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This paper considers a special class of continuous-time Markov chains of level-dependent M/G/1-type, where block matrices representing downward jumps in the infinitesimal generator are nonsingular. This special class naturally arises in the analysis of BMAP/M/$$\infty $$ź queues and BMAP/M/c queues with exponential impatience times (BMAP/M/c+M). We first formulate the boundary probability vector in terms of a solution of a system of infinitely many linear inequalities. We then reveal that in the above special class, this infinite system is regarded as a nested sequence of simplices, and we identify their vertices. Based on these results, we develop a simple yet efficient computational algorithm for the stationary distribution conditioned that the level is not greater than a predefined N. Note that for a large N, the conditional distribution will provide a good approximation to the stationary distribution. Some numerical examples for BMAP/M/$$\infty $$ź and BMAP/M/c+M queues are shown.

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.

This paper studies a system consisting of two parallel queues with transfers of customers. In the system, one queue is called main queue and the other one is called auxiliary queue. The main queue is monitored at exponential time instances. At a monitoring instant, if the number of customers in main queue reaches L\((>K)\), a batch of \(L-K\) customers is transferred from the main queue to the auxiliary queue, and if the number of customers in main queue is less than or equal to K, the transfers will not happen. For this system, by using a Foster-Lyapunov type condition, we establish a sufficient stability condition. Then, we provide a sufficient condition under which, for any fixed number of customers in the auxiliary queue, the stationary probability of the number of customers in the main queue has an exact geometric tail asymptotic as the number of customers in main queue increases to infinity. Finally, we give some numerical results to illustrate the impact of some critical model parameters on the decay rate.

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

We consider a single-server polling system consisting of two queues of fluid with arrival process generated by a big number of heavy-tailed On/Off sources, and application in road traffic and communication systems. Class-j fluid is assigned to queue j, $$j=1,2$$j=1,2. Server 2 visits both queues to process or let pass the corresponding fluid class. If there is class-2 fluid in the system, it is processed by server 2 until the queue is empty, and only then server 2 visits queue 1, revisiting queue 2 and restarting the cycle as soon as new class-2 fluid arrives, with zero switchover times. Server 1 is an "extra" server which continuously processes class-1 fluid (if there is any). During the visits of server 2 to queue 1, class-1 fluid is simultaneously processed by both servers (possibly at different speeds). We prove a heavy-traffic limit theorem for a suitable workload process associated with this model. Our limit process is a two-dimensional reflected fractional Brownian motion living in a convex polyhedron. A key ingredient in the proof is a version of the Invariance Principle of Semimartingale reflecting Brownian motions which, in turn, is also proved.

This paper considers a discrete-time Geo/G/1 queue under the Min(N,D)-policy in which the idle server resumes its service if either N customers accumulate in the system or the total backlog of the service times of the waiting customers exceeds D, whichever occurs first (Min(N,D)-policy). By using renewal process theory and total probability decomposition technique, the authors study the transient and equilibrium properties of the queue length from the beginning of the arbitrary initial state, and obtain both the recursive expression of the z-transformation of the transient queue length distribution and the recursive formula for calculating the steady state queue length at arbitrary time epoch n +. Meanwhile, the authors obtain the explicit expressions of the additional queue length distribution. Furthermore, the important relations between the steady state queue length distributions at different time epochs n -, n and n + are also reported. Finally, the authors give numerical examples to illustrate the effect of system parameters on the steady state queue length distribution, and also show from numerical results that the expressions of the steady state queue length distribution is important in the system capacity design.

This paper studies a finite buffer size Geo X / G / 1 / N queue with single working vacation and different input rates. By combining two classic methods of supplementary variable and embedded Markov chain, the matrix-form solutions of queue-length distributions at departure, arbitrary, pre-arrival and outside observer's observation epochs are obtained. Based on these queue-length distributions, some performance measures are also derived and some numerical experiments are carried out. Then, an optimal control on state-dependent operating cost of a production to order is presented. HighlightsAnalyze a new queue model raised from practical production system.Obtain the solutions in form of formulae for queue length distributions at different epochs.Optimal analysis for the state-dependent cost of system.

Multi-server queueing systems with retrials are widelyused to model problems in a call center. We present an explicitformula for an approximation of the queue length distribution in amulti-server retrial queue, by using the Lerch transcendent. Accu-racy of our approximation is shown in the numerical examples. 1. IntroductionRetrial queues are queueing systems in which arriving customerswho find all servers occupied may retry for service again after a ran-dom amount of time. Retrial queues have been widely used to modelmany problems/situations in telephone systems, call centers, telecom-munication networks, computer networks and computer systems, and indaily life. For an overview regarding retrial queues, refer to the sur-veys [9, 13, 14]. For further details, refer to the books [7, 10] and thebibliographies [3, 4, 5].Typically a call center consists of a finite number of servers that an-swer customer’s calls, and it can be modelled as a queueing system. In aqueueing model of a call center, the customers are callers and the serversare either telephone operators or communication equipment. Queues areformed by callers who are waiting service. The call center can be de-scribed as follows: When a customer’s call arrives, it will be servedimmediately if a server is available. However, if all servers are busy at

This paper explores a discrete-time Geo / G / 1 queueing system with N-policy and D-policy wherein customers arrive at the system in variable input rates according to the states of the server. When the system becomes empty, the server stays idle until the number of waiting customers reaches N or the sum of the service times of the waiting customers in the system reaches or exceeds a predetermined positive integer D, whichever happens first (referred to as Min(N, D)-policy). Employing the renewal theory, the total probability decomposition law and the probability generating function technique we obtain the probability generating function for the transient queue size distribution at time epoch $$n^+$$ . Also, the explicit recursive formulas of the steady-state queue length distribution at time epochs $$n^+$$ , n, $$n^-$$ and outside observer’s time epoch are derived, respectively. Finally, some numerical experiments are conducted to investigate the sensitivity of system performance measures and the optimization of system capacity.

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.

In this paper, a priority polling system consisting of three M / M / 1 queues, served by a single server is investigated. Queue 1 has the Head-of-Line (HoL) priority and Queue 2 has a higher priority over Queue 3 with threshold N. All the switches are instantaneous and preempting. Using the Kernel method we derive the probability of generating functions of the stationary joint queue-length distributions, which yields the mean queue lengths and the mean sojourn times. Furthermore, we consider the limit behaviors in the light-traffic and heavy-traffic scenarios. And an interpolation approximation for the sojourn times utilizing the light and heavy traffic limits are illustrated. To test the validity, we also undertake some simulation works.

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.

A discrete-timeGeo/G/1queue with vacations in random environment is analyzed. Using the method of supplementary variable, we give the probability generating function (PGF) of the stationary queue length distribution at arbitrary epoch. The PGF of the stationary sojourn time distribution is also derived. And we present the various performance measures such as mean number of customers in the system, mean length of the type-icycle, and mean time that the system resides in phase0. In addition, we show that theM/G/1queue with vacations in random environment can be approximated by its discrete-time counterpart. Finally, we present some special cases of the model and numerical examples.

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.

A network constructed according to the IMS architecture consists of different modules that sequentially process signaling messages transferred during providing communication services. Delays that occur when processing signaling messages determine the quality of service for subscribers. International recommendations define the quality-of-service metrics, which include not only the mean, but also the 95% quantile. The paper proposes an approach to analyzing the quality-of-service characteristics. It is assumed that IMS-network nodes work as multi-server queues with general service time distribution.

A generalised Pollaczek–Khinchin formula for the Polya/G/1 queue, with a Polya peaked arrival process, general distributed service times, and infinite number of waiting positions, is obtained. It is shown that the peakedness of the number of arrivals and the variance of the service time lead to a significant increase in the service delay and queue length.

International audience; Motivated by experiments on customers' behavior in service systems, we consider a queueing model with event-dependent arrival rates. Customers' arrival rates depend on the last event which may either be a service departure or an arrival. We derive explicitly the performance measures and analyze the impact of the event-dependency. In particular, we show that this queueing model, in which a service completion generates a higher arrival rate than an arrival, performs better than a system in which customers are insensitive to the last event. Moreover, contrary to the M/G/1 queue, we show that the coefficient of variation of the service does not necessarily deteriorate the system performance. Next, we show that this queueing model may be the result of customer's strategic behavior when only the last event is known. Finally, we investigate the historical admission control problem. We show that under certain conditions a determin-istic policy with two thresholds may be optimal. This new policy is easy to implement and provides an improvement compared to the classical one-threshold policy.

We investigate the transient and stationary queue-length distributions of a class of service systems with correlated service times. The classical MX/G/1 queue with semi-Markov service times is the most prominent example in this class and serves as a vehicle to display our results. The sequence of service times is governed by a modulating process J(t). The state of J(⋅) at a service initiation time determines the joint distribution of the subsequent service duration and the state of J(⋅) at the next service initiation. Several earlier works have imposed technical conditions on the zeros of a matrix determinant arising in the analysis, that are required in the computation of the stationary queue length probabilities. The imposed conditions in several of these articles, are difficult or impossible to verify. Without such assumptions, we determine both the transient and the steady-state joint distribution of the number of customers immediately after a departure and the state of the process J(t) at the start of the next service. We numerically investigate how the mean queue length is affected by variability in the number of customers that arrive during a single service time. Our main observations here are that increasing variability may reduce the mean queue length, and that the Markovian dependence of service times can lead to large queue lengths, even if the system is not in heavy traffic.

In this short communication we study a fluid queue with a finite buffer. The performance measure we are interested in is the occupation time over a finite time period, i.e., the fraction of time the workload process is below some fixed target level. We construct an alternating sequence of sojourn times $D_1,U_1,...$ where the pairs $(D_i,U_i)_{i\in\mathbb{N}}$ are i.i.d. random vectors. We use this sequence to determine the distribution function of the occupation time in terms of its double transform.

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.

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.

We study the paper of Gao and Liu (2013). In the paper, the authors pay little attention to the customer’s sojourn time in their model although both the queue length and the busy cycle are seriously analyzed. The objective of this note is to derive the sojourn time distribution of Gao and Liu’s model without considering Bernoulli schedule. A simple numerical example is also given to demonstrate our result.

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.

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 study the optimal buffer capacity K for the M/M/1/K queue under some standard cost and reward structures by comparing various Markov reward processes. Using explicit expressions for the deviation matrix of the underlying Markov chains, we find the bias optimal value for K in the case of a tie between two consecutive optimal gain policies. We show that the bias optimal value depends both on whether the reward is granted upon arrival or departure of the customers, and on the initial queue size. Moreover, we demonstrate that in some specific cases the optimal policy is threshold-based with respect to the initial queue 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.

This paper studies an M/G/1 queue in a multi-phase random environment. When in operative phase i, i=1,2,…,n, the system is subject to disastrous interruptions, causing all present customers (waiting and served) to leave the system. At an exponential failure instant, the server abandons the service and the system goes directly to repair phase. After an exponential repair time, the system moves to operative phase i with probability qi, i=1,2,…,n. Using the supplementary variable technique, we obtain the distribution for the stationary queue at an arbitrary epoch. We also derive results of the cycle analysis, the sojourn time distribution and the length of the server's working time in a service cycle. In addition, some special cases and numerical examples are presented.

In this paper we consider the MMAP/PH/1 priority queue, both the case of preemptive resume and the case of non-preemptive service. The main idea of the presented analysis procedure is that the sojourn time of the low priority jobs in the preemptive case (and the waiting time distribution in the non-preemptive case) can be represented by the duration of the busy period of a special Markovian fluid model. By making use of the recent results on the busy period analysis of Markovian fluid models it is possible to calculate several queueing performance measures in an efficient way including the sojourn time distribution (both in the time domain and in the Laplace transform domain), the moments of the sojourn time, the generating function of the queue length, the queue length moments and the queue length probabilities.

A queueing system with 2 parallel workstations is common in the field. Typically, the workstations have different features in terms of the inter arrival times of customers and the service times for the customers. Computer simulation study on the optimal server allocation for parallel heterogeneous queueing systems with fixed number of identical servers is presented in this paper. The queueing system is optimized with respect to minimizing the weighted system time of the customers served by 2 parallel workstations. The system time formula for the M/M/c systems in Kendall’s notation is known. Thus, we first compute the optimal allocation for parallel M/M/c systems, comparing the results with those from the computer simulation experiments, and have the same results. The CETI rule is devised through optimizing M/M/c cases, which allocates the servers based on Close or Equal Traffic Intensities between workstations. Traffic intensity is defined as the arrival rate divided by the service rate times the number of servers. The CETI rule is shown to work for M/G/c, G/M/c queueing systems by numerous computer simulation experiments, even if the rule cannot be proven analytically. However, the CETI rule is shown not to work for some of G/G/c systems.

We analyze an M/G/1/K queueing system with queue-length dependent service and arrival rates. There are a single server and a buffer with finite capacity K including a customer in service. The customers are served by a first-come-first-service basis. We put two thresholds and ( ) on the buffer. If the queue length at the service initiation epoch is less than the threshold , the service time of customers follows with a mean of and the arrival of customers follows a Poisson process with a rate of . When the queue length at the service initiation epoch is equal to or greater than and less than , the service time is changed to with a mean of . The arrival rate is still . Finally, if the queue length at the service initiation epoch is greater than , the arrival rate of customers are also changed to a value of and the mean of the service times is . By using the embedded Markov chain method, we derive queue length distribution at departure epochs. We also obtain the queue length distribution at an arbitrary time by the supplementary variable method. Finally, performance measures such as loss probability and mean waiting time are presented.

This article considers a finite-source queueing model of M/G/1 type in which a customer, arriving at a moment of a busy server, is not allowed either to queue or to do repetitions. Instead, for an exponentially distributed time interval he is blocked in the orbit of inactive customers. We carry out a steady state analysis of the system and compare it with the corresponding system with retrials. Optimization problems are considered and formulas for the Laplace–Stieltjes transform of the busy period length are obtained.

This paper investigates the interrelationship between the randomized F- policy and randomized N- policy in discrete-time queues with start-up time. The F-policy queuing system deals with the issue of controlling arrivals. The N-policy queueing system deals with the issue of controlling service, that is, when all the customers are served in the queue, the server is deactivated until N customers are accumulated in the queue. Using the recursive method, the steady-state probabilities are determined for both models. The relationships between the discrete-time G e o/G e o/1/K queues with (p, F)- and (q, N)-policies are established by a series of propositions. The benefit made by interrelationship is that the solution of one queue can be deduced from the other queue readily. Various performance measures and numerical results are carried out for illustration purposes. These queues are common in wireless communication systems, production systems, manufacturing systems.

In this paper we consider the stationary PH/M/c queue with deterministic impatience times (PH/M/c+D). We show that the probability density function of the virtual waiting time takes the form of a matrix exponential whose exponent is given explicitly by system parameters.

We consider an $$\textit{M/G/1}$$ retrial queueing system with two classes of customers, in which the service time distributions are different for both classes of customers. When the server is unavailable, an arriving class-1 customer is queued in the queue with infinite capacity, whereas class-2 customer enters the retrial group. In this paper, we are concerned with the analysis of the waiting time distribution. We obtain the joint transform of the waiting time of a class-2 customer and the number of class-2 customers as well as the Laplace–Stieltjes transform of the waiting time of a class-1 customer. We also obtain all the moments of the waiting time distributions of class-1 and class-2 customers.

This paper studies a single server M/G/1 stochastic clearing queue operating in a 3-phase environment, where the time length of the first and third phase are assumed to follow exponential distributions, and the time length of the second phase is a constant value. At the completion of phase 1, the system moves to phase 2, and after a fixed time length, the system turns to phase 3. At the end of phase 3, all present customers in the system are forced to leave the system, then the system moves to phase 1 and restarts a new service cycle. Using the supplementary variable technique, we obtain the distribution for the stationary queue at an arbitrary epoch. We also derive the sojourn time distribution and the length of the server’s working time in a cycle.

We consider the Erlang A model, or \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M/M/m+M$$\end{document}M/M/m+M queue, with Poisson arrivals, exponential service times, and m parallel servers, and the property that waiting customers abandon the queue after an exponential time. The queue length process is in this case a birth–death process, for which we obtain explicit expressions for the Laplace transforms of the time-dependent distribution and the first passage time. These two transient characteristics were generally presumed to be intractable. Solving for the Laplace transforms involves using Green’s functions and contour integrals related to hypergeometric functions. Our results are specialized to the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M/M/\infty $$\end{document}M/M/∞ queue, the M / M / m queue, and the M / M / m / m loss model. We also obtain some corresponding results for diffusion approximations to these models.

We study the stationary distribution of a system of two parallel M/M/∞ queues managed by the Join the Shortest Queue load balancing policy; one motivation for characterizing the efficiency of that policy is its potential application to resource allocation issues in cloud computing. For the general system with distinct service rates, we first show that the tail of each marginal queue length distribution exhibits a much faster decay than that of a Poisson distribution. Second, the determination of the joint stationary distribution is shown to reduce to the resolution of a pair of linear integral equations. In the case when service rates are identical (“symmetric case”), that pair of integral equations simplifies to a single Fredholm integral equation of the first kind whose solution is explicitly given in terms of Legendre polynomials; this enables us to entirely determine the stationary distribution of the system. We provide, in particular, asymptotics for the second moment and the tail of the queue length...

Recently, queues with speed scaling have received considerable attention due to their applicability to data centers, enabling a better balance between performance and energy consumption. This paper proposes a new model where blocked customers must leave the service area and retry after a random time, with retrial rate either varying proportionally to the number of retrying customers (linear retrial rate) or non-varying (constant retrial rate). For both, we study the case without and with setup time. In all four cases, we obtain an exact solution for the stationary queue length distribution. This document presents the resulting expressions as well as their derivation.

This paper considers stationary MAP/M/c and M/PH/c queues with constant impatience times. In those queues, waiting customers leave the system without receiving their services if their elapsed waiting times exceed a predefined deterministic threshold. For the MAP/M/c queue with constant impatience times, Choi et al. (Math Oper Res 29:309---325, 2004) derive the virtual waiting time distribution, from which the loss probability and the actual waiting time distribution are obtained. We first refine their result for the virtual waiting time and then derive the stationary queue length distribution. We also discuss the computational procedure for performance measures of interest. Next we consider the stationary M/PH/c queue with constant impatience times and derive the loss probability, the waiting time distribution, and the queue length distribution. Some numerical results are also provided.

The performance analysis of the classical M / G / 1 queue, under a general mixed joining/balking strategy was carried out recently by Kerner (Stoch Mod 24:364–375, 2008), who used an analytic approach based on the supplementary variable method. The tractability of the corresponding queueing system with state-dependent arrival rates is particularly significant, as it has important applications in situations where the customers are strategic. In this paper, we present an alternative path for the analysis of the same system, using purely probabilistic arguments.

This paper presents a discrete-time single-server finite-buffer queue with Markovian arrival process and generally distributed batch-size-dependent service time. Given that infinite service time is not commonly encountered in practical situations, we suppose that the distribution of the service time has a finite support. Recently, a similar continuous-time system with Poisson input process was discussed by Banerjee and Gupta (2012). But unfortunately, their method is hard to apply in the analysis of discrete-time case with versatile Markovian point process due to the fact that the difference equation governing the boundary state probabilities is more complex than the continuous one. If we follow their ideas, we will eventually find that some important joint queue length distributions cannot be computed and thus some key performance measures cannot be derived. In this paper, replacing the finite support renewal distribution with an appropriate phase-type distribution, the joint state probabilities at various time epochs (arbitrary, pre-arrival and departure) have been obtained by using matrix analytic method and embedded Markov chain technique. Furthermore, UL-type RG-factorization is employed in numerical computation of block-structured Markov chains with finitely-many levels. Some numerical examples are presented to demonstrate the feasibility of the proposed algorithm for several service time distributions. Moreover, the impact of the correlation factor on loss probability and mean sojourn time is also investigated.

In this paper, we are concerned with the analysis of the queue length and waiting time distributions in a batch arrival $$M^X/G/1$$ retrial queue. Necessary and sufficient conditions are obtained for the existence of the moments of the queue length and waiting time distributions. We also provide recursive formulas for the higher order moments of the queue length and waiting time distributions.

We consider the loss probability in the stationary M/G/1+G queue, i.e., the stationary M/G/1 queue with impatient customers whose impatience times are generally distributed. It is known that the loss probability is given in terms of the probability density function v(x) of the virtual waiting time and that v(x) is given by a formal series solution of a Volterra integral equation. In this paper, we show that the series solution of v(x) can be interpreted as the probability density function of a random sum of dependent random variables and we reveal its dependency structure through the analysis of a last-come first-served, preemptive-resume M/G/1 queue with workload-dependent loss. Furthermore, based on this observation, we show some properties of the loss probability.

In this paper, we carry out an analysis for a single server queue with impatient customers and multiple vacations where customers impatience is due to an absentee of servers upon arrival. Customers arrive at the system according to a Poisson process and exponential service times. Explicit expressions are obtained for the time dependent probabilities, mean and variance of the system size in terms of the modified Bessel functions, by employing the generating functions along with continued fractions and the properties of the confluent hypergeometric function. Finally, some numerical illustrations are provided.

This paper provides the first exact analysis of a preemptive M/M/c queue with two priority classes having different service rates. To perform our analysis, we introduce a new technique to reduce the two-dimensionally infinite Markov chain (MC), representing the two class state space, into a one-dimensionally infinite MC, from which the generating function (GF) of the number of low-priority jobs can be derived in closed form. (The high-priority jobs form a simple M/M/c system and are thus easy to solve.) We demonstrate our methodology for the c = 1, 2 cases; when c > 2, the closed-form expression of the GF becomes cumbersome. We thus develop an exact algorithm to calculate the moments of the number of low-priority jobs for any c ≥ 2. Numerical examples demonstrate the accuracy of our algorithm and generate insights on (i) the relative effect of improving the service rate of either priority class on the mean sojourn time of low-priority jobs; (ii) the performance of a system having many slow servers compared with one having fewer fast servers; and (iii) the validity of the square root staffing rule in maintaining a fixed service level for the low-priority class. Finally, we demonstrate the potential of our methodology to solve other problems using the M/M/c queue with two priority classes, where the high-priority class is completely impatient.

We consider an M/M/1 queueing system with vacations and impatient customers. Whenever a customer arrives at the system, it activates an random ``impatience timer". If the customer's service has not been completed before the customer's impatience timer expires, the customer abandons the queue, and never returns. It is assumed that the impatience timer depends on the server's states. We analyze both multiple and single vacation scenarios and derive the probability generating functions of the number of customers in the system when the server is in vacation period and busy period. Then, we obtain explicit expressions for various performance measures such as the mean system sizes when the server is either on vacation or busy, the proportion of customers served, and the average rate of abandonments due to impatience. We present some numerical results for multiple vacation scenario to show the effects of the parameters of impatience timers on some performance measures. Finally, we show some inequalities on some performances under the single vacation policy and under multiple vacation policy.