In this paper we consider the workload of a storage system with the unconventional feature that the arrival times, rather than the interarrival times, are independent and identically distributed samples from a given distribution. We start by analyzing the ‘base model’ in which the arrival times are exponentially distributed, leading to a closed-form characterization of the queue’s workload at a given moment in time (i.e. in terms of Laplace–Stieltjes transforms), assuming the initial workload was 0. Then we consider four more general models, each of them having a specific additional feature: (a) the initial workload being allowed to have any arbitrary non-negative value, (b) an additional stream of Poisson arrivals, (c) phase-type arrival times, (d) balking customers. For all four variants the transform of the transient workload is identified in closed form.

Polling systems are queueing systems consisting of multiple queues served by a single server. In this paper we analyze two types of preemptive time-limited polling systems, the so-called pure and exhaustive time-limited disciplines. In particular, we derive a direct relation for the evolution of the joint queue length during the course of a server visit. The analysis of the pure time-limited discipline builds on and extends several known results for the transient analysis of an M/G/1 queue. For the analysis of the exhaustive discipline we derive several new results for the transient analysis of the M/G/1 queue during a busy period. The final expressions for both types of polling systems that we obtain generalize previous results by incorporating customer routeing, generalized service times, batch arrivals, and Markovian polling of the server.

The ‘Price of Anarchy’ states that the performance of multi-agent service systems degrades with the agents’ selfishness (anarchy). We investigate a service model in which both customers and the firm are strategic. We find that, for a Stackelberg game in which the server invests in capacity before customers decide whether or not to join, there can be a ‘Benefit of Anarchy’, that is, customers acting selfishly can have a greater overall utility than customers who are coordinated to maximize their overall utility. We also show that customer anarchy can be socially beneficial, resulting in a ‘Social Benefit of Anarchy’. We show that such phenomena are rather general and can arise in multiple settings (e.g. in both profit-maximizing and welfare-maximizing firms, in both capacity-setting and price-setting firms, and in both observable and unobservable queues). However, the underlying mechanism leading to the Benefit of Anarchy can differ significantly from one setting to another.

Batch service has a wide application in manufacturing, communication networks, and cloud computing. In batch service queues with limited resources, one critical issue is to properly schedule the service so as to ensure the quality of service. In this paper we consider an M/G[a,b]/1/N batch service queue with bulking threshold a, max service capacity b, and buffer capacity N, where N can be finite or infinite. Through renewal theory, busy period analysis and decomposition techniques, we demonstrate explicitly how the bulking threshold influences the system performance such as the mean waiting time and time-averaged number of loss customers in batch service queues. We then establish a necessary and sufficient condition on the optimal bulking threshold that minimizes the expected waiting time. Enabled by this condition, we propose a simple algorithm which guarantees to find the optimal threshold in polynomial time. The performance of the algorithm is also demonstrated by numerical examples.

In this paper we study a reflected AR(1) process, i.e. a process (Z n )n obeying the recursion Z n +1= max{aZ n +X n ,0}, with (X n )n a sequence of independent and identically distributed (i.i.d.) random variables. We find explicit results for the distribution of Z n (in terms of transforms) in case X n can be written as Y n −B n , with (B n )n being a sequence of independent random variables which are all Exp(λ) distributed, and (Y n )n i.i.d.; when |a|<1 we can also perform the corresponding stationary analysis. Extensions are possible to the case that (B n )n are of phase-type. Under a heavy-traffic scaling, it is shown that the process converges to a reflected Ornstein–Uhlenbeck process; the corresponding steady-state distribution converges to the distribution of a normal random variable conditioned on being positive.

We study random walks whose increments are α-stable distributions with shape parameter 1<α<2. Specifically, assuming a mean increment size which is negative, we provide series expansions in terms of the mean increment size for the probability that the all-time maximum of an α-stable random walk is equal to 0 and, in the totally skewed-to-the-left case of skewness parameter β=-1, for the expected value of the all-time maximum of an α-stable random walk. Our series expansions generalize previous results for Gaussian random walks. Key ingredients in our proofs are Spitzer's identity for random walks, the stability property of α-stable random variables, and Zolotarev's integral representation for the cumulative distribution function of an α-stable random variable. We also discuss an application of our results to a problem arising in queueing theory.

This paper provides a comparison of the successive lumping (SL) methodology developed in Katehakis et al. (2015) with the popular lattice path counting (Mohanty (1979)) in obtaining rate matrices for queueing models, satisfying the specific quasi birth and death structure as in Van Leeuwaarden et al. (2009) and Van Leeuwaarden and Winands (2006). The two methodologies are compared both in terms of applicability requirements and numerical complexity by analyzing their performance for the same classical queueing models considered in Van Leeuwaarden et al. (2009). The main findings are threefold. First, when both methods are applicable, the SL-based algorithms outperform the lattice path counting algorithm (LPCA). Second, there are important classes of problems (for example, models with (level) nonhomogenous rates or with finite state spaces) for which the SL methodology is applicable and for which the LPCA cannot be used. Third, another main advantage of SL algorithms over lattice path counting is that the former includes a method to compute the steady state distribution using this rate matrix.

We develop diffusion approximations for parallel-queueing systems with the randomized longest-queue-first scheduling (LQF) algorithm by establishing new mean-field limit theorems as the number of buffers n → ∞. We achieve this by allowing the number of sampled buffers d = d(n) to depend on the number of buffers n, which yields an asymptotic 'decoupling' of the queue length processes. We show through simulation experiments that the resulting approximation is accurate even for moderate values of n and d(n). To the best of the authors' knowledge, this is the first derivation of diffusion approximations for a queueing system in the large-buffer mean-field regime. Another noteworthy feature of our scaling idea is that the randomized LQF algorithm emulates the LQF algorithm, yet is computationally more attractive. The analysis of the system performance as a function of d(n) is facilitated by the multi-scale nature in our limit theorems: the various processes we study have different space scalings. This allows us to show the trade-off between performance and complexity of the randomized LQF scheduling algorithm.

We show how to write the solution to the generalized drift Skorokhod problem in one-dimension in terms of the supremum of the solution of a tractable unrestricted integral equation (that is, an integral equation with no boundaries). As an application of our result, we equate the transient distribution of a reflected Ornstein–Uhlenbeck (OU) process to the first hitting time distribution of an OU process (that is not reflected). Then, we use this relationship to approximate the transient distribution of the GI/GI/1 + GI queue in conventional heavy traffic and the M/M/N/N queue in a many-server heavy traffic regime.

We study the best achievable performance (in terms of the average queue size and delay) in a stochastic and dynamic version of the bin-packing problem. Items arrive to a queue according to a Poisson process with rate 2ρ, where ρ ∈ (0, 1). The item sizes are independent and identically distributed (i.i.d.) with a uniform distribution in [0, 1]. At each time unit, a single unit-size bin is available and can receive any of the queued items, as long as their total size does not exceed 1. Coffman and Stolyar (1999) and Gamarnik (2004) have established that there exist packing policies under which the average queue size is finite for every ρ ∈ (0, 1). In this paper we study the precise scaling of the average queue size, as a function of ρ, with emphasis on the critical regime where ρ approaches 1. Standard results on the probabilistic (but static) bin-packing problem can be readily applied to produce policies under which the queue size scales as O(h2), where h = 1 / (1 - ρ), which raises the question of whether this is the best possible. We establish that the average queue size scales as Ω(hlogh), under any policy. Furthermore, we provide an easily implementable policy, which packs at most two items per bin. Under that policy, the average queue size scales as O(hlog3/2h), which is nearly optimal. On the other hand, if we impose the additional requirement that any two items packed together must have near-complementary sizes (in a sense to be made precise), we show that the average queue size must scale as Θ(h2).

This paper deals with censoring of infinite-state banded Markov chains. Censoring involves reducing the time spent in states outside a certain set of states to 0 without affecting the number of visits within this set. We show that, if all states are transient, there is, besides the standard censored Markov chain, a nonstandard censored Markov chain which is stochastic. Both the stochastic and the substochastic solutions are found by censoring a sequence of finite transition matrices. If all matrices in the sequence are stochastic, the stochastic solution arises in the limit, whereas the substochastic solution arises if the matrices in the sequence are substochastic. We also show that, if the Markov chain is recurrent, the only solution is the stochastic solution. Censoring is particularly fruitful when applied to quasi-birth-and-death (QBD) processes. It turns out that key matrices in such processes are not unique, a fact that has been observed by several authors. We note that the stochastic solution is important for the analysis of finite queues.

We study a service facility modeled as a queueing system with finite or infinite capacity. Arriving customers enter if there is room in the facility and if they are willing to pay the price posted by the service provider. Customers belong to one of a finite number of classes that have different willingnesses-to-pay. Moreover, there is a penalty for congestion in the facility in the form of state-dependent holding costs. The service provider may advertise class-specific prices that may fluctuate over time. We show the existence of a unique optimal stationary pricing policy in a continuous and unbounded action space that maximizes the long-run average profit per unit time. We determine an expression for this policy under certain conditions. We also analyze the structure and the properties of this policy.

This paper analyzes adiscrete-time multi-server queue in which service capacity of eachserver is a minimum of one and a maximum of b customers. Theinterarrival- and service-times are assumed to be independent andgeometrically distributed. The queue is analyzed under theassumptions of early arrival system and late arrival system withdelayed access. Besides, obtaining state probabilities atarbitrary and outside observer's observation epochs, someperformance measures and waiting-time distribution in the queuehave also been discussed. Finally, it is shown that in limitingcase the results obtained in thispaper tend to the continuous-time counterpart.

Consider a continuous-time Markov chain evolving in a random environment. We study certain forms of interaction between the process of interest and the environmental process, under which the stationary joint distribution is tractable. Moreover, we obtain necessary and sufficient conditions for a product-form stationary distribution. A number of examples that illustrate the applicability of our results in queueing and population growth models are also included.

In this paper, we analyze the diffusion limit of a discrete-time queueing system with constant service rate and connections that randomly enter and depart from the system. Each connection generates periodic traffic while it is active, and a connection's lifetime has finite mean. This can model a time division multiple access system with constant bit-rate connections. The diffusion scaling retains semiperiodic behavior in the limit, allowing for both short-time analysis (within one frame) and long-time analysis (over multiple frames). Weak convergence of the cumulative arrival process and the stationary buffer-length distribution is proved. It is shown that the limit of the cumulative arrival process can be viewed as a discrete-time stationary-increment Gaussian process interpolated by Brownian bridges. We present bounds on the overflow probability of the limit queueing process as functions of the arrival rate and the connection lifetime distribution. Also, numerical and simulation results are presented for geometrically distributed connection lifetimes.

The issue of ‘fairness’ is raised frequently in the context of evaluating queueing policies, notably in relation to telecommunications and computer systems where it may be of no lesser importance than the conventional measures of performance. Comparisons of the fairness of various systems and policies are often awkward due to lack of generally accepted definitions and measures for this important property. The purpose of this work is to propose possible fairness measures enabling us to quantitatively measure and compare the level of fairness associated with G/G/R queueing systems. We define and discuss order (of service) fairness and use an axiomatic approach for developing a measure for it in the G/D/1 case. The measure obtained for the G/D/1 system is then generalized and applied to the G/G/R class of systems. A practical implication of this work is that, for a wide class of service disciplines, the variance of the waiting time can be used as a yardstick for comparing fairness levels.

We study a fork-join processing network in which jobs arrive according to a Poisson process and each job splits into m tasks, which are simultaneously assigned to m nodes that operate like M/M/s queueing systems. When all of its tasks are finished, the job is completed. The main result is a closed-form formula for approximating the distribution of the network's response time (the time to complete a job) in equilibrium. We also present an analogous approximation for the distribution of the equilibrium queue length (the number of jobs in the system), when each node has one server. Kolmogorov-Smirnov statistical tests show that these formulae are good fits for the distributions obtained from simulations.

We consider a Markov decision process for an MX/M/1 queue that iscontrolled by batches of negative customers. More specifically, we deriveconditions that imply threshold-type optimal policies, under either thetotal discounted cost criterion or the average cost criterion. Theperformance analysis of the model when it operates under a giventhreshold-type policy is also studied. We prove a stability condition and acomplete stochastic comparison characterization for models operating underdifferent thresholds. Exact and asymptotic results concerning thecomputation of the stationary distribution of the model are also derived.

We study a service system in which, in each service period, the server performs the current set B of tasks as a batch, taking time s(B), where the function s(·) is subadditive. A natural definition of ‘traffic intensity under congestion’ in this setting is ρ := limt→∞t-1Es (all tasks arriving during time [0,t]). We show that ρ > 1 and a finite mean of individual service times are necessary and sufficient to imply stability of the system. A key observation is that the numbers of arrivals during successive service periods form a Markov chain {An}, enabling us to apply classical regenerative techniques and to express the stationary distribution of the process in terms of the stationary distribution of {An}.

We consider a reflected superposition of a Brownian motion and a compound Poisson process as a model for the workload process of a queueing system with two types of customers under heavy traffic. The distributions of the duration of a busy cycle and the maximum workload during a cycle are determined in closed form.