It is interesting and challenging to study double-ended queues with First-Come-First-Match discipline under customers' impatient behavior and non-Poisson inputs. The system stability can be guaranteed by the customers' impatient behavior, while the existence of impatient customers makes analysis of such double-ended queues more difficult or even impossible to find an explicitly analytic solution, thus it becomes more and more important to develop effective numerical methods in a variety of practical matching problems. This paper studies a block-structured double-ended queue, whose block structure comes from two independent Markovian arrival processes (MAPs), which are non-Poisson inputs. We show that such a queue can be expressed as a new bilateral quasi birth-and-death (QBD) process which has its own interest. Based on this, we provide a detailed analysis for both the bilateral QBD process and the double-ended queue, including the system stability, the queue size distributions, the average stationary queue lengths, and the sojourn time of any arriving customers. Furthermore, we develop three effective algorithms for computing the performance measures (i.e., the probabilities of stationary queue lengths, the average stationary queue lengths, and the average sojourn times) of the double-ended queue with non-Poisson inputs. Finally, we use some numerical examples in tabular and graphical to illustrate how the performance measures are influenced by some key system parameters. We believe that the methodology and results described in this paper can be applicable to deal with more general double-ended queues in practice, and develop some effective algorithms for the purpose of many actual uses.
We investigate Markovian queues that are examined by a controller at random times determined by a Poisson process. Upon examination, the controller sets the service speed to be equal to the minimum of the current number of customers in the queue and a certain maximum service speed; this service speed prevails until the next examination time. We study the resulting two-dimensional Markov process of queue length and server speed, in particular two regimes with time scale separation, specifically for infinitely frequent and infinitely long examination times. In the intermediate regime the analysi…
The mathematics of the finite single server queue with Poisson input and semi-Markov service times($M/SM/1/b$) is similar to that used for $BMAP/G/1/b$ systems. This observation results in new analytical formulas for a queue size in the $M/SM/1/b$ system. Both stationary and the transient solutions are considered
In this article it is shown that if the busy period of a MGinf queue system is PME distributed, the service time is a random variable with a long tail distribution. The result is obtained through Laplace transforms analysis.
The queue system,with Poisson arrivals,constant service time and infinite servers, busy period distribution is intensively studied because, due to its probability density function quite easy interpretation, it may serve as a clue to interpret the queue systems, with Poisson arrivals, infinite servers, and other service time distributions, busy period distributions. In this work the queue system, with Poisson arrivals,constant service time and infinite servers, busy period distribution parameters are exactly computed and its probability density function is interpreted and explained. The only pr…
This paper purpose is to investigate exponential behavior conditions for the infinite servers queue with Poisson arrivals busy period length distribution. It is presented a general theoretical result that is the basis of this work. The complementary analysis rely on the infinite servers queue with Poisson arrivals busy period length distribution moments computation. In infinite servers queue with Poisson arrivals practical applications, in economic, management, and business areas, the management of the effective number of servers is essential since the physical presence of infinite servers is …
This paper considers the Poisson equation for general state-space Markov chains in continuous time. The main purpose of this paper is to present specific bounds for the solutions of the Poisson equation for general state-space Markov chains. The solutions of the Poisson equation are unique in the sense that they are expressed in terms of a certain probabilistically interpretable solution (called the {\it standard solution}). Thus, we establish some specific bounds for the standard solution under the $f$-modulated drift condition (which is a kind of Foster-Lyapunov-type condition) and some mode…
In the infinite servers queue with Poisson arrivals real life practical applications, the busy period and the busy cycle probabilistic study is of main importance. But it is a very difficult task. In this text, we show that by solving a Riccati equation induced by this queue transient probabilities monotony study as time functions, we obtain a collection of service length distribution functions, for which both the busy period and the busy cycle have lengths with quite simple distributions, generally given in terms of exponential distributions and the degenerate at the origin distribution.
In this paper, we consider the occupancy distribution for an open network of infinite server queues with multivariate batch arrivals following a non-homogeneous Poisson process, and general service time distributions. We derive a probability generating function for the transient occupancy distribution of the network, and prove that it is necessary and sufficient for ergodicity that the expected occupancy time for each batch be finite. Further, we recover recurrence relations for the transient probability mass function formulated in terms of a distribution obtained by compounding the batch size…
The busy period length distribution function knowledge is important for any queue system, and for the MGINF queue. But the mathematical expressions are in general very complicated, with a few exceptions, involving usually infinite sums and multiple convolutions. So, in this work are deduced some bounds for the MMINF system busy period length distribution function, meaning the second M exponential service time, which analytic expressions are simpler than the exact one. As a consequence, also some bounds for the MMINF system busy cycle length distribution function are presented.
The infinite servers queue with Poisson arrivals state transient probabilities, considering the time origin at the beginning of a busy period, mean and variance monotony as time functions is studied. These studies, for which results it is determinant the hazard rate function service time length, induce the consideration of two differential equations, one related with the mean monotony study and another with the variance monotony study, which solutions lead to some particular service time distributions, for which those parameters present specific behaviors as time functions.
This article provides entropic inequalities for binomial-Poisson distributions, derived from the two point space. They appear as local inequalities of the M/M/$\infty$ queue. They describe in particular the exponential dissipation of $Φ$-entropies along this process. This simple queueing process appears as a model of ``constant curvature'', and plays for the simple Poisson process the role played by the Ornstein-Uhlenbeck process for Brownian Motion. Some of the inequalities are recovered by semi-group interpolation. Additionally, we explore the behaviour of these entropic inequalities under a…
We study a queueing network with a single shared server that serves the queues in a cyclic order. External customers arrive at the queues according to independent Poisson processes. After completing service, a customer either leaves the system or is routed to another queue. This model is very generic and finds many applications in computer systems, communication networks, manufacturing systems, and robotics. Special cases of the introduced network include well-known polling models, tandem queues, systems with a waiting room, multi-stage models with parallel queues, and many others. A complicat…
In discrete time, customers arrive at random. Each waits until one of two servers is available; each thereafter departs at random. We seek the distribution of maximum line length of idle customers. In the context of an emergency room (for medical treatment), the virtue of one fast doctor over two slow doctors is explored. Via limiting argument to continuous time, we study likewise the M/M/2 queue.
In continuous time, customers arrive at random. Each waits until one of $c$ servers is available; each thereafter departs at random. The distribution of maximum line length of idle customers was studied over 25 years ago. We revisit two good approximations of this, employing a discrete Gumbel formulation and detailed graphics to describe simulation outcomes.
In discrete time, $\ell$-blocks of red lights are separated by $\ell$-blocks of green lights. Cars arrive at random. We seek the distribution of maximum line length of idle cars, and justify conjectured probabilistic asymptotics for $2 \leq \ell \leq 3$.
This paper considers a discrete-time single-server queue with a single acceptance period for a Poissonian population of homogeneous customers. Customers are served on a first-come first-served (FCFS) basis, and their service times are independent and identically distributed with a general distribution. We assume that each customer chooses her/his arrival-time slot with the goal of minimizing her/his expected waiting time in competition with other customers. For this queueing game, we derive a symmetric (mixed-strategy) Nash equilibrium; that is, an equilibrium arrival-time distribution of homo…
This paper aims at semi-parametrically estimating the input process to a Lévy-driven queue by sampling the workload process at Poisson times. We construct a method-of-moments based estimator for the Lévy process' characteristic exponent. This method exploits the known distribution of the workload sampled at an exponential time, thus taking into account the dependence between subsequent samples. Verifiable conditions for consistency and asymptotic normality are provided, along with explicit expressions for the asymptotic variance. The method requires an intermediate estimation step of estimatin…
Motivated by applications that involve setting proper staffing levels for multi-server queueing systems with batch arrivals, we present a thorough study of the queue-length process $\{Q(t); t \geq 0\}$, departure process $\{D(t); t \geq 0\}$, and the workload process $\{W(t); t \geq 0\}$ associated with the M$_{t}^{B_{t}}$/G$_{t}$/$\infty$ queueing system. With two fundamental assumptions of (non-stationary) Poisson arrivals and infinitely many servers, we otherwise maintain a highly general model, in which the service duration and batch size distributions may depend on time and, moreover, whe…
We study a queueing network with a single shared server, that serves the queues in a cyclic order according to the gated service discipline. External customers arrive at the queues according to independent Poisson processes. After completing service, a customer either leaves the system or is routed to another queue. This model is very generic and finds many applications in computer systems, communication networks, manufacturing systems and robotics. Special cases of the introduced network include well-known polling models and tandem queues. We derive exact limits of the mean delays under both …
We study a double-ended queue which consists of two classes of customers. Whenever there is a pair of customers from both classes, they are matched and leave the system immediately. The matching follows first-come-first-serve principle. If a customer from one class cannot be matched immediately, he/she will stay in a queue and wait for the upcoming arrivals from the other class. Thus there cannot be non-zero numbers of customers from both classes simultaneously in the system. We also assume that each customer can leave the queue without being matched because of impatience. The arrival processe…
We consider a queue where the server is the Euclidean space, and the customers are random closed sets (RACS) of the Euclidean space. These RACS arrive according to a Poisson rain and each of them has a random service time (in the case of hail falling on the Euclidean plane, this is the height of the hailstone, whereas the RACS is its footprint). The Euclidean space serves customers at speed 1. The service discipline is a hard exclusion rule: no two intersecting RACS can be served simultaneously and service is in the First In First Out order: only the hailstones in contact with the ground melt …
We consider Poisson's equation for quasi-birth-and-death processes (QBDs) and we exploit the special transition structure of QBDs to obtain its solutions in two different forms. One is based on a decomposition through first passage times to lower levels, the other is based on a recursive expression for the deviation matrix. We revisit the link between a solution of Poisson's equation and perturbation analysis and we show that it applies to QBDs. We conclude with the PH/M/1 queue as an illustrative example, and we measure the sensitivity of the expected queue size to the initial value.
Queues that feature multiple entities arriving simultaneously are among the oldest models in queueing theory, and are often referred to as "batch" (or, in some cases, "bulk") arrival queueing systems. In this work we study the affect of batch arrivals on infinite server queues. We assume that the arrival epochs occur according to a Poisson process, with treatment of both stationary and non-stationary arrival rates. We consider both exponentially and generally distributed service durations and we analyze both fixed and random arrival batch sizes. In addition to deriving the transient mean, vari…
In this paper we generalize known workload decomposition results for Lévy queues with secondary jump inputs and queues with server vacations or service interruptions. Special cases are polling systems with either compound Poisson or more general Lévy inputs. Our main tools are new martingale results, which have been derived in a companion paper.
We consider Poisson streams of exponentially distributed jobs arriving at each edge of a hypergraph of queues. Upon arrival, an incoming job is rooted 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 provide a generic condition for stability of this model. We show that some graph topologies lead to a loss of capacity, implying more restrictive stability conditions than in, e.g., complete graphs.
We study a double-ended queue where buyers and sellers arrive to conduct trades. When there is a pair of buyer and seller in the system, they immediately transact a trade and leave. Thus there cannot be non-zero number of buyers and sellers simultaneously in the system. We assume that sellers and buyers arrive at the system according to independent renewal processes, and they would leave the system after independent exponential patience times. We establish fluid and diffusion approximations for the queue length process under a suitable asymptotic regime. The fluid limit is the solution of an o…
We consider a Markovian queue subject to Poisson generated catastrophes. Whenever a catastrophe occurs, all customers are forced to abandon the system, the server is rendered inoperative and an exponential repair time is set on. We assume that the arriving customers decide whether to join the system or balk, based on a natural reward-cost structure. We study the balking behavior of the customers and derive the corresponding Nash equilibrium strategies.
We consider a controlled double-ended queue consisting of two classes of customers, labeled sellers and buyers. The sellers and buyers arrive in a trading market according to two independent renewal processes. Whenever there is a seller and buyer pair, they are matched and leave the system instantaneously. The matching follows first-come-first-match service discipline. Those customers who cannot be matched immediately need to wait in the designated queue, and they are assumed to be impatient with generally distributed patience times. The control problem is concerned with the trade-off between …
In Internet environment, traffic flow to a link is typically modeled by superposition of ON/OFF based sources. During each ON-period for a particular source, packets arrive according to a Poisson process and packet sizes (hence service times) can be generally distributed. In this paper, we establish heavy traffic limit theorems to provide suitable approximations for the system under first-in first-out (FIFO) and work conserving service discipline, which state that, when the lengths of both ON- and OFF-periods are lightly tailed, the sequences of the scaled queue length and workload processes c…
Consider a system performing a continuous-time random walk on the integers, subject to catastrophes occurring at constant rate, and followed by exponentially-distributed repair times. After any repair the system starts anew from state zero. We study both the transient and steady-state probability laws of the stochastic process that describes the state of the system. We then derive a heavy-traffic approximation to the model that yields a jump-diffusion process. The latter is equivalent to a Wiener process subject to randomly occurring jumps, whose probability law is obtained. The goodness of th…
The problem of appropriately matching items subject to compatibility constraints arises in a number of important applications. While most of the literature on matching theory focuses on a static setting with a fixed number of items, several recent works incorporated time by considering a stochastic model in which items of different classes arrive according to independent Poisson processes and assignment constraints are described by an undirected non-bipartite graph on the classes. In this paper, we prove that the Markov process associated with this model has the same transition diagram as in a…
We study a token-based central queue with multiple customer types. Customers of each type arrive according to a Poisson process and have an associated set of compatible tokens. Customers may only receive service when they have claimed a compatible token. If upon arrival, more than one compatible token is available, an assignment rule determines which token will be claimed. The service rate obtained by a customer is state-dependent, i.e., it depends on the set of claimed tokens and on the number of customers in the system. Our first main result shows that, provided the assignment rule and the s…
In this paper, we consider a discrete-time preemptive priority queue with different service rates for two classes of customers, one with high-priority and the other with low-priority. This model corresponds to the classical preemptive priority queueing system with two classes of independent Poisson customers and a single exponential server. Due to the possibility of customers' arriving and departing at the same time in a discrete-time queue, the model considered in this paper is more complicated. In this model, we focus on the characterization of exact tail asymptotics for the joint stationary…
In this paper we study a two-queue polling model with zero switch-over times and $k$-limited service (serve at most $k_i$ customers during one visit period to queue $i$, $i=1,2$) in each queue. The arrival processes at the two queues are Poisson, and the service times are exponentially distributed. By increasing the arrival intensities until one of the queues becomes critically loaded, we derive exact heavy-traffic limits for the joint queue-length distribution using a singular-perturbation technique. It turns out that the number of customers in the stable queue has the same distribution as th…
We analyze the behavior of closed product-form queueing networks when the number of customers grows to infinity and remains proportionate on each route (or class). First, we focus on the stationary behavior and prove the conjecture that the stationary distribution at non-bottleneck queues converges weakly to the stationary distribution of an ergodic, open product-form queueing network. This open network is obtained by replacing bottleneck queues with per-route Poissonian sources whose rates are determined by the solution of a strictly concave optimization problem. Then, we focus on the transie…
In this paper we derive exact large-buffer asymptotics for a two-class Generalized Processor Sharing (GPS) model, under the assumption that the input traffic streams generated by both classes correspond to heavy-tailed Lévy processes. Four scenarios need to be distinguished, which differ in terms of (i)~the level of heavy-tailedness of the driving Lévy processes as well as (ii)~the values of the corresponding mean rates relative to the GPS weights. The derived results are illustrated by two important special cases, in which the queues' inputs are modeled by heavy-tailed compound Poisson proces…
In the Hammersley-Aldous-Diaconis process infinitely many particles sit in R and at most one particle is allowed at each position. A particle at x$ whose nearest neighbor to the right is at y, jumps at rate y-x to a position uniformly distributed in the interval (x,y). The basic coupling between trajectories with different initial configuration induces a process with different classes of particles. We show that the invariant measures for the two-class process can be obtained as follows. First, a stationary M/M/1 queue is constructed as a function of two homogeneous Poisson processes, the arriv…
There are $n$ queues, each with a single server. Customers arrive in a Poisson process at rate $λn$, where $0<λ<1$. Upon arrival each customer selects $d\geq2$ servers uniformly at random, and joins the queue at a least-loaded server among those chosen. Service times are independent exponentially distributed random variables with mean 1. We show that the system is rapidly mixing, and then investigate the maximum length of a queue in the equilibrium distribution. We prove that with probability tending to 1 as $n\to\infty$ the maximum queue length takes at most two values, which are $\ln\ln n/\l…
We consider a single-server GI/GI/1 queueing system with feedback. We assume the service times distribution to be (intermediate) regularly varying. We find the tail asymptotics for a customer's sojourn time in two regimes: the customer arrives in an empty system, and the customer arrives in the system in the stationary regime. In particular, in the case of Poisson input we use the branching processes structure and provide more precise formulae. As auxiliary results, we find the tail asymptotics for the busy period distribution in a single-server queue with an intermediate varying service times…
We consider a queueing system with $n$ parallel queues operating according to the so-called "supermarket model" in which arriving customers join the shortest of $d$ randomly selected queues. Assuming rate $nλ_{n}$ Poisson arrivals and rate $1$ exponentially distributed service times, we consider this model in the heavy traffic regime, described by $λ_{n}\uparrow 1$ as $n\to\infty$. We give a simple expectation argument establishing that majority of queues have steady state length at least $\log_d(1-λ_{n})^{-1} - O(1)$ with probability approaching one as $n\rightarrow\infty$, implying the same …
Motivated by demand prediction for the custodial prison population in England and Wales, this paper describes an approach to the study of service systems using infinite server queues, where the system has non-empty initial state and the elapsed time of individuals initially present is not known. By separating the population into initial content and new arrivals, we can apply several techniques either separately or jointly to those sub-populations, to enable both short-term queue length predictions and longer-term considerations such as managing congestion and analysing the impact of potential …
Using a result of Blanchet and Wallwater (2015: Exact sampling of stationary and time-reversed queues. ACM TOMACS, 25, 26) for exactly simulating the maximum of a negative drift random walk queue endowed with independent and identically distributed (iid) increments, we extend it to a multi-dimensional setting and then we give a new algorithm for simulating exactly the stationary distribution of a first-in-first-out (FIFO) multi-server queue in which the arrival process is a general renewal process and the service times are iid; the FIFO GI/GI/c queue with 2 \le c < 1. Our method utilizes domin…
We consider a polling system where a group of an infinite number of servers visits sequentially a set of queues. When visited, each queue is attended for a random time. Arrivals at each queue follow a Poisson process, and service time of each individual customer is drawn from a general probability distribution function. Thus, each of the queues comprising the system is, in isolation, an M/G/$\infty$-type queue. A job that is not completed during a visit will have a new service time requirement sampled from the service-time distribution of the corresponding queue. To the best of our knowledge, …
Consider a surplus process which both of collected premium and payed claim size are two independent compound Poisson processes. This article derives two approximated formulas for the ruin probability of such surplus process, say double stochastic compound poisson process. More precisely, it provides two mixture exponential approximations for ruin probability of such double stochastic compound poisson process. Applications to long_term Bonus_Malus systems and a heavy-tiled claim size distribution have been given. Improvement of our findings compared to the Cramer- Lundberg upper bound has been …
This paper studies a discrete model of a large dam where the difference between lower and upper levels, $L$, is assumed to be large. Passage across the levels leads to damage, and the damage costs of crossing the lower or upper level are proportional to the large parameter $L$. Input stream of water is described by compound Poisson process, and the water cost depends upon current level of water in the dam. The aim of the paper is to choose the parameters of output stream (specifically defined in the paper) minimizing the long-run expenses that include the damage costs and water costs. The pres…
This paper studies an infinite buffer single server queueing model with exponentially distributed service times and negative arrivals. The ordinary (positive) customers arrive in batches of random size according to renewal arrival process, and joins the queue/server for service. The negative arrivals are characterized by two independent Poisson arrival processes, a negative customer which removes the positive customer undergoing service, if any, and a disaster which makes the system empty by simultaneously removing all the positive customers present in the system. Using the supplementary varia…
In this paper we prove the Poisson Hypothesis for the limiting behavior of the large queueing systems in some simple ("mean-field") cases. We show in particular that the corresponding dynamical systems, defined by the non-linear Markov processes, have a line of fixed points which are global attractors. To do this we derive the corresponding non-linear integral equation and we explore its self-averaging properties. Our derivation relies on a solution of a combinatorial problem of rode placements.
The single server queue with multiple customer types and semi-Markovian service times, sometimes referred to as the $M/SM/1$ queue, has been well-studied since its introduction by Neuts in 1966. In this paper, we apply an extension of this model, with batch arrivals and exceptional first service, to road traffic situations involving multiple streams of conflicting traffic. In particular, we use it in the context of gap acceptance models where low-priority traffic needs to cross (or, depending on the application, merge with) another traffic flow of higher priority. Traditionally, gap acceptance…
We consider a queueing system consisting of two non-identical exponential servers, where each server has its own dedicated queue and serves the customers in that queue FCFS. Customers arrive according to a Poisson process and join the queue promising the shortest expected delay, which is a natural and near-optimal policy for systems with non-identical servers. This system can be modeled as an inhomogeneous random walk in the quadrant. By stretching the boundaries of the compensation approach we prove that the equilibrium distribution of this random walk can be expressed as a series of product-…
In the supermarket model there are n queues, each with a unit rate server. Customers arrive in a Poisson process at rate λn, where 0<λ<1. Each customer chooses d > 2 queues uniformly at random, and joins a shortest one. It is known that the equilibrium distribution of a typical queue length converges to a certain explicit limiting distribution as n -> oo. We quantify the rate of convergence by showing that the total variation distance between the equilibrium distribution and the limiting distribution is essentially of order n^{-1}; and we give a corresponding result for systems starting from q…
In this paper we study the Poisson Hypothesis, which is a device to analyze approximately the behavior of large queueing networks. We prove it in some simple limiting cases. We show in particular that the corresponding dynamical system, defined by the non-linear Markov process, has a line of fixed points which are global attractors. To do this we derive the corresponding non-linear equation and we explore its self-averaging properties. We also argue that in cases of havy-tail service times the PH can be violated.
