Abstract: Many service systems use technology to notify customers about their expected waiting times or queue lengths via delay announcements However, in many cases, either the information might be delayed or customers might require time to travel to the queue of their choice, thus causing a lag in information In this paper, we construct a neutral delay differential equation model for the queue length process and explore the use of velocity information in our delay announcement Our results illustrate that using velocity information can have either a beneficial or detrimental impact on the system Thus, it is important to understand how much velocity information a manager should use In some parameter settings, we show that velocity information can eliminate oscillations created by delays in information We derive a fixed point equation for determining the optimal amount of velocity information that should be used and find closed-form upper and lower bounds on its value When the oscillations cannot be eliminated altogether, we identify the amount of velocity information that minimizes the amplitude of the oscillations However, we also find that using too much velocity information can create oscillations in the queue lengths that would otherwise be stable

Abstract: We analyze the time-dependent behavior of various types of infinite-server queueing systems, where, within each system we consider, jobs interact with one another in ways that induce batch departures from the system. One example of such a queue was introduced in the recent paper of Frolkova and Mandjes (Stochastic Models, 2019) in order to model a type of one-sided communication between two users in the Bitcoin network: here we show that a time-dependent version of the distributional Little’s law can be used to study the time-dependent behavior of this model, as well as a related model where blocks are communicated to a user at a rate that is allowed to vary with time. We also show that the time-dependent behavior of analogous infinite-server queueing systems with batch arrivals and exponentially distributed services can be analyzed just as thoroughly.

Abstract: We consider Poisson streams of exponentially distributed jobs arriving at each edge of a hypergraph of queues. Upon arrival, an incoming job is routed to the shortest queue among the corresponding vertices. This generalizes many known models such as power-of-d load balancing and JSQ (join the shortest queue) on generic graphs. We prove that stability in this model is achieved if and only if there exists a stable static routing policy. This stability condition is equivalent to that of the JSW (join the shortest workload) policy. We show that some graph topologies lead to a loss of capacity, implying more restrictive stability conditions than in, for example, complete graphs.

Abstract: We consider a decentralised multi-access algorithm, motivated primarily by the control of transmissions in a wireless network. For a finite single-hop network with arbitrary interference constraints, we prove stochastic stability under the natural conditions. For infinite and finite single-hop networks, we obtain broad rate-stability conditions. We also consider symmetric finite multi-hop networks and show that the natural condition is sufficient for stochastic stability.

Abstract: This paper considers an unobservable two-site tandem queueing system attended by an alternating server. We study the strategic customer behaviour under two threshold-based operating policies, applied by a profit-maximizing server, while customers’ strategic behaviour and server’s switching costs are taken into account. Under the Exact-N policy, in each cycle the server first completes service of N customers in the first stage ( $$Q_1$$ ), then switches to the second stage ( $$Q_2$$ ) and then serves those N customers before switching back to $$Q_1$$ to start a new cycle. This policy leads to a mixture of Follow-the-Crowd and Avoid-the-Crowd customer behaviour. In contrast, under the N-Limited policy, the server switches from $$Q_1$$ to $$Q_2$$ also when the first queue is emptied, making this regime work-conserving and leading only to Avoid-the-Crowd behaviour. Performance measures are obtained using matrix geometric methods for both policies and any threshold N, while for sequential service ( $$N=1$$ ) explicit expressions are derived. It is shown that the system’s stability condition is independent of N, and of the switching policy. Optimal performances in equilibrium, under each of these switching policies, are analysed and compared through a numerical study.

Abstract: Stochastic networks with complex structures are key modelling tools for many important applications. In this paper, we consider a specific type of network: retrial queueing systems with priority. This type of queueing system is important in various applications, including telecommunication and computer management networks with big data. The system considered here receives two types of customers, of which Type-1 customers (in a queue) have non-pre-emptive priority to receive service over Type-2 customers (in an orbit). For this type of system, we propose an exhaustive version of the stochastic decomposition approach, which is one of the main contributions made in this paper, for the purpose of studying asymptotic behaviour of the tail probability of the number of customers in the steady state for this retrial queue with two types of customers. Under the assumption that the service times of Type-1 customers have a regularly varying tail and the service times of Type-2 customers have a tail lighter than Type-1 customers, we obtain tail asymptotic properties for the numbers of customers in the queue and in the orbit, respectively, conditioning on the server’s status, in terms of the exhaustive stochastic decomposition results. These tail asymptotic results are new, which is another main contribution made in this paper. Tail asymptotic properties are very important, not only on their own merits but also often as key tools for approximating performance metrics and constructing numerical algorithms.

Abstract: We analyze a discrete-time two-class queueing system with a single server which is alternately available for only one customer class. The server is each time allocated to a customer class for a geometrically distributed amount of time. Service times of the customers are deterministically equal to 1 time slot each. During each time slot, both classes can have at most one arrival. The bivariate process of the number of customers of both classes can be considered as a two-dimensional nearest-neighbor random walk. The generating function of this random walk has to be obtained from a functional equation. This type of functional equation is known to be difficult to solve. In this paper, we obtain closed-form expressions for the joint probability distribution for the number of customers of both classes, in steady state.

Abstract: The idea behind the recently introduced “age-of-information” performance measure of a network message processing system is that it indicates our knowledge regarding the “freshness” of the most recent piece of information that can be used as a criterion for real-time control. In this foundational paper, we examine two such measures, one that has been extensively studied in the recent literature and a new one that could be more relevant from the point of view of the processor. Considering these measures as stochastic processes in a stationary environment (defined by the arrival processes, message processing times and admission controls in bufferless systems), we characterize their distributions using the Palm inversion formula. Under renewal assumptions, we derive explicit solutions for their Laplace transforms and show some interesting decomposition properties. Previous work has mostly focused on computation of expectations in very particular cases. We argue that using bufferless or very small buffer systems is best and support this by simulation. We also pose some open problems including assessment of enqueueing policies that may be better in cases where one wishes to minimize more general functionals of the age-of-information measures.

Abstract: This paper studies an extension of Naor’s model in which there is parameter uncertainty. In particular, the arrival rate is known, to customers and system managers, only through its distribution. For the observable case, the relationship between the optimal individual threshold and the thresholds for a social optimizer or revenue maximizer does not change from the classical model with a known arrival rate. However, in the unobservable case, it is shown that the decisions of the social optimizer and revenue maximizer no longer coincide. Furthermore, in the unobservable case with arrival rate uncertainty, the social optimizer induces a lower expected arrival rate than the revenue maximizer. This stands in contrast to the observable case, in which the social optimizer prefers a more congested system.

Abstract: This paper studies stationary customer flows in an open queueing network. The flows are the processes counting customers flowing from one queue to another or out of the network. We establish the existence of unique stationary flows in generalized Jackson networks and convergence to the stationary flows as time increases. We establish heavy-traffic limits for the stationary flows, allowing an arbitrary subset of the queues to be critically loaded. The heavy-traffic limit with a single bottleneck queue is especially tractable because it yields limit processes involving one-dimensional reflected Brownian motion. That limit plays an important role in our new nonparametric decomposition approximation of the steady-state performance using indices of dispersion and robust optimization.

Abstract: Motivated by service systems, such as telephone call centers and emergency departments, we consider admission control for a two-class multi-server loss system with periodically varying parameters and customers who may abandon from service. Assuming mild conditions for the parameters, a dynamic programming formulation is developed. We show that under the infinite horizon discounted problem, there exists an optimal threshold policy and provide conditions for a customer class to be preferred for each fixed time, extending stationary results to the non-stationary setting. We approximate the non-stationary problem by discretizing the time horizon into equally spaced intervals and examine how policies for this approximation change as a function of time and parameters numerically. We compare the performance of these approximations with several admission policies used in practice in a discrete-event simulation study. We show that simpler admission policies that ignore non-stationarity or abandonments lead to significant losses in rewards.

Abstract: We study a system of taxis and customers with Poisson arrivals and exponential patience times. We model a delayed matching process between taxis and customers using a matching rate $$\theta $$ as follows: if there are i taxis and j customers in the system, the next pairing will occur after an exponential amount of time with rate $$\theta i^{\delta _1}j^{\delta _2}$$ ( $$\delta _1, \delta _2 \in (0,+\infty $$ )). We formulate the system as a CTMC and study the fluid and diffusion approximations for this system, which involve the solutions to a system of differential equations. We consider two approximation methods: Kurtz’s method (KA) derived from Kurtz’s results (Kurtz in J Appl Probab 7(1):49–58, 1970; Kurtz in J Appl Probab 8(2):344–356, 1971) and Gaussian approximation (GA) that works for the case $$\delta _1 = \delta _2 = 1$$ (we call this the bilinear case) based on the infinitesimal analysis of the CTMC. We compare their performance numerically with simulations and conclude that GA performs better than KA in the bilinear case. We next formulate an optimal control problem to maximize the total net revenue over a fixed time horizon T by controlling the arrival rate of taxis. We solve the optimal control problem numerically and compare its performance to the real system. We also use Markov decision processes to compute the optimal policy that maximizes the long-run revenue rate. We finally propose a heuristic control policy (HPKA) and show that its expected regret is a bounded function of T. We also propose a version of this policy (HPMDP) that can actually be implemented in the real queueing system and study its performance numerically.

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.

Abstract: We study a model of wireless networks where users move at speed $$\theta \ge 0$$, which has the original feature of being defined through a fixed-point equation. Namely, we start from a two-class processor-sharing queue to model one representative cell of this network: class 1 users are patient (non-moving) and class 2 users are impatient (moving). This model has five parameters, and we study the case where one of these parameters is set as a function of the other four through a fixed-point equation. This fixed-point equation captures the fact that the considered cell is in balance with the rest of the network. This modeling approach allows us to alleviate some drawbacks of earlier models of mobile networks. Our main and surprising finding is that for this model, mobility drastically improves the heavy traffic behavior, going from the usual $$\frac{1}{1-\varrho }$$ scaling without mobility (i.e., when $$\theta = 0$$) to a logarithmic scaling $$\log (1/(1-\varrho ))$$ as soon as $$\theta > 0$$. In the high load regime, this confirms that the performance of mobile systems benefits from the spatial mobility of users. Finally, other model extensions and complementary methodological approaches to this heavy traffic analysis are discussed.

Abstract: A Markovian single-server queue is studied in an interactive random environment. The arrival and service rates of the queue depend on the environment, while the transition dynamics of the random environment depend on the queue length. We consider in detail two types of Markov random environments: a pure jump process and a reflected jump diffusion. In both cases, the joint dynamics are constructed so that the stationary distribution can be explicitly found in a simple form (weighted geometric). We also derive an explicit estimate for the exponential rate of convergence to the stationary distribution via coupling.

Abstract: In standard queues, when there are waiting customers, service completions are followed by service commencements. In retrial queues, this is not the case. In such systems, customers try to receive service at a time of their choosing, or the server seeks the next customer for a non-negligible time. In this note, we consider a hybrid model with both a finite standard queue and an orbit. While in the orbit, customers try to join the standard queue in their own time. We assume that the retrial rate is a decision variable, and study both the Nash equilibrium and the socially optimal retrial rates, under a cost model that considers both waiting costs and retrial costs.

Abstract: Bulk-service multi-server queues with heterogeneous server capacity and thresholds are commonly seen in several situations such as passenger transport or package delivery services. In this paper, we develop a novel decomposition-based solution approach for such queues using arguments from renewal theory. We then obtain the distribution of the waiting time measure for multi-type server systems. We also obtain other useful performance measures such as utilization, expected throughput time, and expected queue lengths.

Abstract: We investigate the stability condition of a multiserver queueing system. Each customer needs simultaneously a random number of servers to complete the service. The times taken by each server are independent. The input flow is assumed to be a regenerative one. The service time has an exponential, phase-type or hyper-exponential distribution. The stability criteria for the models are established. It turns out that the stability conditions do not depend on the structure of the input flow, but only on the rate of the process. However, the distribution of the service times is a very important factor. We give examples which show that the stability condition cannot be expressed only in terms of the mean of the service time.

Abstract: The aim of this work concerns the performance analysis of systems with interacting queues under the join the shortest queue policy. The case of two interacting queues results in a two-dimensional random walk with bounded transitions to non-neighboring states, which in turn results in complicated boundary behavior. Although this system violates the conditions of the compensation approach due to the transitions to non-neighboring states, we show how to extend the framework of the approach and how to apply it to the system at hand. Moreover, as an additional level of theoretic validation, we have compared the results obtained with the compensation approach with those obtained using the power series algorithm and we have shown that the compensation approach outperforms the power series algorithm in terms of numerical efficiency. In addition to the fundamental contribution, the results obtained are also of practical value, since our analysis constitutes a first attempt toward gaining insight into the performance of such interacting queues under the join the shortest queue policy. To fully comprehend the benefits of such a protocol, we compare its performance to the Bernoulli routing scheme as well as to that of the single relay system. Extensive numerical results show interesting insights into the system’s performance.

Abstract: Polling systems are systems consisting of multiple queues served by a single server. In this paper, we analyze polling systems with a server that is self-ruling, i.e., the server can decide to leave a queue, independent of the queue length and the number of served customers, or stay longer at a queue even if there is no customer waiting in the queue. The server decides during a service whether this is the last service of the visit and to leave the queue afterward, or it is a regular service followed, possibly, by other services. The characteristics of the last service may be different from the other services. For these polling systems, we derive a relation between the joint probability generating functions of the number of customers at the start of a server visit and, respectively, at the end of a server visit. We use these key relations to derive the joint probability generating function of the number of customers and the Laplace transform of the workload in the queues at an arbitrary time. Our analysis in this paper is a generalization of several models including the exponential time-limited model with preemptive-repeat-random service, the exponential time-limited model with non-preemptive service, the gated time-limited model, the Bernoulli time-limited model, the 1-limited discipline, the binomial gated discipline, and the binomial exhaustive discipline. Finally, we apply our results on an example of a new polling discipline, called the 1 + 1 self-ruling server, with Poisson batch arrivals. For this example, we compute numerically the expected sojourn time of an arbitrary customer in the queues.

Abstract: Donsker Theorem is perhaps the most famous invariance principle result for Markov processes. It states that when properly normalized, a random walk behaves asymptotically like a Brownian motion. This approach can be extended to general Markov processes whose driving parameters are taken to a limit, which can lead to insightful results in contexts like large distributed systems or queueing networks. The purpose of this paper is to assess the rate of convergence in these so-called diffusion approximations, in a queueing context. To this end, we extend the functional Stein method introduced for the Brownian approximation of Poisson processes, to two simple examples: the single-server queue and the infinite-server queue. By doing so, we complete the recent applications of Stein's method to queueing systems, with results concerning the whole trajectory of the considered process, rather than its stationary distribution.

Abstract: This paper considers a multi-type fluid queue with priority service. The input fluid rates are modulated by a Markov chain, which is common for all fluid types. The service rate of the queue is constant. Various performance measures are derived, including the Laplace–Stieltjes transform and the moments of the stationary waiting time of the fluid drops and the queue length distributions. An Erlangization-based numerical method is also provided to approximate the waiting time and the queue length distributions up to arbitrary precision. All performance measures are formulated as reward accumulation problems during busy periods of simple Markovian fluid flow models, for which efficient matrix-analytic solutions are provided, enabling us to solve large models with several hundred states.

Abstract: In this note, we prove that the speed of convergence of the workload of a Levy-driven queue to the quasi-stationary distribution is of order 1/t. We identify also the Laplace transform of the measure giving this speed and provide some examples.

Abstract: We consider a model for transitory queues in which only a finite number of customers can join. The queue thus operates over a finite time horizon. In this system, also known as the $$\Delta _{(i)}/G/1$$ queue, the customers decide independently when to join the queue by sampling their arrival time from a common distribution. We prove that, when the queue satisfies a certain heavy-traffic condition and under the additional assumption that the second moment of the service time is finite, the rescaled queue length process converges to a reflected Brownian motion with parabolic drift. Our result holds for general arrival times, thus improving on an earlier result Bet et al. (Math Oper Res 2019, https://doi.org/10.1287/moor.2018.0947) which assumes exponential arrival times.

Abstract: A result of Ward and Glynn (Queueing Syst 50(4):371–400, 2005) asserts that the sequence of scaled offered waiting time processes of the $$GI/GI/1+GI$$ queue converges weakly to a reflected Ornstein–Uhlenbeck process (ROU) in the positive real line, as the traffic intensity approaches one. As a consequence, the stationary distribution of a ROU process, which is a truncated normal, should approximate the scaled stationary distribution of the offered waiting time in a $$GI/GI/1+GI$$ queue; however, no such result has been proved. We prove the aforementioned convergence, and the convergence of the moments, in heavy traffic, thus resolving a question left open in 2005. In comparison with Kingman’s classical result (Kingman in Proc Camb Philos Soc 57:902–904, 1961) showing that an exponential distribution approximates the scaled stationary offered waiting time distribution in a GI / GI / 1 queue in heavy traffic, our result confirms that the addition of customer abandonment has a non-trivial effect on the queue’s stationary behavior.

Abstract: A mean-field extension of the queueing system (GI/GI/1) is considered. The process is constructed as a Markov solution of a martingale problem. Uniqueness in distribution is also established under a slightly different set of assumptions on intensities in comparison with those required for existence.

Abstract: We consider a general k-dimensional discounted infinite server queueing process (alternatively, an incurred but not reported claim process) where the multivariate inputs (claims) are given by a k-dimensional finite-state Markov chain and the arrivals follow a renewal process. After deriving a multidimensional integral equation for the moment-generating function jointly to the state of the input at time t given the initial state of the input at time 0, asymptotic results for the first and second (matrix) moments of the process are provided. In particular, when the interarrival or service times are exponentially distributed, transient expressions for the first two moments are obtained. Also, the moment-generating function for the process with deterministic interarrival times is considered to provide more explicit expressions. Finally, we demonstrate the potential of the present model by showing how it allows us to study semi-Markovian modulated infinite server queues where the customers (claims) arrival and service (reporting delay) times depend on the state of the process immediately before and at the switching times.

Abstract: Men and jobs alike are characterized by a single trait, which may take on categorical values according to given population frequencies. Men arrive to the system following a Poisson process and wait till jobs are assigned to them. Jobs arrive to the system following another, independent, Poisson process. An arriving job must be assigned to a waiting man immediately, or be discarded, ensuing no gain. An assignment of a job to a man yields a higher gain if they match in trait, and a lower one if not. Each man waits a limited time for a job and leaves the system if unassigned by that time limit. It is stipulated that a man who arrives first has priority to either accept the pending job, or to pass it to the next man, who makes a similar decision. The last man in the line takes the job, or it is discarded. The individually optimal policy for each man is defined by some critical time for accepting a mismatched job. We solve for the critical times, depending on the mens’ place in the queue, and obtain expressions for the ensuing optimal value functions of this system, for expected gain. The model originates from the utility-equity dilemma in assigning live organs to patients on the national waiting list. The paper reports numerical comparison of the above policy with alternative ones, for several performance measures.