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In this paper, we consider the Bayesian inference of M/M/𝑅 queue with 𝑅 heterogeneous servers with service rates \mu_{1},\mu_{2},\ldots,\mu_{R}, where \mu_{1}>\mu_{2}>\cdots>\mu_{R}. Assuming multivariate gamma prior distribution for service rates and gamma prior distribution for arrival rate 𝜆, we derive the conditional posterior densities of mean arrival rate and mean service rates. We apply the Markov chain Monte Carlo method and compute the Bayes estimates and credible interval for the M/M/3 queue, as a particular case of the M/M/𝑅 queue under squared error loss function, entropy loss function and linex loss function corresponding to a different set of hyperparameters.

In this paper, we consider an M/M/ ∞ queue with catastrophes. We study the model and obtain the time dependent solution by using continued fraction techniques. The corresponding steady stat...

Central-limit (Brownian) approximations are widely used for the performance analysis and optimization of queueing networks because of their tractability relative to the original queueing models. The stationary distributions of the approximations are used as proxies for those of the queues. The convergence of suitably scaled and centered processes provides mathematical support for the use of these Brownian models. As with the central limit theorem, to establish convergence, one must impose assumptions directly on the primitives or indirectly on the parameters of a related optimization problem. These assumptions reflect an interpretation of the underlying parameters—a classification into so-called heavy-traffic regimes that specify a scaling relationship between the utilization and the arrival rate. Here, it matters whether a utilization of 90% in a queue with an arrival rate of λ = 100 is read as [Formula: see text] or as ρ(λ) ≡ 0.9, because different interpretations lead to different limits and, in turn, to different approximations. However, from a heuristic point of view, there is an immediate Brownian (i.e., normal) analogue of the queueing model that is derived directly from the primitives and requires no scaling interpretation of the parameters. In this model, the drift is that of the original queue, and the noise term is replaced by a Brownian motion with the same variance. This is intuitive and appealing as a tool, but it lacks mathematical justification. In this paper, we prove that for the fundamental M/GI/1 + GI queue, this direct intuitive approach works: the Brownian model is accurate uniformly over a family of patience distributions and universally in the heavy-traffic regime. The validity of this approach extends to dynamic control in that the solution of the directly derived diffusion control problem is universally accurate. To build mathematical support for the accuracy of this model, we introduce a framework built around “queue families” that allows us to treat various patience distributions simultaneously, and it uncovers the role of a concentration property of the queue. The e-companion is available at https://doi.org/10.1287/opre.2017.1715 .

We consider a non-preemptive priority M / M / m queue with two classes of customers and multiple vacations. Service times for all customers are exponentially distributed with the same mean, and vacation times follow an exponential distribution. We obtain the vector probability generating function for the stationary distribution of the number of customers in the queue for each class. This is established by deriving a matrix equation for the vector probability generating function of the stationary distribution of the censored Markov process and then studying the analytical properties of the matrix generating function. We also obtain exact expressions for the first two moments of the number of customers in the queue for each class. Finally, as an application, we investigate a customer’s equilibrium strategy and the optimal priority fee associated with social cost minimization for an unobservable M / M / m queue with two priority classes and multiple vacations.

We consider an M X /M/c queue with catastrophes and state-dependent control at idle time. Properties of the queues which terminate when the servers become idle are first studied. Recurrence, equilibrium distribution, and equilibrium queue-size structure are studied for the case of resurrection and no catastrophes. All of these properties and the first effective catastrophe occurrence time are then investigated for the case of resurrection and catastrophes. In particular, we obtain the Laplace transform of the transition probability for the absorbing M X /M/c queue.

This paper is concerned with an M/PH/K queue with customer abandonment, constant impatient time, and many servers. By combining the method developed in Choi et al. (Math Oper Res 29:309–325, 2004) and Kim and Kim (Perform Eval 83–84:1–15, 2015) and the state space reduction method introduced in Ramaswami (Stoch Models 1:393–417, 1985), the paper develops an efficient algorithm for computing performance measures for the queueing system of interest. The paper shows a number of properties associated with matrices used in the development of the algorithm, which make it possible for the algorithm, under certain conditions, to handle systems with up to one hundred servers. The paper also obtains analytical properties of performance measures that are useful in gaining insight into the queueing system of interest.

We offer a new profit-maximizing mechanism for Naor’s M/M/1 model, and bound the loss incurred when the waiting room’s size is limited or the server is restricted to a static price and FCFS discipline.

We propose a vacation queue model for a signalized traffic intersection to elucidate intra-cycle queue size variations. Each incoming street to the intersection is modeled to experience Poisson arrivals, and to have finite queue capacity; vehicles arriving to full queue are dropped. Under a fixed-time policy, the queue size dynamics on each street are decoupled. Motivated by vacation queues, an imbedded Markov chain corresponding to queue sizes at the end of cycles is considered, whose transition probabilities are computed from analytical transient solutions of M/D/1/N queues. Sufficient conditions for convergence to steady-state distributions are provided for fixed-time policy. Simulations suggest consistency between the queue sizes computed by the proposed model, and the Webster and time-dependent traffic queue models.

In this paper, we study the steady state behaviour of an M/G/1 queue with two types of general heterogeneous service and optional repeated service subject to server’s breakdowns occurring randomly ...

A queueing system (QS) with finite size holds a current number N, and maximum number M, of present customers (waiting or being served), N ∊ {0, 1, … , M}. We model the system as family of Birth–Death Processes in equilibrium, with size M + 1, indexed by the system utilization parameter. Depending on the concepts of information i and entropy S = E(i) of QS, we promote an information analysis of queueing systems. Introducing the information stiffness of QS, the system information (entropy) consists of three basic parts: null, elastic (Hooke’s), and synchronization part. A QS can function in information elastic, antielastic, or inelastic regime. We use the coefficients of information elasticity and of unpredictability to determine the optimal arrival/service mode of customers. This analysis is illustrated on the information linear (i.e. M/M/1/M and M/M/1), harmonically encouraged arrivals (or discouraged service), and M/M/1/M/M type QS. Also, short remarks are given for the Erlang loss, Poisson, and ...

The paper considers retrial queueing system M /M /1/ 0 with combined service discipline, namely, a customer from the orbit is serviced in its turn, but in case of a free channel an arrival from the original flow is serviced immediately. The author obtains the expressions for state probabilities as well as ergodicity conditions. The system is compared with the Lakatos-type system.

A discrete time Geo/Geo/1 queue with (m, N)-policy is considered in this paper. There are three operation periods being considered: high speed, low speed service periods and idle periods. With double thresholds policy, the server begins to take a working vacation when the number of customers is below m after a service and there is one customer in the system at least. What’s more, if the system becomes empty after a service, the server will take an ordinary vacation. Otherwise, high speed service continues if the number of customers still exceeds m after a service. At the vacation completion instant, servers resume their service if the quantity of customers exceeds N. Vacations can also be interrupted when the system accumulate customers more than the prefixed threshold. Using the quasi birth-death process and matrix-geometric solution methods, we derive the stationary queue length distribution and some system characteristics of interest. Based on these, we apply the queue to a virtual channel switching system and present various numerical experiments for the system. Finally, numerical results are offered to illustrate the optimal (m, N)-policy to minimize cost function and obtain practical consequence on the operation of double thresholds policy.

We present an analysis of the queueing system in which arriving jobs are dropped with probability depending on the queue size. The arrivals are assumed to be autocorrelated and they are modeled by the Markov-modulated Poisson process. Both transient and stationary distributions of the queue size, as well as the system loss ratio and throughput are obtained. The analytical results are accompanied with numerical examples based on the autocorrelated traffic recorded in an IP computer network.

In this paper, we study a single server GI/M/1 queue in a multi-phase service environment with disasters, where the disasters occur only when the server is busy serving customers. Whenever a disaster occurs in an operative service phase, all present customers are forced to leave the system simultaneously, the server abandons the service and an exponential repair time is set on. After the system is repaired, the server resumes his service and moves to service phase i immediately with probability q i ,i = 1,2, ... ,N . Using the matrix analytic approach and semi-Markov process, we obtain the stationary queue length distribution at both arrival and arbitrary epochs. After introducing tagged customers and the concept of a cycle, we also derive the sojourn time distribution, the duration of a cycle, and the length of the server’s working time in a service cycle. In addition, numerical examples are presented to illustrate the impact of some critical model parameters on performance measures.

This paper analyzes the customers’ equilibrium strategic behavior in the M/M/1 queue subject to Poisson generated partial failures and repairs. In such a queueing system, whenever a partial failure occurs, the system does not admit any customer and continues the service at a lower rate instead of stopping working completely. This goes on till the system becomes empty and, at that point, an exponential repair time is sent on. The arriving customer’s dilemma is whether to enter or balk based on a linear reward-cost structure. We discuss the fully observable and fully unobservable queues respectively. For each type of queue, the corresponding equilibrium balking strategies of customers and the expected social benefits per time unit are derived. Finally, we illustrate the theoretical results by presenting some numerical examples.

An analysis of the open queueing network MAP − ( G I / ∞ ) K is presented in this paper. The MAP − ( G I / ∞ ) K network implements Markov routing, general service time distribution, and an infinite number of servers at each node. Analysis is performed under the condition of a growing fundamental rate for the Markovian arrival process. It is shown that the stationary probability distribution of the number of customers at the nodes can be approximated by multi-dimensional Gaussian distribution. Parameters of this distribution are presented in the paper. Numerical results validate the applicability of the obtained approximations under relevant conditions. The results of the approximations are applied to estimate the optimal number of servers for a network with finite-server nodes. In addition, an approximation of higher-order accuracy is derived.

This paper investigates the M/M/s queueing model to predict an estimate for the proximity of the performance measures of queues with arrival processes that are slightly different from the Poisson process. Specifically, we use the strong stability method to obtain perturbation bounds on the effect of perturbing the arrival process in the M/M/s queue. Therefore, we build some algorithms based on strong stability method to predict stationary performance measures of the GI/M/s queue. Some numerical examples are sketched out to illustrate the accuracy of the proposed method.

This article develops a computational algorithm for the loss probability in the stationary M/G/1 queue with impatient customers whose impatience times follow a phase-type distribution (M/G/1+PH). T...

Bayes estimator of different queueing performance measures are derived in steady state by recording system size from each of n iid M/M/1 queues. The Bayes estimators are obtained under both squared error loss function and precautionary loss function with a bivariate distribution beta stacy as prior, with natural restriction 0 < λ < µ where λ and µ are arrival rate and service rate respectively. A comprehensive simulation results are also shown at the last section.

We establish a limit theorem supporting a Poisson approximation for the departure process from a multi-server queue that tends to have many busy servers. This limit can support approximating a flow out of such a queue in a complex queueing network by an independent Poisson source. The main ideas are: (i) to scale time so that previous many-server heavy-traffic limits can be applied and (ii) for time-varying arrival-rate functions, to scale (spread out) time by a large factor about each fixed time.

This note discusses the inter-loss time ofan M/G/1/1 queuing system. The inter-loss time is defined as the time duration between two consecutive losses of arriving customers. In this study, we present the explicit Laplace transform of the inter-loss time distribution of an M/G/1/1 queuing system.

We propose a method to analyze M/G/1/m queuing systems with the function of random dropping of customers and service time distribution dependent on the queue length. We obtain formulas to determine the Laplace transforms of the distribution of the number of customers in the system during busy period and of the distribution function of the busy period and to calculate the stationary characteristics. The relations for the stationary characteristics are tested using simulation models constructed with the use of the GPSS World tools. The results of the use of various control tools of system parameters are compared in the example.

An M / M / C queueing system operating in a Markovian environment is studied. This paper focuses on the stationary behavior and presents the theoretical framework. For a special case, analytical results are derived that are analogous to the classical solutions for the simple M / M / C queue. The elaborate analysis of a specific case is given to illustrate the basic idea of the framework. A technical proof with respect to the existence of d − 1 roots is displayed to sustain the corresponding theory.

This paper deals with the study of M (t)/M/1 queueing system with customers arrive to the system with sinusoidal arrival rate function λ (t) and are served exponentially with the rate μ. On formulating the mathematical model, we obtain the expressions for mean waiting time in the queue, mean time spent in the system, mean number of customers in the queue and in the system by using recursive method. Some numerical illustrations are also obtained by using computing software so as to show the applicability of the model under study.Journal of the Institute of Engineering, 2015, 11(1): 120-127

Burke's theorem is a well-known fundamental result in queueing theory, stating that a stationary M/M/1 queue has a departure process that is identical in law to the arrival process and, moreover, for each time t, the following three random objects are independent: the queue length at time t, the arrival process after t and the departure process before t. Burke's theorem also holds for a stationary Brownian queue. In particular, it implies that a certain "complicated" functional derived from two independent Brownian motions is also a Brownian motion. The aim of this overview paper is to present an independent complete explanation of this phenomenon.

This paper focuses on status age, which is a metric for measuring the freshness of a continually updated piece of information (i.e., status) as observed at a remote monitor. In paper, we study a system in which a sensor sends random status updates over a dynamic network to a monitor. For this system, we consider the impact of having messages take different routes through the network on the status age. First, we consider a network with plentiful resources (i.e., many nodes that can provide numerous alternate paths), so that packets need not wait in queues at each node in a multihop path. This system is modeled as a single queue with an infinite number of servers, specifically as an $M/M/\infty $ queue. Packets routed over a dynamic network may arrive at the monitor out of order, which we account for in our analysis for the $M/M/\infty $ model. We then consider a network with somewhat limited resources, so that packets can arrive out of order but also must wait in a queue. This is modeled as a single queue with two servers, specifically an $M/M/2$ queue. We present the exact approach to computing the analytical status age, and we provide an approximation that is shown to be close to the simulated age. We also compare both models with $M/M/1$ , which corresponds to severely limited network resources, and we demonstrate the tradeoff between the status age and the unnecessary network resource consumption.

This paper deals with optimal control points of M(t)/M/c/c queues with periodic arrival rates and two levels of the number of servers. We use the results of this model to build a Markov decision process (MDP). The problem arose from a case study in the Kelowna General Hospital (KGH). The KGH uses surge beds when the emergency room is overcrowded which results in having two levels for the number of the beds. The objective is to minimize a cost function. The findings of this work are not limited to the healthcare; They may be used in any stochastic system with fluctuation in arrival rates and/or two levels of the number of servers, i.e., call centers, transportation, and internet services. We model the situation and define a cost function which needs to be minimized. In order to find the cost function we need transient solutions of the M(t)/M/c/c queue. We modify the fourth-order Runge–Kutta to calculate the transient solutions and we obtain better solutions than the existing Runge–Kutta method. We show that the periodic variation of arrival rates makes the control policies time-dependent and periodic. We also study how fast the policies converge to a periodic pattern and obtain a criterion for independence of policies in two sequential cycles.

A performance model for a two-node computing cluster is studied.Customers' impatience is considered for an M/M/2 queue with synchronous single working vacation.Explicit expressions are derived by using hypergeometric function and the generating function approach.The steady-state probabilities of the server, mean waiting time in the queue and queue length are obtained.Key performance measures such as sojourn time of a customer and the proportion of customers served are obtained. This paper proposes a queueing model for the performance evaluation of a computing cluster. Requests arrive to the configurations according to a Poisson process at rate λ. There is a load balancing hardware in front of two servers to route requests to a free server if any. The service is commenced according to the first-come-first-served (FCFS) principle. The system can be either in the normal state or in the working vacation state. The service time of each customer during the normal state is exponentially distributed. We develop the balance equations for the steady-state probabilities and solve the equations by using the generating function and hypergeometric function. We obtain some performance measures for the system, such as the steady-state probabilities of the servers, the mean system sizes, the mean sojourn time of a customer served as well as the proportion of customers served and the rate of abandonment due to impatience. Finally, some numerical results are presented to demonstrate effects of some parameters on these performance measures of the system.

An elegant and simple solution to determine the distributions of queue length at different epochs and the waiting time for the model $$GI^{X}/M/c$$GIX/M/c is presented. In the past, the model $$GI^{X}/M/c$$GIX/M/c has been extensively analyzed using various techniques by many authors. The purpose of this paper is to present a simple and effective derivation of the analytic solution for pre-arrival epoch probabilities as a linear combination of specific geometric terms (except for the boundary probabilities when the number of servers is greater than the maximum batch size) involving the roots of the underlying characteristic equation. The solution is then leveraged to compute the waiting-time distributions of both first and arbitrary customers of an incoming batch. Numerical examples with various arrival patterns and batch size distributions are also presented. The method that is being proposed here not only gives an alternate solution to the existing methods, but it is also analytically simple, easy to implement, and computationally efficient. It is hoped that the results obtained will prove beneficial to both theoreticians and practitioners.

This paper deals with the stationary analysis of a fluid queue driven by anM/M/1queueing model subject to Bernoulli-Schedule-Controlled Vacation and Vacation Interruption. The model under consideration can be viewed as a quasi-birth and death process. The governing system of differential difference equations is solved using matrix-geometric method in the Laplacian domain. The resulting solutions are then inverted to obtain an explicit expression for the joint steady state probabilities of the content of the buffer and the state of the background queueing model. Numerical illustrations are added to depict the convergence of the stationary buffer content distribution to one subject to suitable stability conditions.

In this work, we are interested in the approximation of the stationary characteristics of the GI/M/∞ system by those of an M/M/∞ system. In other words, we propose to study the strong stability of the M/M/∞ system (ideal system) when the arrivals flow is subject to a small perturbation (the GI/M/∞ is the resulting perturbed system). For this purpose, we first determine the approximation conditions of the characteristics of the perturbed queuing system, and under these conditions we obtain the stability inequalities of the stationary distribution of the queue size. To evaluate the performance of the proposed method, we develop an algorithm which allows us to compute the various theoretical results and which is executed on some systems (Coxian2/M/∞ and E2/M/∞) in order to compare its output results with those of simulation.

International audience; This paper deals with a retrial queuing system M/M/C/K with exponential abandonment at which positive and negative primary customers arrive according to Poisson processes. This model is of practical interest: it permits to analyze the performance in call centers or multiprocessor computer systems. For model under study, we find the ergodicity condition and also the approximate solution by applying Value Extrapolation method which includes solving of some algebraic system of equations. To this end, we have resolved the algebraic system in question by different numerical methods. We present also numerical results to analyze the system performance.

A Markovian single-server queueing model with Poisson arrivals and state-dependent service rates, characterized by a logarithmic steady-state distribution, is considered. The Laplace transforms of the transition probabilities and of the densities of the first-passage time to zero are explicitly evaluated. The performance measures are compared with those ones of the well-known M / M / 1 queueing system. Finally, the effect of catastrophes is introduced in the model and the steady-state distribution, the asymptotic moments and the first-visit time density to zero state are determined.

In this paper, we consider a GI/M/1 queue in a multi-phase service environment with disasters and working breakdowns. When the server is working in any normal service phases, it may suffer disastrous interruptions, causing all present customers to leave the system. At an exponential failure instant, the system becomes defective, and goes directly to repair phase. During the repair period, instead of stopping service completely, the system is equipped with a substitute server which continues to provide service to the arriving customers. After an exponential repair time, the substitute server stops service and the system moves to normal operative phase i with probability qi,i=1,2,…,N. Using the matrix analytic approach and semi-Markov process, we obtain the stationary queue length distribution at both arrival and arbitrary epochs. We also provide the elaborate analysis of some performance measures and sojourn time distribution of an arbitrary customer. In addition, some numerical examples are presented.

We consider the M/M/c/c queues in which the customers blocked to enter the service facility retry after a random amount of time and some of idle servers can leave the vacation. The vacation time and retrial time are assumed to be of phase type distribution. Approximation formulae for the distribution of the number of customers in service facility and the mean number of customers in orbit are presented. We provide an approximation for M/M/c/c queue with general retrial time and general vacation time by approximating the general distribution with phase type distribution. Some numerical results are presented.

In this paper we consider an M/M/c queue modified to allow both mass arrivals when the system is empty and the workload to be removed. Properties of queues which terminate when the server becomes idle are firstly developed. Recurrence properties, equilibrium distribution, and equilibrium queue-size structure are studied for the case of resurrection and no mass exodus. All of these results are then generalized to allow for the removal of the entire workload. In particular, we obtain the Laplace transformation of the transition probability for the absorptive M/M/c queue.

A single server Markovian queueing model is considered. The arrivals are allowed to join the queue according to a Poisson distribution and the service takes place according to an exponential distribution. Whenever the system is empty, the server goes for a vacation and return back to the system after N or more customers are found in the system. If the number of customers in the system is less than ‘’ then the server takes another vacation. In this paper, we obtain explicit expressions for the time dependent system size probabilities of such a model using Laplace transform and generating function techniques. Numerical illustrations are added to support the theoretical results obtained.

We consider the loss probability P loss in the stationary M/G/1 queue with generally distributed impatience times (M/G/1+G queue). Recently, it was shown that P loss increases with service times in the convex order. In this paper, we show that P loss also increases with impatience times in the excess wealth order. With these results, we show that P loss in the M/D/1+D queue is smallest among all M/G/1+G queues with the same and finite arrival rate, mean service time, and mean impatience time.

In this paper we analyze the dynamics of a P2P file exchange swarm from a queueing standpoint. In such systems, the service rate a peer receives depends on one mostly fixed component (servers or seeders), and another that scales with the number of peers present. We analyze a class of M / G Processor Sharing queues that describe populations and residual workloads in this situation, characterizing its stationary regime in the case of a fixed population of servers; the result behaves like a combination of M / G / 1 and M / G / ∞ queues. We apply scaling limits to this queue and identify two limiting regimes, depending on whether the server or peer contribution becomes dominant. For the latter, more important case we refine the fluid limit description of the download profile with a suitable functional Gaussian approximation. We also analyze the case of a slowly varying population of servers, extending the fixed case through a quasi-stationary analysis. For practical validation we offer comparisons with detailed packet simulations.

We consider a state-dependent $$M_{n}$$Mn/$$G_{n}$$Gn/1 queueing system with both finite and infinite buffer sizes. We allow the arrival rate of customers to depend on the number of people in the system. Service times are also state dependent and service rates can be modified at both arrivals and departures of customers. We show that the steady-state solution of this system at arbitrary times can be derived using the supplementary variable method, and that the system's state at arrival epochs can be analyzed using an embedded Markov chain. For the system with infinite buffer size, we first obtain an expression for the steady-state distribution of the number of customers in the system at both arbitrary and arrival times. Then, we derive the average service time of a customer observed at both arbitrary times and arrival epochs. We show that our state-dependent queueing system is equivalent to a Markovian birth-and-death process. This equivalency demonstrates our main insight that the $$M_{n}$$Mn/$$G_{n}$$Gn/1 system can be decomposed at any given state as a Markovian queue. Thus, many of the existing results for systems modeled as an M / M / 1 queue can be carried through to the much more practical M / G / 1 model with state-dependent arrival and service rates. Then, we extend the results to the $$M_{n}$$Mn/$$G_{n}$$Gn/1 queueing systems with finite buffer size.

We consider an M/M/1 queue with two vacation policies which comprise single working vacation and multiple vacations, denoted by M/M/1/SMV+MV. Using two methods called R-matrix method and G-matrix method, we obtain the stationary distribution of queue length including the customer being in service and make further analysis on the stationary numbers of customers in the working vacation and vacation period, respectively. The stochastic decomposition results of stationary queue length and the sojourn time of a customer are also derived. Meanwhile, we show that a simple and direct method of decomposition developed in Liu et al. [Stochastic decompositions in the M/M/1 queue with working vacations, Oper. Res. Lett. 35 2007, pp. 595–600] is also applicable to our model. Furthermore, busy period is analysed by the limiting theorem of alternative renewal process. Finally, some boundary properties and numerical analysis on performance measures are presented.