We consider an interacting particle system, which generalizes the classical totally asymmetric simple exclusion process (TASEP), in that each site can contain up to a fixed finite number of particles, and the particle movement is governed by a back-pressure (BP) algorithm (also often called MaxWeight). There are N sites (with N finite or infinite), each may contain at most c particles, 1 ≤ c < ∞. New particles enter the system at the left-most site 1 as a Poisson process of rate α≤ 1, unless site 1 has c particles. Particles (if any) are removed from the right-most site N as a Poisson process of rate β≤ 1. The left-to-right movement of particles between neighboring sites is governed by the BP rule: one particle moves from site n to n+1 at epochs of a rate 1 Poisson process, as long as the former site has strictly more particles than the latter. When c=1, this is the standard TASEP. Our main results address the asymptotics of the stationary distribution of a finite system, and especially the limit of the flux (current) as N→∞. In particular, we prove that interesting non-trivial phase transitions take place in a system with c>1. For example, if c>1 and 1/2 ≤ β≤ 1, the maximum limiting flux 1/4 is achieved as long as α≥ α_c^*, where α_c^* < 1/2 is some non-trivial threshold. (For the standard TASEP the threshold is 1/2.) We also put forward a general conjecture about the stationary distribution asymptotics under an arbitrary parameter setting. We illustrate our formal results and the conjecture by simulations, and identify interesting directions for further research.

We consider a class of multi-agent distributed synchronization systems, which are modeled as n particles moving on the real line. This class generalizes the model of a multi-server queueing system, considered in Stolyar (Stoch. Syst. 12:340–372, 2022), employing so-called cancel-on-completion (c.o.c.) redundancy mechanism, but is motivated by other applications as well. In the multi-server queueing system a particle location represents a server workload. Under c.o.c. mechanism, when a job of class j arrives, it selects $$d_j$$ d j particles uniformly at random, which try to jump forward, by random distances, but their advance is truncated at the new location of the $$k_j$$ k j -th left-most selected particle ( $$k_j \le d_j$$ k j ≤ d j ). Between jumps all particles move to the left at constant speed, but cannot cross point 0 (workload cannot be less than 0). Thus, the multi-server queueing system is modeled as a particle system, regulated at the left boundary point. The more general model of this paper is such that particles evolve the same way as the in left-regulated system, but we allow regulation boundaries on either side, or both sides, or no regulation at all. We consider the mean-field asymptotic regime, when the number of particles n and the job arrival rates go to infinity, while the job arrival rates per particle remain constant. The system state for a given n is the empirical distribution of the particles’ locations. Our results include: the existence/uniqueness of fixed points of mean-field limits (ML), which describe the limiting dynamics of the system; conditions for the steady-state asymptotic independence (concentration, as $$n \rightarrow \infty $$ n → ∞ , of the stationary distribution on a single state, which is necessarily an ML fixed point); the limits, as $$n \rightarrow \infty $$ n → ∞ , of the average velocity at which unregulated (free) particle system advances. In particular, our results for the left-regulated system unify and generalize the corresponding results in Stolyar (Stoch. Syst. 12:340–372, 2022). Our technical development is such that the systems with different types of regulation are analyzed within a unified framework. In particular, these systems are used as tools for analysis of each other.

A service system with multiple types of customers, arriving as Poisson processes, is considered. The system has infinite number of servers, ranked by 1,2,3, …; a server rank is its “location." Each customer has an independent exponentially distributed service time, with the mean determined by its type. Multiple customers (possibly of different types) can be placed for service into one server, subject to “packing” constraints. Service times of different customers are independent, even if served simultaneously by the same server. The large-scale asymptotic regime is considered, such that the mean number of customers r goes to infinity. We seek algorithms with the underlying objective of minimizing the location (rank) U of the right-most (highest ranked) occupied (non-empty) server. Therefore, this objective seeks to minimize the total number Q of occupied servers and keep the set of occupied servers as far at the “left” as possible, i.e., keep U close to Q. In previous work, versions of Greedy Random (GRAND) algorithm have been shown to asymptotically minimize Q/r as r→∞. In this paper we show that when these algorithms are combined with the First-Fit rule for “taking” empty servers, they asymptotically minimize U/r as well.

Recent research provided proof-of-concept that the randomness of lead times in inventory systems can be exploited to achieve large-potentially unlimited-performance improvements, compared to the case of constant lead time. Specifically, the Generalized Base Stock (GBS) policy serves as such proof-of-concept-it can deliver unlimited improvements within a certain class of models, when the ratio of the minimum lead time to the mean lead time can be arbitrarily small. In this paper, we explore what improvements are actually achievable under practical system constraints, most importantly-in discrete-time systems, where the minimum-to-mean lead time ratio is lower bounded by a positive constant; and also, which policies both allow significant improvements and are attractive for practical use. We consider a discrete-time version of GBS and introduce two new discrete-time policies, labeled ADAPTIVE and PIPELINE. We prove the stochastic stability and finiteness of average inventory level under GBS, ADAPTIVE, and PIPELINE policies, in the important special case of bounded lead time. We use simulations to evaluate the performance of the three policies and their dependence on lead time distributions. We observe that the performance improvements, provided by our policies under practical constraints, can indeed be very significant, and they are larger when the lead time "randomness" (say, variance) is larger. It also appears that the PIPELINE policy typically has the best performance and is robust from the practical use point of view, in the sense that it applies to a wide range of practical scenarios and does not require careful parameter tuning.

We consider a classic joint pricing and inventory control problem with lead times, which is extensively studied in the literature but is notoriously difficult to solve because of the complex structure of the optimal policy. In this work, rather than analyzing the optimal policy, we propose a class of constant-order dynamic pricing policies, which are fundamentally different from base-stock list price policies, the primary emphasis in the existing literature. Under such a policy, a constant-order amount of new inventory is ordered every period, and a pricing decision is made based on the inventory level. The policy is independent of the lead time. We prove that the best constant-order dynamic pricing policy is asymptotically optimal as the lead time grows large, which is exactly the setting in which the problem becomes computationally intractable because of the curse of dimensionality. As our main methodological contributions, we establish the convergence to a long-run average random yield inventory model with zero lead time and ordering capacities by its discounted counterpart as the discount factor goes to one, nontrivially extending the previous results in Federgruen and Yang that analyze a similar model but without capacity constraints. Funding: Research of X. Chen and L. Xin was partly supported by the National Science Foundation [Grant CMMI-1635160].

We revisit a classical problem in dynamic storage allocation. Items arrive in a linear storage medium, modeled as a half-axis, at a Poisson rate r and depart after an independent exponentially distributed unit mean service time. The arriving item sizes (lengths) are assumed to be independent and identically distributed (i.i.d.) from a common distribution H. A widely employed algorithm for allocating the items is the "first-fit" discipline, namely, each arriving item is placed in the left-most vacant interval large enough to accommodate it. In a seminal 1985 paper, Coffman, Kadota, and Shepp ([6]) proved that in the special case of unit length items (i.e. degenerate H), as r tends towards infinity, the first-fit algorithm is asymptotically optimal in the following sense: the steady-state ratio of expected "empty space" (gaps between items) to expected occupied space tends towards 0. In a sequel to [6], Coffman, Kadota, and Shepp ([5]) conjectured that the first-fit discipline is also asymptotically optimal for non-degenerate H. In this paper we provide the first proof of first-fit asymptotic optimality for non-degenerate distributions H of item sizes. Our main result is for the case when H is concentrated on countably many positive real sizes forming an increasing sequence that is either finite or goes to infinity, with the average item size being finite. We prove that under the first-fit discipline, as r tends towards infinity, the steady-state packing configuration (scaled down by r) converges in distribution to the limiting packing configuration with smaller items on the left, larger items on the right, and with no gaps between. In particular, this proves asymptotic optimality of first-fit in the sense that in steady-state the empty space (scaled down by r) vanishes.

We study a system consisting of n particles, moving forward in jumps on the real line. Each particle can make both independent jumps, whose sizes have some distribution, and 'synchronization' jumps, which allow it to join a randomly chosen other particle if the latter happens to be ahead of it. The system state is the empirical distribution of particle locations. We consider the mean-field asymptotic regime where $n\to\infty$ . We prove that $v_n$ , the steady-state speed of advance of the particle system, converges, as $n\to\infty$ , to a limit $v_{**}$ which can easily be found from a minimum speed selection principle. Also we prove that as $n\to\infty$ , the system dynamics converges to that of a deterministic mean-field limit (MFL). We show that the average speed of advance of any MFL is lower-bounded by $v_{**}$ , and the speed of a 'benchmark' MFL, resulting from all particles initially being co-located, is equal to $v_{**}$ . In the special case of exponentially distributed independent jump sizes, we prove that a traveling-wave MFL with speed v exists if and only if $v\ge v_{**}$ , with $v_{**}$ having a simple explicit form; we also show the existence of traveling waves for the modified systems with a left or right boundary moving at a constant speed v. We provide bounds on an MFL's average speed of advance, depending on the right tail exponent of its initial state. We conjecture that these results for exponential jump sizes extend to general jump sizes.

A generic way to verify asymptotic optimality of semi-open-loop policies for a wide class of MDPs with large lead times. In many real-life inventory models, order lead times can result in uncertain effects of inventory decisions. However, as the lead time grows large, one would naturally postulate that the effect of the delayed order depends weakly on the current inventory level and, thus, intuit that decoupling the delayed order with the current inventory level may provide good heuristics. Motivated by these examples, in “Asymptotic Optimality of Semi-open-Loop Policies in Markov Decision Processes with Large Lead Times,” Bai et al. consider a generic Markov decision process (MDP) with one delayed control and one immediate control. For MDPs defined on general spaces with uniformly bounded cost functions and a fast mixing property, they construct a periodic semi-open-loop policy for each lead time value and show that these policies are asymptotically optimal as the lead time goes to infinity. For MDPs defined on Euclidean spaces with linear dynamics and convex structures, they impose another set of conditions under which constant delayed-control policies are asymptotically optimal.

We consider the following network model motivated, in particular, by blockchains and peer-to-peer live streaming. Data packet flows arrive at the network nodes and need to be disseminated to all other nodes. Packets are relayed through the network via links of finite capacity. A packet leaves the network when it is disseminated to all nodes. Our focus is on two communication disciplines, which determine the order in which packets are transmitted over each link, namely Random-Useful (RU) and Oldest-Useful (OU). We show that RU has the maximum stability region in a general network. For the OU we demonstrate that, somewhat surprisingly, it does not in general have the maximum stability region. We prove that OU does achieve maximum stability in the important special case of a symmetric network, given by the full graph with equal capacities on all links and equal arrival rates at all nodes. We also give other stability results, and compare different disciplines’ performances in a symmetric system via simulation. Finally, we study the cumulative delays experienced by a packet as it propagates through the symmetric system, specifically the delay asymptotic behavior as N →∞ . We put forward some conjectures about this behavior, supported by heuristic arguments and simulation experiments.

Inventory models with large and uncertain lead times are notoriously difficult to manage due to the curse of dimensionality. Recent works suggest that in inventory models with large deterministic lead times, semiopen-loop policies are asymptotically optimal. In this paper, we provide a theoretical foundation for the superior performance of semi-open-loop policies in inventory models where the lead times are not only large but also exhibit high variability. In the single-sourcing lost-sales inventory model with divisible products, we show that the optimality gap of constant-order policies decays exponentially fast as the lead time increases. In the single-sourcing lost-sales inventory model with indivisible products, under the assumption that the placed orders cannot cross in time, we propose a bracket policy, which alternates deterministically between two consecutive integer order quantities, and prove that the bracket policy is asymptotically optimal. In the dual-sourcing backlog inventory model with divisible products, we show that a semi-open-loop policy, which places a constant order from the regular supplier in each period, and implements a state-dependent modified base-stock policy from the emergency supplier, is asymptotically optimal, and we also extend our analysis to the joint pricing and inventory model. Finally, we provide a comprehensive numerical study to demonstrate the good performance of the proposed policies, and derive further managerial insights.

In many real-life situations, the inventory record may not match the actual stock perfectly. This can happen due to distortion of inventory data, such as transaction errors, misplaced inventories, and spoilage. In these cases, because the decision maker only has incomplete information about the inventory levels, many well-known inventory policies are not even admissible, and our understanding of the optimal policies, even their existence, is very limited. In “Average Cost Optimality in Partially Observable Lost-Sales Inventory Systems,” Bai et al. consider the classical lost-sales inventory model, in which the inventory level is only observed when it becomes zero. They formulate the cost-minimization problem as a partially observable Markov decision process. By exploiting the vanishing discount factor approach, they provide a way to verify the existence of optimal policies under the average cost criterion. The key step in their analysis is the construction of a valid policy, which, in a certain sense, copies the actions of another policy for the process starting from another initial state.

AbstractWe use probabilistic methods to study properties of mean-field models, which arise as large-scale limits of certain particle systems with mean-field interaction. The underlying particle system is such that n particles move forward on the real line. Specifically, each particle ‘jumps forward’ at some time points, with the instantaneous rate of jumps given by a decreasing function of the particle’s location quantile within the overall distribution of particle locations. A mean-field model describes the evolution of the particles’ distribution when n is large. It is essentially a solution to an integro-differential equation within a certain class. Our main results concern the existence and uniqueness of—and attraction to—mean-field models which are traveling waves, under general conditions on the jump-rate function and the jump-size distribution.

We consider a system consisting of n particles, moving forward in jumps on the real line. System state is the empirical distribution of particle locations. Each particle “jumps forward” at some time points, with the instantaneous rate of jumps given by a decreasing function of the particle’s location quantile within the current state (empirical distribution). Previous work on this model established, under certain conditions, the convergence, as [Formula: see text], of the system random dynamics to that of a deterministic mean-field model (MFM), which is a solution to an integro-differential equation. Another line of previous work established the existence of MFMs that are traveling waves, as well as the attraction of MFM trajectories to traveling waves. The main results of this paper are: (a) We prove that, as [Formula: see text], the stationary distributions of (recentered) states concentrate on a (recentered) traveling wave; (b) we obtain a uniform across n moment bound on the stationary distributions of (recentered) states; and (c) we prove a convergence-to-MFM result, which is substantially more general than that in previous work. Results (b) and (c) serve as “ingredients” of the proof of (a), but also are of independent interest.

Taking Advantage of the Lead Time Randomness in Supply Chains Randomness in lead times is a major—and increasingly important—issue of inventory management, as a variety of risk factors motivate companies to diversify their supply sources and rely on distributed networks of suppliers. In “Exploiting Random Lead Times for Significant Inventory Cost Savings,” A. Stolyar and Q. Wang show that, surprisingly, instead of being a damaging factor to supply chain performance, randomness may be harnessed for potentially very substantial reductions of inventory costs. Specifically, the theoretical analysis and simulation results in the paper demonstrate that, under certain conditions, appropriately designed novel policies can significantly outperform the conventional base stock policies.

A service system with multiple types of arriving customers is considered. There is an infinite number of homogeneous servers. Multiple customers can be placed for simultaneous service into one server, subject to general packing constraints. The service times of different customers are independent even if they are served simultaneously by the same server; the service time distribution depends on the customer type. Each new arriving customer is placed for service immediately into either an occupied server, that is, one already serving other customers, as long as packing constraints are not violated or into an empty server. After service completion, each customer leaves its server and the system. The basic objective is to minimize the number of occupied servers in steady state. We study a greedy random (GRAND) placement (packing) algorithm, introduced in our previous work. This is a simple online algorithm that places each arriving customer uniformly at random into either one of the already occupied servers that can still fit the customer or one of the so-called zero servers, which are empty servers designated to be available to new arrivals. In our previous work, a version of the algorithm, labeled GRAND(aZ), is considered, in which the number of zero servers is aZ with Z being the current total number of customers in the system and positive a being an algorithm parameter. GRAND(aZ) is shown in our previous work to be asymptotically optimal in the following sense: (a) the steady-state optimality gap grows linearly in the system scale r (the mean total number of customers in service), that is, as c(a)r for some positive c(a), and (b) c(a) vanishes as a goes to zero. In this paper, we consider the GRAND(Z p ) algorithm, in which the number of zero servers is Z p , where p < 1 is a fixed parameter, sufficiently close to 1. We prove the asymptotic optimality of GRAND(Z p ) in the sense that the steady-state optimality gap is sublinear in the system scale r. This is a stronger form of asymptotic optimality than that of GRAND(aZ).

We study the following interacting particle system. There are $\rho n$ particles, $\rho < 1$, moving clockwise ("right"), in discrete time, on $n$ sites arranged in a circle. Each site may contain at most one particle. At each time, a particle may move to the right-neighbor site according to the following rules. If its right-neighbor site is occupied by another particle, the particle does not move. If the particle has unoccupied sites ("holes") as neighbors on both sides, it moves right with probability $1$. If the particle has a hole as the right-neighbor and an occupied site as the left-neighbor, it moves right with probability $0 h$, a {\em condensation} phenomenon occurs, namely the formation and persistence of large particle clusters; in particular, the typical flux in this case is $p(1-\rho) < h < \rho$, which differs from the formal flux when $h < \rho < 1/2$. Our results include both steady-state and transient analysis. In particular, we derive a version of the Ballot Theorem, and show that the key "reason" for large cluster formation for densities $\rho > h$ is described by this theorem.

A broad class of parallel server systems is considered, for which we prove the steady-state asymptotic independence of server workloads, as the number of servers goes to infinity, while the system load remains sub-critical. Arriving jobs consist of multiple components. There are multiple job classes, and each class may be of one of two types, which determines the rule according to which the job components add workloads to the servers. The model is broad enough to include as special cases some popular queueing models with redundancy, such as cancel-on-start and cancel-on-completion redundancy. Our analysis uses mean-field process representation and the corresponding mean-field limits. In essence, our approach relies almost exclusively on three fundamental properties of the model: (a) monotonicity, (b) work conservation and (c) the property that, on average, "new arriving workload prefers to go to servers with lower workloads."

We consider a parallel server system with so-called cancel-on-completion redundancy. There are n servers and multiple job classes j. An arriving class j job consists of d j components placed on a randomly selected subset of servers; the job service is complete as soon as k j components out of d j (with [Formula: see text]) complete their service, at which point the unfinished service of all remaining [Formula: see text] components is canceled. The system is in general non-work-conserving in the sense that the average amount of new workload added to the system by an arriving class j job is not defined a priori—it depends on the system state at the time of arrival. This poses the main challenge for the system analysis. For the system with a fixed number of servers n, our main results include: the stability properties; the property that the stationary distributions of the relative server workloads remain tight uniformly in the system load. We also consider the mean-field asymptotic regime when [Formula: see text] while each job class arrival rate per server remains constant. The main question we address here is: under which conditions the steady-state asymptotic independence (SSAI) of server workloads holds and, in particular, when the SSAI for the full range of loads (SSAI-FRL) holds. (Informally, SSAI-FRL means that SSAI holds for any system load less than one.) We obtain sufficient conditions for SSAI and SSAI-FRL. In particular, we prove that SSAI-FRL holds in the important special case when job components of each class j are independent and identically distributed with an increasing-hazard-rate distribution.

We study networks of interacting queues governed by utility-maximising service-rate allocations in both discrete and continuous time. For finite networks we establish stability and some steady-state moment bounds under natural conditions and rather weak assumptions on utility functions. These results are obtained using direct applications of Lyapunov-Foster-type criteria, and apply to a wide class of systems, including those for which fluid-limit-based approaches are not applicable. We then establish stability and some steady-state moment bounds for two classes of infinite networks, with single-hop and multi-hop message routes. These results are proved by considering the infinite systems as limits of their truncated finite versions. The uniform moment bounds for the finite networks play a key role in these limit transitions.

We consider the auto-scaling problem for application hosting in a cloud, where applications are elastic and the number of requests changes over time. The application requests are serviced by Virtual Machines (VMs), which reside on Physical Machines (PMs) in a cloud. We aim to minimize the number of hosting PMs by intelligently packing VMs into PMs, while the VMs are auto-scaled, i.e., dynamically acquired and released, to accommodate varying application needs. We consider a shadow routing based approach for this problem. The proposed shadow algorithm employs a specially constructed virtual queueing system to dynamically produce an optimal solution that guides the VM auto-scaling and the VM-to-PM packing. The proposed algorithm runs continuously without the need to re-solve the underlying optimization problem “from scratch”, and adapts automatically to the changes in the application demands. We prove the asymptotic optimality of the shadow algorithm. The simulation experiments further demonstrate the algorithm's good performance and high adaptivity.