Abstract: We provide a novel expression of the scale function for a Lévy process with negative phase-type jumps. It is in terms of a certain transition rate matrix which is explicit up to a single positive number. A monotone iterative scheme for the calculation of the latter is presented and it is shown that the error decays exponentially fast. Our numerical examples suggest that this algorithm allows us to employ phase-type distributions with hundreds of phases, which is problematic when using the known formula for the scale function in terms of roots. Extensions to other distributions, such as matrix-exponential and infinite-dimensional phase-type, can be anticipated. PubDate: 2021-03-08

Abstract: The Adaptive MaxWeight policy achieves optimal throughput for switches with nonzero reconfiguration delay and has been shown to have good delay performance in simulation. In this paper, we analyze the queue length behavior of a switch with nonzero reconfiguration delay operating under the Adaptive MaxWeight. We first show that the Adaptive MaxWeight policy exhibits a weak state space collapse behavior in steady state, which can be viewed as an inheritance of a similar property of the MaxWeight policy in a switch with zero reconfiguration delay. The weak state space collapse result is then utilized to obtain an asymptotically tight bound on an expression involving the steady-state queue length and the probability of reconfiguration for the Adaptive MaxWeight policy in the heavy traffic regime. We then derive the relation between the expected schedule duration and the steady-state queue length through Lyapunov drift analysis and characterize bounds for the expected steady-state queue length. While the resulting queue length bounds are not asymptotically tight, they suggest an approximate queue length scaling behavior, which approaches the optimal scaling with respect to the traffic load and the reconfiguration delay when the hysteresis function of the Adaptive MaxWeight policy approaches a linear function. PubDate: 2021-03-05

Abstract: Pay-for-priority is a common practice in congestion-prone service systems. The extant literature on this topic restricts attention to the case where the only epoch for customers to purchase priority is upon arrival, and if customers choose not to upgrade when they arrive, they cannot do so later during their wait. A natural alternative is to let customers pay and upgrade to priority at any time during their stay in the queue, even if they choose not to do so initially. This paper builds a queueing-game-theoretic model that explicitly captures self-interested customers’ dynamic in-queue priority-purchasing behavior. When all customers (who have not upgraded yet) simultaneously decide whether to upgrade, we find in our model that pure-strategy equilibria do not exist under some intuitive criteria, contrasting the findings in classical models where customers can only purchase priority upon arrival. However, when customers sequentially decide whether to upgrade, threshold-type pure-strategy equilibria may exist. In particular, under sufficiently light traffic, if the number of ordinary customers accumulates to a certain threshold, then it is always the second last customer who upgrades, but in general, it could be a customer from another position, and the queue-length threshold that triggers an upgrade can also vary with the traffic intensity. Finally, we find that in-queue priority purchase subject to the sequential rule yields less revenue than upon-arrival priority purchase in systems with small buffers. PubDate: 2021-03-05

Abstract: Admission control and service rate speedup may be used during periods of congestion to minimize customer waiting in different service settings. In a healthcare setting, this can mean sending patients to alternative care facilities that may take more time and/or provide less ideal treatment. While waiting can be detrimental to patient outcomes, strategies used to control congestion can also be costly. In this work, we examine a multi-server queueing system that considers both admission control and speedup. We use dynamic programming to characterize properties of the optimal control and find that in some instances the optimal policy has a simple form of a threshold policy. Leveraging this insight, we examine a queueing system where speedup is used when the number of customers (patients) in the system exceeds some threshold and admission control is used when that number exceeds some (potentially different) threshold. Using a fluid model and a stochastic loss model, we develop a methodology to derive approximations for the probability that speedup will be applied, the probability that admission control will be applied and the expected queue length customers experience. We use the approximations as the basis for a greedy heuristic to derive a near optimal solution to the original stochastic optimization problem. We use simulation to demonstrate the quality of these approximations and find that they can be quite accurate and robust. This analysis can provide insight to managers deciding how to balance admission control and speedup in service settings: when and to what extent to use each. PubDate: 2021-02-24

Abstract: The traditional approach when looking for a symmetric equilibrium behavior in queueing models with strategic customers who arrive according to some stationary arrival process is to look for a strategy which, if used by all, is also a best response of an individual customer under the resulting stochastic steady-state conditions. This description lacks a key component: a proper definition of the set of players. Hence, many of these models cannot be defined as proper noncooperative games. This limitation raises two main concerns. First, it is hard to formulate these models in a canonical way, and hence, results are typically limited to a specific model or to a narrow class of models. Second, game theoretic results cannot be applied directly in the analysis of such models and call for ad hoc adaptations. We suggest a different approach, one that is based on the stationarity of the underlying stochastic processes. In particular, instead of considering a system that is functioning from time immemorial and has already reached stochastic steady-state conditions, we look at the process as a series of isolated, a priori identical, and independent strategic situations with a random, yet finite, set of players. Each of the isolated games can be analyzed using existing tools from the classical game theoretic literature. Moreover, this approach suggests a canonic definition of strategic queueing models as properly defined mathematical objects. A significant advantage of our approach is its compatibility with models in the existing literature. This is exemplified in detail for the famous model of the unobservable “to queue or not to queue” problem and other related models. PubDate: 2021-02-20

Abstract: We consider an acyclic network of single-server queues with heterogeneous processing rates. It is assumed that each queue is fed by the superposition of a large number of i.i.d. Gaussian processes with stationary increments and positive drifts, which can be correlated across different queues. The flow of work departing from each server is split deterministically and routed to its neighbors according to a fixed routing matrix, with a fraction of it leaving the network altogether. We study the exponential decay rate of the probability that the steady-state queue length at any given node in the network is above any fixed threshold, also referred to as the ‘overflow probability’. In particular, we first leverage Schilder’s sample-path large deviations theorem to obtain a general lower bound for the limit of this exponential decay rate, as the number of Gaussian processes goes to infinity. Then, we show that this lower bound is tight under additional technical conditions. Finally, we show that if the input processes to the different queues are nonnegatively correlated, non-short-range dependent fractional Brownian motions, and if the processing rates are large enough, then the asymptotic exponential decay rates of the queues coincide with the ones of isolated queues with appropriate Gaussian inputs. PubDate: 2021-02-18

Abstract: We present a broad literature survey of parameter and state estimation for queueing systems. Our approach is based on various inference activities, queueing models, observations schemes, and statistical methods. We categorize these into branches of research that we call estimation paradigms. These include: the classical sampling approach, inverse problems, inference for non-interacting systems, inference with discrete sampling, inference with queueing fundamentals, queue inference engine problems, Bayesian approaches, online prediction, implicit models, and control, design, and uncertainty quantification. For each of these estimation paradigms, we outline the principles and ideas, while surveying key references. We also present various simple numerical experiments. In addition to some key references mentioned here, a periodically updated comprehensive list of references dealing with parameter and state estimation of queues will be kept in an accompanying annotated bibliography. PubDate: 2021-02-17

Abstract: 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.” PubDate: 2021-02-17

Abstract: We consider a discrete-time d-dimensional process \(\{{\varvec{X}}_n\}=\{(X_{1,n},X_{2,n},\ldots ,X_{d,n})\}\) on \({\mathbb {Z}}^d\) with a background process \(\{J_n\}\) on a countable set \(S_0\) , where individual processes \(\{X_{i,n}\},i\in \{1,2,\ldots ,d\},\) are skip free. We assume that the joint process \(\{{\varvec{Y}}_n\}=\{({\varvec{X}}_n,J_n)\}\) is Markovian and that the transition probabilities of the d-dimensional process \(\{{\varvec{X}}_n\}\) vary according to the state of the background process \(\{J_n\}\) . This modulation is assumed to be space homogeneous. We refer to this process as a d-dimensional skip-free Markov-modulated random walk. For \({\varvec{y}}, {\varvec{y}}'\in {\mathbb {Z}}_+^d\times S_0\) , consider the process \(\{{\varvec{Y}}_n\}_{n\ge 0}\) starting from the state \({\varvec{y}}\) and let \({\tilde{q}}_{{\varvec{y}},{\varvec{y}}'}\) be the expected number of visits to the state \({\varvec{y}}'\) before the process leaves the nonnegative area \({\mathbb {Z}}_+^d\times S_0\) for the first time. For \({\varvec{y}}=({\varvec{x}},j)\in {\mathbb {Z}}_+^d\times S_0\) , the measure \(({\tilde{q}}_{{\varvec{y}},{\varvec{y}}'}; {\varvec{y}}'=({\varvec{x}}',j')\in {\mathbb {Z}}_+^d\times S_0)\) is called an occupation measure. Our primary aim is to obtain the asymptotic decay rate of the occupation measure as \({\varvec{x}}'\) goes to infinity in a given direction. We also obtain the convergence domain of the matrix moment generating function of the occupation measure. PubDate: 2021-02-01

Abstract: We study the problem of strategic choice of arrival time to a single-server queue with opening and closing times when there is uncertainty regarding service speed. A Poisson population of customers choose their arrival time with the goal of minimizing their expected waiting times and are served on a first-come first-served basis. There are two types of customers that differ in their beliefs regarding the service-time distribution. The inconsistent beliefs may arise from randomness in the server state along with noisy signals that customers observe. Customers are aware of the two types of populations with differing beliefs. We characterize the Nash equilibrium dynamics for exponentially distributed service times and show how they substantially differ from the model with homogeneous customers. We further provide an explicit solution for a fluid approximation of the game. For general service-time distributions we provide an algorithm for computing the equilibrium in a discrete-time setting. We find that in equilibrium customers with different beliefs arrive during different (and often disjoint) time intervals. Numerical analysis further shows that the mean waiting time increases with the coefficient of variation of the service time. Furthermore, we present a learning agent-based model (ABM) in which customers make joining decisions based solely on their signals and past experience. We numerically compare the long-term average outcome of the ABM with that of the equilibrium and find that the arrival distributions are quite close if we assume (for the equilibrium solution) that customers are fully rational and have knowledge of the system parameters, while they may greatly differ if customers have limited information or computing abilities. PubDate: 2021-01-28

Abstract: Datacenter operations today provide a plethora of new queueing and scheduling problems. The notion of a “job” has become more general and multi-dimensional. The ways in which jobs and servers can interact have grown in complexity, involving parallelism, speedup functions, precedence constraints, and task graphs. The workloads are vastly more variable and more heavy-tailed. Even the performance metrics of interest are broader than in the past, with multi-dimensional service-level objectives in terms of tail probabilities. The purpose of this article is to expose queueing theorists to new models, while providing suggestions for many specific open problems of interest, as well as some insights into their potential solution. PubDate: 2021-01-27

Abstract: This paper introduces non-cooperative games on a network of single server queues with fixed routes. A player has a set of routes available and has to decide which route(s) to use for its customers. Each player’s goal is to minimize the expected sojourn time of its customers. We consider two cases: a continuous strategy space, where each player is allowed to divide its customers over multiple routes, and a discrete strategy space, where each player selects a single route for all its customers. For the continuous strategy space, we show that a unique pure-strategy Nash equilibrium exists that can be found using a best-response algorithm. For the discrete strategy space, we show that the game has a Nash equilibrium in mixed strategies, but need not have a pure-strategy Nash equilibrium. We show the existence of pure-strategy Nash equilibria for four subclasses: (i) N-player games with equal arrival rates for the players, (ii) 2-player games with identical service rates for all nodes, (iii) 2-player games on a \(2\times 2\) -grid, and (iv) 2-player games on an \(A\times B\) -grid with small differences in the service rates. PubDate: 2021-01-25

Abstract: The workload of a generalized n-site asymmetric simple inclusion process (ASIP) is investigated. Three models are analyzed. The first model is a serial network for which the steady-state Laplace–Stieltjes transform (LST) of the total workload in the first k sites ( \(k\le n\) ) just after gate openings and at arbitrary epochs is derived. In a special case, the former (just after gate openings) turns out to be an LST of the sum of k independent random variables. The second model is a 2-site ASIP with leakage from the first queue. Gate openings occur at exponentially distributed intervals, and the external input processes to the stations are two independent subordinator Lévy processes. The steady-state joint workload distribution right after gate openings, right before gate openings and at arbitrary epochs is derived. The third model is a shot-noise counterpart of the second model where the workload at the first queue behaves like a shot-noise process. The steady-state total amount of work just before a gate opening turns out to be a sum of two independent random variables. PubDate: 2021-01-02 DOI: 10.1007/s11134-020-09678-4

Abstract: We consider an infinite sequence consisting of agents of several types and goods of several types, with a bipartite compatibility graph between agent and good types. Goods are matched with agents that appear earlier in the sequence using FCFS matching, if such are available, and are lost otherwise. This model may be used for two-sided queueing applications such as ride sharing, Web purchases, organ transplants, and for parallel redundant service queues. For this model, we calculate matching rates and delays. These can be used to obtain waiting times and help with design questions for related service systems. We explore some relations of this model to other FCFS stochastic matching models. PubDate: 2020-12-01 DOI: 10.1007/s11134-020-09676-6

Abstract: This paper studies tight upper bounds for the mean and higher moments of the steady-state waiting time in the GI/GI/1 queue given the first two moments of the interarrival-time and service-time distributions. We apply the theory of Tchebycheff systems to obtain sufficient conditions for classical two-point distributions to yield the extreme values. These distributions are determined by having one mass at 0 or at the upper limit of support. PubDate: 2020-11-09 DOI: 10.1007/s11134-020-09675-7

Abstract: We consider an \(M/M/1/{\overline{N}}\) observable non-customer-intensive service queueing system with unknown service rates consisting of strategic impatient customers who make balking decisions and non-strategic patient customers who do not make any decision. In the queueing game amongst the impatient customers, we show that there exists at least one pure threshold strategy equilibrium in the presence of patient customers. As multiple pure threshold strategy equilibria exist in certain cases, we consider the minimal pure threshold strategy equilibrium in our sensitivity analysis. We find that the likelihood ratio of a fast server to a slow server in an empty queue is monotonically decreasing in the proportion of impatient customers and monotonically increasing in the waiting area capacity. Further, we find that the minimal pure threshold strategy equilibrium is non-increasing in the proportion of impatient customers and non-decreasing in the waiting area capacity. We also show that at least one pure threshold strategy equilibrium exists when the waiting area capacity is infinite. PubDate: 2020-10-14 DOI: 10.1007/s11134-020-09671-x