaining the complex supply externalities between stations, which is the hallmark of riesharing systems. Unfortunately, however, even after this simplification, the setting still oes not amit any amenable way to characterize the performance of complex assignment policies. To circumvent this we make a secon simplification, we stuy the performance of assignment policies as the number of supply units K grow to infinity, while fixing all other parameters. Formal System Definition: We efine φ R n n to be the arrival rate matrix with a row for each origin an a column for each estination, normalize 4 such that T φ =. We enote the i-th column i.e., the arrival rates from ifferent origins of customers to estination i) as φ i), an the transpose of the i -th row of the arrival matrix i.e., the arrival rates of customers to noe i with ifferent estination noes) as φ i. Thus, the probability a customer arrives at noe i is T φ i, an, assuming all customers are matche, the rate of supply units arriving at noe i is T φ i). As mentione above, we consier a sequence of systems parameterize by the number of supplies K. For the K-th system, its state at any time t N is given by X K [t], a vector that tracks the 4 This can always be achieve by appropriately re-scaling the arrival rates { ˆφ ij}, which prouces an equivalent setting since pickups an ropoffs are instantaneous.

10 Article submitte to Stochastic Systems; manuscript no. MS T φ ) T φ T φ 2) 2 2 T φ 2 T φ n) n n T φ n Figure The bipartite compatibility graph for the assignment problem: On the left are supply noes i, an on the right are eman noes i. Customers arrive to i with istributions chosen accoring to φ i ; the total probability of a customer arrival at i is thus T φ i. Similarly, assuming no eman is roppe, the total probability a supply unit arrives at i is T φ i). The eges entering a noe i encoe compatible i.e., nearby) noes that can supply noe i. number of supplies at each location in time-slot t. The state space of the K-th system is thus given by: Ω K {x R n T x = K} N n. Note that the normalize state X K /K lies in the n-simplex Ω = {x R n x 0, T x = }. Henceforth, we rop explicit epenence on t when clear from context Assignment Policies an System Dynamics The main iea behin the assignment problem in riesharing is that most arriving customers have a maximum tolerance say 7 minutes) for the pickup time or ETA i.e., expecte time of arrival) of a matche supply unit, but are essentially inifferent if the ETA is less than that. Thus when a customer arrives at a noe i, then any supply unit locate at a noe which is within 7 minutes of i is a feasible match, while other vehicles which are further away are infeasible. This suggests a natural moel for assignment via a bipartite compatibility graph, as epicte in Figure. Compatibility Graph: For peagogical reasons, we move to a setup where we istinguish formally between eman locations V D where customers arrive an supply locations V S where supply units wait an where customers are roppe off). We a a prime symbol to the inices of noes in V D to istinguish between the two.

11 0 Article submitte to Stochastic Systems; manuscript no. MS We encoe the compatibility graph as a bipartite graph GV S V D, E), wherein each station i V is replicate as a supply noe i V S an a eman noe i V D. An ege i, j ) E represents a compatible pair of supply an eman noes, i.e., supplies statione at i can serve eman arriving at j. We enote the neighborhoo of a supply noe i V S resp. eman noe j V D ) in G as i) resp. j )); thus, for a supply noe i, its compatible emans are given by i) = {j V D i, j ) E}, an similarly for each eman noe. Moreover, for any set of supply noes A V S, we also use A) to enote its eman neighborhoo an vice versa). We make some mil assumptions on arrival rates φ: Assumption The following hols:. Connecteness: Matrix [φ ij ] is irreucible Non-triviality: There exists an origin-estination pair i V D an j V S such that j / i ) an φ i j > 0. Remark The irreucibility assumption is useful for charactering the system s steay state behavior uner our policies. The secon assumption is mae to ensure that the assignment control problem at han is non-trivial. If it is violate, an if the system starts with at least one supply unit in each location, then we can reserve a supply unit for serving each eman origin location i V D, an each such car will never leave the corresponing neighborhoo i ), ensuring that no eman is ever roppe.) Assignment policies: Given the above setting, the problem we want to stuy is how to esign assignment policies which minimize the probability of ropping eman. For fixe K, this problem can be formulate as an average cost Markovian ecision process on finite state space, an is thus known to amit an optimal stationary policy i.e., ones where assignment ecisions at time t only epen on the system state X K [t]; see Proposition 5..3 in Bertsekas 995)). Let U K be the set of stationary policies for the K-th system. For each t N, i V D, an assignment policy U U inclues a series of mappings U K U K for each system K =, 2,, which maps the current queue-length to U K [X K [t]]i ) i ) { }. Here U K [X K [t]]i ) = j enotes that given the current state X K [t], we assign a supply unit from j i ) to fulfill eman at i, an U K [X K [t]]i ) = means that the platform oes not assign supplies to i an hence any arriving eman at i is roppe. For simplicity of notation, we refer to the policies by U instea of U K. 5 Replacing non-zero entries in the matrix by one, an viewing the matrix as the ajacency matrix of a irecte graph, the matrix is irreucible if an only if this irecte graph is strongly connecte.

12 Article submitte to Stochastic Systems; manuscript no. MS System Evolution: We can now formally efine the evolution of the Markov chain we want to stuy. At the beginning of time-slot t, the state of the system is X K [t ]; note that this incorporates the state-change ue to serving the eman in time-slot t. Now suppose the platform uses an assignment policy U, an in time-slot t, a customer arrives at origin noe o[t] with estination [t] chosen from arrival matrix φ). If U K [X K [t]]o[t]), then a supply unit from U K [X K [t]]o[t]) will pick up the eman an relocate to [t] instantly. Let S[t] U K [X K [t]]o[t]) be the chosen supply noe potentially ). Formally, we have { X K XK [t ] e [t] = S[t] + e [t], if S[t] V S. X K [t ], if S[t] = Performance Measure The platform s goal is to fin an assignment policy that rops as few emans as possible in steaystate. In this section we formally efine the performance measure for all assignment policies. We first introuce some necessary notations. For each subset A V S resp. A V D ), efine: A c {v V S : v / A} resp. A ) c {v V D : v / A }). ) Define the scale state as X K K XK Ω. Now for a given state x Ω, we efine the set of empty stations as We now have the following simple observation: I em x) {v V S : x v = 0}. Observation If the scale state at time t is X K [t] = x, then 6 P[eman roppe at t + ] = i : i ) I emx),j V S φ i j = T {i : i ) I emx)}φ. This follows from observing that eman arriving in perio t to stations in {i : i) I em X K [t])} must be roppe. Given the above setting, a natural performance measure is the long-run average eman-rop probability. Formally, for U U we efine P K,U o min X K,U [0] Ω K E lim T T T t=0 ] [ T{i: i) IemX φ K,U [t])}, 2) 6 The equality hols for non-iling policies, i.e. policies that on t rop a customer at i whenever there are supplies in i). For other policies, the equality becomes large than or equal to. This oesn t affect our result because )for achievability result, the policies we propose are non-iling; 2)for converse benchmark, the inequality will only make our result stronger.

13 2 Article submitte to Stochastic Systems; manuscript no. MS P K,U p max X K,U [0] Ω K E lim T T T t=0 ] [ T{i: i) IemX φ K,U [t])}. 3) Here 2) is an optimistic subscript o for optimistic) performance measure uner-estimates eman-rop probability), 3) is a pessimistic subscript p for pessimistic) performance measure over-estimates eman-rop probability). When establishing the optimality of our policy, we will compare its pessimistic measure against other policies optimistic measure, so we are not cheating. Since U K U K is a stationary policy, the limits in 2) an 3) exist. We will rop the expectation sign. One issue however is that the exact expression of 2) an 3) may be complicate for a fixe K. To this en, the main performance measures of interest in this work are the ecay rates of P K,U o an P K,U p as K : γ o U) = lim inf K γ p U) = lim sup K K,U log Po, 4) K K,U log Pp. 5) K For brevity, we henceforth refer to these as the eman-rop exponents. Again the efinition is suite to strong converse results, whereas for our positive results the lim sup will, in fact, be the limit, an hence the same as the lim inf. For any given policy U, the eman-rop exponent can be simplifie further in terms of the long-run average behavior of X K,U [t]. Now let D A {x Ω : x i = 0 for i A, x j > 0 for j / A}, an let A = {A V S : i V D such that A i )} an let D A A D A. In wors, D is the set of normalize) states satisfying the following requirement that there is at least one location where eman arrives with positive rate which is currently starve of supply at compatible locations. The eman-rop exponent can now be simplifie by the following lemma: Lemma γ o U) = lim inf K γ p U) = lim sup K K log K log min X K,U [0] Ω K max X K,U [0] Ω K lim inf T T lim sup T T T { XK,U [t] D } a.s. 6) t=0 T { XK,U [t] D } a.s. 7) Lemma says that the probability of eman ropping has the same exponential ecay rate as the probability that there exists a noe without any supply unit at a compatible location. The key t=0

14 Article submitte to Stochastic Systems; manuscript no. MS iea of the proof is to boun the ratio of the two probabilities by a constant that oesn t scale with K. Finally, the benchmark we use over all policies is: since no policy can achieve a larger eman-rop exponent Sample Path Large Deviation Principle max γo U) ), 8) U U Our result relies on classical large eviation theory, which we briefly introuce in this subsection. For each fixe K N +, efine ĀK [ ] L [0, T ]) n2 where ĀK [0] = 0. For t = /K, 2/K,..., KT /K, Ā K ij [t] K Kt τ= {o[τ] = i, [τ] = j}. Ā K [t] is efine by linear interpolation for other t s. Here ĀK [t] is the flui-scale accumulate eman arrival process of the K-th system. Let µ K Ā Ā []: be the law of ĀK [ ] in L [0, T ]) n2. Let Λλ) be the cumulant generating function of n Λλ) log Ee λ,ā = log i= j= n φ ij e λ ij λ R n n. Let Λ f) be the Legenre-Fenchel transform of Λ ), then { Λ DKL f φ) if f 0, f) sup λ, f Λλ) = T f = λ R n otherwise. Here D KL f φ) is Kullback-Leibler ivergence efine as: n n D KL f φ) = f ij log f ij. φ ij i= j= For any set Γ, efine Γ as its closure, Γ o as its interior. Below is the sample path large eviation principle a.k.a. Mogulskii s theorem) see Dembo an Zeitouni 998): Fact For measures {µ K } efine above, an any arbitrary measurable set Γ L [0, T ]) n2, we have inf I Ā Γ o T Ā) lim inf K where the rate function is: K log µ KΓ) lim sup K K log µ KΓ) inf I Ā), T Ā Γ { T I Ā) = T 0 Λ tāt)) t, if Ā ) AC[0, T ], Ā0) = 0, otherwise. Here AC[0, T ] is the space of absolutely continuous functions on [0, T ].

15 4 Article submitte to Stochastic Systems; manuscript no. MS Remark 2 Since absolutely continuous functions are ifferentiable almost everywhere, the rate function is well-efine. 3. Scale MaxWeight Policies We now introuce the family of scale MaxWeight SMW) policies. The traitional MaxWeight policy hereafter referre to as vanilla MaxWeight) is a ynamic scheuling rule that allocates the service capacity to the queues) with largest weight where weight can be any relevant parameter such as queue-length, sum of queue-lengths, hea-of-the-line waiting time, etc.). In our setting, vanilla MaxWeight woul correspon to assigning from the compatible location with most supplies with appropriate tie-breaking rules). The popularity of MaxWeight scheuling stems from the fact that it is known to be optimal for ifferent metrics in various problem settings e.g., Stolyar 2003, 2004, Shi et al. 205, Maguluri an Srikant 206). However, in our setting, vanilla MaxWeight is suboptimal an further, we will show that it oes not achieve the optimal exponent). The suboptimality of vanilla MaxWeight can be seen from the following simple example. Example Consier a network with two noes {, 2}, compatibility graph G = V S V D, E) = {, 2} {, 2 }, {, 2, 22 }) an eman arrival rates φ = 3/8, φ 2 = /8, φ 2 = φ 2 2 = /4, as shown in Figure 2. Suppose at time t we have X [t] > X 2 [t] an a eman arrives at noe 2. Uner vanilla MaxWeight policy, we woul assign from noe since there are more supply units there. However, we claim that vanilla MaxWeight is ominate in terms of minimizing eman ropping probability) by another policy where one always assign from noe 2 to serve eman at 2 as long as X 2 > 0. We call this policy the priority policy. To see this intuitively, note that uner both policies eman at noe 2 will never be roppe, hence eman ropping happens if an only if supply at noe is eplete. The priority policy tries to keep all the servers at noe while vanilla MaxWeight tries to equal the number of servers on both noes, hence the priority policy rops less eman. In fact, as we will show formally later, the exponent of eman ropping uner the priority policy is twice as large as the exponent uner vanilla MaxWeight. To eal with this issue, we slightly generalize vanilla MaxWeight by attaching a positive scaleown parameter w i to each queue i V, an assign from the compatible queue with largest weighte queue length X i /w i. Without loss of generality, we normalize w s.t. T w =, or equivalently, w relintω). We call this family of policies Scale MaxWeight SMW) policies, an enote SMW with parameter w as SMWw). Going back to Example, we can approximate the priority policy by attaching a much larger scale-own parameter at noe than at noe 2.

16 Article submitte to Stochastic Systems; manuscript no. MS Supply Deman X t) φ = 3/8 φ 2 = /8 X 2 t) 2 2 φ 2 = /4 φ 2 2 = /4 Figure 2 An example of the sub-optimality of vanilla MaxWeight policy. The formal efinition of SMW is as follows. Definition Scale MaxWeight) Given system state X[t ] at the start of t-th perio an for eman arriving at o[t], SMWw) assigns from argmax i o[t]) X i [t ] w i X if max i [t ] i o[t]) w i > 0; otherwise the eman is roppe. If there are ties when etermining the argmax, assign from the location with highest inex. The following fact which we formalize in Section 5, on the way to proving our main result) gives some intuition about SMW policies. Remark 3 Resting point of state uner SMWw)) If Assumption 2 hols, the SMWw) policy causes the normalize system state X K /K to rift towars w, where all the scale queue lengths are equal to ). 4. Optimality of Scale MaxWeight In this section we present our main result. 4.. Complete Resource Pooling Conition The following is the main assumption of this paper. Assumption 2 We assume that for any J V D where J, T φ i) > T φ j. 9) i J) j J

17 6 Article submitte to Stochastic Systems; manuscript no. MS The intuition behin this assumption is clear: it assumes the system is balanceable in that for each subset J V D of eman locations we have enough supply at neighboring locations to meet the eman. Assumption 2 is equivalent to a strict version of the conition in Hall s marriage theorem. It is also closely relate to Complete Resource Pooling CRP) conition in queueing literature Harrison an López 999, Stolyar 2004, Ata an Kumar 2005, Gurvich an Whitt 2009). This assumption marks the limit of assignment policies no assignment policy can achieve exponentially ecaying eman ropping probability when Assumption 2 is violate. if the sign of the inequality is reverse in 9) for any subset J, then it is easy to see that, in fact, an Ω) fraction of eman must be roppe uner any policy). Proposition For any G an φ s such that Assumption 2 is violate, it hols that for any policy U, the eman ropping probability oes not ecay exponentially, i.e., γ o U) = γ p U) = 0 where γ o U) an γ p U) was efine in 4) an 5). In fact, if the inequality 9) in Assumption 2 is strictly reverse for some J V D, then we have lim sup K P K,U p ɛ = j J T φ j i J) T φ i). lim inf K P K,U o ɛ > 0 for In other wors, if Assumption 2 is violate, this means the system has significant spatial imbalance of eman an stronger forms of control like pricing or repositioning shoul be employe to restore spatial balance Rate Optimality of Scale MaxWeight In the flui limit, it is easy to fin an assignment rule such that no eman is roppe see, e.g., Banerjee et al. 206). Our goal here is to approach this utopian worl as fast as possible as the number of supply units K grows. Define J J V D : i J j / J) φ i j > 0, 0) we have J by Assumption. The following result characterizes the rate at which the fraction of roppe eman falls as K. Theorem Uner Assumptions an 2, the following statements are true:. Achievability: For any w relintω), SMWw) achieves exponential ecay of the eman ropping probability with exponent 7,8 ) γw) = min T J)w log J J i / J i J j J) φ i j j / J) φ i j ) > 0. ) 7 We emphasize that for SMW policies, the lim inf an lim sup in 4) an 5) are limits, an we show that they are equal. Thus, we are not cheating on either count. 8 Note that the argument of the logarithm has a larger numerator than enominator for every J V D since Assumption 2 hols, implying that γw) is the minimum of several positive numbers, an hence is positive. Also see Remark 4.)

18 Article submitte to Stochastic Systems; manuscript no. MS Converse: Uner any policy U, it must be that where γ p U) γ o U) γ, γ = sup w relintω) γw). 2) In wors, there is an SMW policy that achieves an exponent arbitrarily close to the optimal one. We will prove claim of the theorem in section 5.3, claim 2 in section 6.. The first part of the theorem states that for any SMW policy with w relintω), the policy achieves an explicitly specifie positive exponent γw) such that the eman ropping probability ecreases at the rate Θe γw)k ) as K. The secon part of the theorem provies a universal upper boun γ on the exponent that any policy can achieve, i.e., for any assignment policy U, eman ropping probability must be Ωe γ K ). Crucially, γ is in fact ientical to the supremum over w of γw). In other wors, there is an almost) exponent optimal SMW policy, an moreover, this policy can be obtaine as the solution to an analytically specifie optimization problem. We note that this result is somewhat ifferent from the numerous results showing optimality of maximum weight matching in various open queuing network settings. Here, our objective is a natural objective that is symmetric in all the queues. Our result says that there is an optimal maximum weight policy, but it is not symmetric; rather, it is asymmetric via specific scaling factors in a way that optimally accounts for the primitives of the setting at han by protecting structurally unersupplie locations. The following remark provies some intuition regaring the expression for γw). Remark 4 Intuition for γw)) Consier the expression for γw) in ). It is a minimum of a robustness terms for each subset J J of eman locations. For subset J, the robustness of SMWw) s ability to serve eman arising in J is the prouct of two terms: Robustness arising from w: At the resting point w see Remark 3) of SMWw), the supply at neighboring locations is T J) w), an the larger that is, the more unlikely it is that the subset will be eprive of supply. Robustness arising from excess supply: The logarithmic term captures how vulnerable that subset is to being raine of supply. The numerator of the argument of the logarithm can be interprete as maximum 9 average rate at which supply can come in to J) from V S \ J), whereas the enominator can be interprete as the minimum 9 average rate at which supply must go from J) to 9 Which arises if the assignment policy uses supply at J) only to serve eman in J, i.e., the policy is maximally protecting J.

19 8 Article submitte to Stochastic Systems; manuscript no. MS V S \ J), unless eman is roppe the larger the ratio, the more oversupplie an hence robust J is). To see why the term appears in this form, recall that the equilibrium istribution of the length of a stable M/M/ queue is geometric, an hence has an exponentially ecaying tail with the exponent corresponing to logservice rate/arrival rate). The magic here is that SMWw) achieves an exponent such that it suffers no loss from the nee to protecting multiple J s simultaneously. The converse result is somewhat more intuitive, pursuing this reasoning further.) w. Similarly, we also try an give some intuition regaring the optimal choice of the resting state Remark 5 Intuition for optimal w) To evelop some intuition regaring the optimal choice of w, consier informally) the special case of a heavy traffic type setting where there is just one subset J which has a vanishing logarithmic term because it is only very slightly oversupplie, the numerator being only slightly more than the enominator), whereas each other subset of V D has a logarithmic term that converges to some positive number. Then the optimal choice of w will satisfy T J) w, i.e., all but a vanishing fraction of the weight will go to supply locations that can serve J. The intuition is that the ranom walk for the supply at J) has only slightly positive rift even if the assignment protects it, an hence it is optimal to keep the total supply at these locations at a high resting point, to minimize the likelihoo that the supply at these locations will be eplete. We think an optimal policy for such a special case is itself interesting; what is more remarkable is that the optimal policy characterize in Theorem solves the general n-imensional problem consiering all subsets simultaneously. Another feature of our result is the novel Lyapunov analysis for a close queueing network. A key technical challenge we face in our close queueing network setting is that it is a priori unclear what the best state is for the system to be in. This is in contrast to open queueing network settings in which the best state is typically the one in which all queues are empty, an the Lyapunov functions consiere typically achieve their minimum at this state. We get aroun this issue via an innovative approach where we efine a tailore Lyapunov function that achieves its minimum at the resting point of the SMW policy we are analyzing, an use this Lyapunov function to characterize its exponent γw). Moreover, given the optimal choice of w, our tailore Lyapunov function corresponing to this choice of helps us establish our converse result. Our analysis is escribe in the next section.

20 Article submitte to Stochastic Systems; manuscript no. MS Analysis of Scale MaxWeight Policies In this section, we first formally characterize the system behavior uner SMW in flui scale through flui sample paths section 5.) an flui limits section 5.2). In section 5.3 we prove achievability boun of SMW policies base on a novel family of Lyapunov functions an large eviation principle Fact ) of flui sample paths. Finally, we provie the performance guarantee of vanilla MaxWeight in section 5.4. Proofs are inclue in the appenices. 5.. Flui Sample Paths Uner any stationary assignment policy U U efine in section 2, the system ynamics is as follows: X K i [t + ] X K i [t] = {o[t + ] = j, [t + ] = i}{u[x K [t]]j ) } j V D 3) {o[t + ] = k, [t + ] = j}{u[x K [t]]k ) = i}. k i),j V S For SMWw), we write the assignment mapping as U w [ X]i ) for scale queue length X Ω an i V D since it only epens on the scale state. Let A j i[t] be the total number of type j, i) emans arriving in system uring the first t perios t =, 2, ), an let Ā K j i[t] K A j i[kt], XK,U j i [t] K XK,U j i [Kt] 4) for t = 0, /K, 2/K,. For other t 0, ĀK j i [t] an X K,U [t] are efine by linear interpolation. Definition 2 Set of eman arrival sample paths) Define Γ K,T C n2 [0, T ] as the set of all scale eman arrival sample paths of the K-th system for T K perios. Mathematically, any Ā[ ] Γ K,T. Ā[0] = 0. satisfies: 2. For any t = 0, /K, 2/K,, T K /K, Ā i j[t] = Āi j[t + /K] for but one pair i 0, j 0 ) = o[kt + )], [Kt + )]). We have Āi 0 j [t + /K] = 0 Āi 0 j [T ] + /K Ā[t] is efine by linear interpolation for other values of t. Definition 3 Queue-length corresponence of SMWw)) For each given eman arrival sample path ĀK [ ] Γ K,T an initial state X K [0], scale queue length X K [ ] is uniquely recursively) efine by equation 3)4). Denote this corresponence by the mapping: Ψ w : C n2 [0, T ] Ω C n [0, T ] ĀK [ ], X K [0]) X K [ ].

21 20 Article submitte to Stochastic Systems; manuscript no. MS To obtain a large eviation result, we nee to look at the queue-length process at the flui scaling. We take the stanar approach of flui sample paths FSP) see Stolyar 2003, Venkataramanan an Lin 203). Definition 4 Flui sample paths) We call a pair Ā[ ], X[ ]) a flui sample path uner SMWw)) if there exists a sequence ĀK [ ], X K [0], Ψ w ĀK [ ], X K [0])) where ĀK [ ] Γ K,T, X K [0] Ω, such that it has a subsequence which converges to Ā[ ], X[0], X[ ]) uniformly on [0, T ]. Remark 6 Since for any such sequence all elements ĀK [ ], X K [0], Ψ w ĀK [ ], X K [0])) are Lipschitz continuous with Lipschitz constant, it must have a subsequence that converges uniformly to a limit by Arzelà-Ascoli theorem. Meanwhile, uniform convergence passes the Lipschitz continuity to the limit see Ruin 964), hence all FSPs are Lipschitz continuous. Remark 7 One must take extra care when stating results involving FSP. Although each ĀK [ ] uniquely efines a queue-length sample path X K [ ], it is not necessarily true that the Ā[ ] component in FSP uniquely efines the X[ ] component. Theoretically, it is possible that for ifferent X [ ] an X 2 [ ], Ā[ ], X [ ]) an Ā[ ], X 2 [ ]) are both FSPs. Contraction principle see Dembo an Zeitouni 998) can rule out such behaviors, but it s technically challenging to prove it for MaxWeight policies see Subramanian 200, for its proof uner a ifferent setting). As a result, we circumvent this technicality by consiering all FSPs in the following proofs. In brief, FSPs inclue both typical an atypical sample paths. Recall Fact, which gives the likelihoo for any atypical sample path to be realize. The following large eviation analysis in section 5.3 will base on fining the most-likely atypical sample path that leas to eman-rop Flui Limits The flui limits of the system characterizes its behavior on the Law of Large Numbers scale, i.e. for typical sample paths. Except for the fact that our queueing network is close rather than open, the results in this section are similar to Dai an Lin 2005). For the K-th system, enote A K j k [t] as the total number of type j, k) emans arriving in system uring the first t perios, enote the number of times i is assigne to serve type j, k) eman as E K [t]; enote the number of times type ij k j, k) eman being roppe as D K j k [t]. We have the following. Proposition 2 The following hols almost surely. Denote X K [t] = K XK [Kt], Ē K [t] = K EK [Kt], DK [t] = K DK [Kt]