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We consider an MX/G/1 retrial queue, where the batch size and service time distributions have finite exponential moments. We show that the tail of the queue size distribution is asymptotically given by a geometric function multiplied by a power function. Our result generalizes the result of Kim et al. (2007) to the MX/G/1 retrial queue.

Queuing models create lot of interest due to their ready applicability in the analysis of several congestion control systems. In this paper, we develop and analyze K-parallel and series queuing systems which is connected in a single network. Here, it is assumed that there are ‘K’ queues Q 1, Q 2, …, Q k which are connected in parallel and connected in series to another queue Q k+1 . Here the arrivals are in bulk and follows a compound Poisson process. It is assumed that the service completions in K nodes is load dependent and follow Poisson processes. Using the difference differential equations the joint probability generating function of number of customers in each queue is derived. The system performance measures such as average number of customers in each queue, the probability of the system emptiness, the average waiting time of a customer, throughput of the queues and the variability of system size distribution in each queue are derived and analyzed through numerical illustrations. The utility of this model in queue line control is demonstrated through applying it at Tirumala Tirupati Devasthanam which deals with pilgrims is also discussed. It is observed that the load dependent strategy reduces the congestion in queues and mean delays.

This paper presents a simple closed-form analysis for evaluating system-length distributions at various epochs of the discrete-time GI/D-MSP/1 queue. The proposed analysis is based on roots of the associated characteristic equation of the vector-generating function of system-length distribution at prearrival epochs. We provide the steady-state system-length distribution at random epoch by using the classical argument based on Markov renewal theory. The queueing-time distribution has also been investigated. Numerical aspects have been tested for a variety of interarrival- and service-time distributions and a sample of numerical outputs is presented.

We consider the output processes of some elementary queueing models such as the M/M/1/K queue and the M/G/1 queue. An important performance measure for these counting processes is their variance curve v(t), which gives the variance of the number of customers in the time interval [0, t]. Recent work has revealed some non-trivial properties dealing with the asymptotic rate at which the variance curve grows. In this paper we add to these results by finding explicit expressions for the intercept term of the linear asymptote. For M/M/1/K queues our results are based on the deviation matrix of the generator. It turns out that by viewing output processes as Markovian Point Processes and considering the deviation matrix, one can obtain explicit expressions for the intercept term, together with some further insight regarding the BRAVO (Balancing Reduces Asymptotic Variance of Outputs) effect. For M/G/1 queues our results are based on a classic transform of D. J. Daley. In this case we represent the intercept term of the variance curve in terms of the first three moments of the service time distribution. In addition we shed light on a conjecture of Daley, dealing with characterization of stationary M/M/1 queues within the class of stationary M/G/1 queues, based on the variance curve.

This paper presents a heuristic approach to derive the Laplace-Stieltjes transform (LST) and the probability generating function (PGF) of the waiting time distributions of a continuous- and a discrete-time GI/G/1 queue, respectively. This is a new idea to derive the well-known results, the waiting time distribution of GI/G/1 queue, in a different way.

This paper is concerned with the stability of multiclass queueing networks of 2 stations and4 buffers under the longest queue first served discipline (LQFS). For this network, the service priority of a customer is determined by the length of the queue that customer resides in at that time. The main result includes two parts. Firstly, the corresponding fluid model is established, and then it is shown that the queueing networks under LQFS are stable whenever the traffic intensity is strictly less than one for each station.

We consider a workload-barrier M/G/1 queue where service times that overshoot the barrier are truncated. We derive the pdf and expected value of an arbitrary service, the expected number served in a busy period, and related quantities. Examples are given.

This article discusses the steady state analysis of the M / G / 2 queuing system with two heterogeneous servers under new queue disciplines when the classical First Come First Served ‘(FCFS)’ queue discipline is to be violated. Customers are served either by server-I according to an exponential service time distribution with mean rate μ or by server-II with a general service time distribution B ( t ) . Sequel to some objections raised in the literature on the use of the classical FCFS queue discipline in heterogeneous service systems, two alternative queue disciplines (Serial and Parallel) are considered in this work with the objective that if the FCFS is violated then the violation is a minimum in the long run. Using the embedded method under the serial queue discipline and the supplementary variable technique under the parallel queue discipline, we present an exact analysis of the steady state number of customers in the system and most importantly, the actual waiting time expectation of customers in the system. Our work shows that one can obtain all stationary probabilities and other vital measures for this queue under certain simple but realistic assumptions.

This paper considers single server retrial queues with setup time. In the basic model, if the server completes a service and there are no customers in the orbit, the server is turned off immediately. Arriving customers that see the server occupied join the orbit and repeat their attempt after some random time. The new feature of our models is that an arriving customer that sees the server off waits at the server and the server is turned on. The server needs some setup time to be active so as to serve the waiting customer. If the server completes a service and the orbit is not empty, it stays idle waiting for either a new customer or a customer from the orbit. For this model, we obtain explicit expressions for the generating functions of the joint queue length. We then consider an extended model where the server stays idle for a while before being turned off for which explicit solution is also obtained.

Closed, finite queueing networks are applicable to many different manufacturing and service system settings. The incorporation of material handling and transportation networks in finite buffer closed queueing networks is studied. A novel queue decomposition approach using state dependent queues to capture the buffer of finite M/M/1/K queues is shown to be a viable approach for modelling these systems. Each M / M / 1 / K queue is replaced with a coupled state dependent queue plus an M/M/1 queue. An extended mean value analysis (MVA) algorithm is employed to demonstrate the integration of the state dependent queues for the buffers in the approach. Under certain restrictions concerning the network population, finite queueing networks with the state dependent queues acting as buffers should have a product form distribution. This paper focuses on M/M/1/K queues and their transformation while future papers will treat the multi-server case. Several different closed series (i.e. cyclic), merge, and split topological systems of finite queues are analyzed and presented.

We consider a bivariate Markov process f (U(t);S(t)); t � 0g , where U(t) (t � 0) takes values in (0; 1 ) and S(t) (t � 0) takes values in a nite set. We assume that U(t) (t � 0) is skip-free to the left, and therefore we call it the M/G/1-type Markov process. The M/G/1-type Markov process was rst introduced as a generalization of the workload process in the MAP/G/1 queue and its stationary distribution was analyzed under a strong assumption that the conditional innitesimal generator of the underlying Markov chain S(t) given U(t) > 0 is irreducible. In this paper, we extend known results for the stationary distribution to the case that the conditional innitesimal generator of the underlying Markov chain given U(t) > 0 is reducible. With this extension, those results become applicable to the analysis of a certain class of queueing models.

We use a probing strategy to estimate the time dependent traffic intensity in an Mt/Gt/1 queue, where the arrival rate and the general service-time distribution change from one time interval to another, and derive statistical properties of the proposed estimator. We present a method to detect a switch from a stationary interval to another using a sequence of probes to improve the estimation. At the end, we compare our results with two estimators proposed in the literature for the M/G/1 queue.

We show how a shot-noise fluid queue can be considered as the limiting case of a sequence of infinite-server queues with batch arrivals. The shot-noise queue we consider receives fluid amounts at the arrival times of a (time-inhomogeneous) Poisson process, the sizes of which are governed by some probability distribution that may also depend on time. The continuous rate at which fluid leaves the queue is proportional to the current content of the queue. Thus, intuitively, one can think of drops of fluid arriving in batches, which are taken into service immediately upon arrival, at an exponential service rate. We show how to obtain the partial differential equation for (the Laplace–Stieltjes transform of) the queue content at time t , as well as its solution, from the corresponding infinite-server systems by taking appropriate limits. Also, for the special case of a time-homogeneous arrival process, we show that the scaled number of occupied servers in the infinite-server system converges as a process to the shot-noise queue content, implying that finite-dimensional distributions also converge.

Concerning the problem of network congestion risk of computer network service system with transmitted priority, a method about non-preemptive limited-priority M/M/n/m queuing system model is proposed. Firstly, by introducing the parameter r of limited-priority into the model, the priority is converted to a limited priority system, then the system state is divided into r+1 kinds of state. Secondly, by computing the transition probability of different states based on Markov transfer theory, three results of the model: the average waiting time, the average dwelling time and the average queue length are obtained. Lastly, by considering the fairness of the order in which different priority queues receive services, the risk of data service in network system can be greatly reduced. Experimental results show that under the limited-priority, the network system yields better stability.

Chen et al. (2015) studied the equilibrium threshold balking strategies for the fully observable and fully unobservable single-server queues with threshold policy and setup times. The server shuts down whenever the system becomes empty, and is only resumed when the number of customers reaches to a given threshold. Customers decide whether to join or to balk the system based on their observations of the queue length and status of the server at arrival instants. This paper aims to study the partially observable case and the unobservable case. The stationary probability distribution, the mean queue length and the social welfare are derived. The equilibrium strategies for the customers and the system performance under these strategies are analyzed.

We analyze the MX/G/1 queue in the framework of Markov decision processes (MDPs). The service times become known upon arrival, and each job incurs a cost according to a given cost function. The value function is a central concept in MDP theory as it characterizes the value of the system's state with respect to future developments. We derive compact expressions for the generating functions for general families of value functions corresponding to often used cost structures defined in terms of waiting and sojourn times. Moreover, we consider systems with and without setup delays.

The paper deals with two-way communication M/M/1/1 retrial queue where the server during its idle time makes outgoing calls of two types - to the customers in orbit and to the customers outside it. Durations of these calls follow two distinct exponential distributions. After completion of the outgoing call to a customer from orbit, this customer with probability p rejoins the orbit, and with its complementary probability leaves the service area. Using generating functions approach we derive explicit and recursive formulas for the stationary system state distribution and its factorial moments.

We consider in this paper retrial queue with one server that serves a finite number of customers, each one producing a Poisson flow of incoming calls. In addition, after some exponentially distributed idle time the server makes outgoing calls of two types - to the customers in orbit and to the customers outside it. The outgoing calls of both types follow the same exponential distribution, different from the exponential service time distribution of the incoming calls. We derive formulas for computing the steady state distribution of the system state as well as formulas expressing the main performance macro characteristics in terms of the server utilization. Numerical examples are presented.

We analyze theMMPP/G/1/K queue with a modified state-dependent service rate. The service time of customers upon service initiation is changed if the number of customers in the system reaches a threshold. Then, the changed service time is continued until the system becomes empty completely, and this process is repeated. We analyze this system using an embedded Markov chain and a supplementary variable method, and present the queue length distributions at a customer’s departure epochs and then at an arbitrary time.

In this paper we present algorithms for the perfect sampling of single-server time-varying queues with periodic Poisson arrivals under the first come first served (FCFS) discipline. The service durations have periodically time-dependent exponential ($$\mathrm M _t/\mathrm M _t/1$$Mt/Mt/1) or homogeneous general ($$\mathrm M _t/\mathrm G /1$$Mt/G/1) distributions. Assuming a cycle length of 1, we construct discrete dominating processes at the integer instants $$n \in \{0, \pm 1, \ldots \}$$n?{0,±1,?}. Perfect sampling of the $$\mathrm M _t/\mathrm M _t/1$$Mt/Mt/1 queue is obtained using dominated CFTP (Kendall and MOller 2000) when the system is relatively lightly loaded or with the regenerative method (Sigman 2012) in the general case. For the $$\mathrm M _t/\mathrm G /1$$Mt/G/1 queue, perfect sampling is achieved with dominated CFTP.

We consider a tandem fluid system composed of multiple buffers connected in a series. The first buffer receives input from a number of independent homogeneous on-off sources and each buffer provides input to the next buffer. The active (on) periods and silent (off) periods follow general and exponential distribution, respectively. Furthermore, the generally distributed active periods are controlled by an exponential timer. Under this assumption, explicit expressions for the distribution of the buffer content for the first buffer fed by a single source is obtained for the fluid queue driven by discouraged arrivals queue and infinite server queue. The buffer content distribution of the subsequent buffers when the first buffer is fed by multiple sources are found in terms of confluent hypergeometric functions. Numerical results are illustrated to compare the trend of the average buffer content for the models under consideration.

In the study of normal queueing systems, the server’s average service times are generally assumed to be constant. However, in numerous applications this assumption may not be valid. To prevent congestion in overload control telecommunication networks, the transmission rates vary depending on the number of packets waiting in the queue. As traffics in telecommunication networks are of bursty nature and correlated, we assume that arrivals follow the discrete-time Markovian arrival process. This paper analyzes a queueing model in which the server changes its service times (rates) only at the beginning of service depending on the number of customers waiting in the queue. We obtain the steady-state probabilities at various epochs and some performance measures. In addition, varieties of numerical results are discussed to display the effect of the system parameters on the performance measures.

An M2θ/G/1/m queueing system with the batch arrival of customers is considered. In the system, a threshold mechanism with two hysteretic loops is applied to control the service time and input flow intensity. The system receives two independent flows of customers. One of the flows is blocked in the overload mode. Operation modes are switched at the instants when the service of customers is completed. Complete blocking of the input flow starts when the queue length reaches number m. An approach based on Korolyuk’s potential method is proposed for determining the probabilistic characteristics of the system. Laplace transforms of the distributions of the number of customers in the system during the busy period and of the distribution function of the busy period are found. The mean duration of the busy period is determined. Formulas are derived for the stationary distribution of the number of customers in the system, for the probability of service, and for the stationary characteristics of a queue. The results obtained are verified with the use of a simulation model developed with the help of the GPSS World tools.

We study the M � /G/1/m and M � /G/1 queuing systems with the function of the random dropping of customers used to ensure the required characteristics of the system. Each arriving packet of customers can be rejected with a probability defined depending on the queue length at the service beginning of each customer. The Laplace transform for the distribution of the number of customers in the system on the busy period is found, the mean duration of the busy period is determined, and formulas for the stationary distribution of the number of customers in the system are derived via the approach based on the idea of Korolyuk's potential method. The obtained results are verified with the help of a simulation model constructed with the assistance of GPSS World tools.

This paper deals with the ( p , N )-policy M/G/1 queue with an unreliable server and single vacation. Immediately after all of the customers in the system are served, the server takes single vacation. As soon as N customers are accumulated in the queue, the server is activated for services with probability p or deactivated with probability (1 − p ). When the server returns from vacation and the system size exceeds N , the server begins serving the waiting customers. If the number of customers waiting in the queue is less than N when the server returns from vacation, he waits in the system until the system size reaches or exceeds N . It is assumed that the server is subject to break down according to a Poisson process and the repair time obeys a general distribution. This paper derived the system size distribution for the system described above at a stationary point of time. Various system characteristics were also developed. We then constructed a total expected cost function per unit time and applied the Tabu search method to find the minimum cost. Some numerical results are also given for illustrative purposes.

We examine the optimal design of measurements on queues with particular reference to the M/M/1 queue. Using the statistical theory of design of experiments, we calculate numerically the Fisher information matrix for an estimator of the arrival rate and the service rate to find optimal times to measure the queue when the number of measurements is limited for both interfering and non-interfering measurements. We prove that in the non-interfering case, the optimal design is equally spaced. For the interfering case, optimal designs are not necessarily equally spaced. We compute optimal designs for a variety of queuing situations and give results obtained under the $$D$$D- and $$D_s$$Ds-optimality criteria.

We present a Markov model to analyze the queueing behavior of the nonstationary G ( t ) / G ( t ) / s ( t ) + G ( t ) queue. We assume an exhaustive service discipline (where servers complete their current service before leaving) and use acyclic phase-type distributions to approximate the general interarrival, service, and abandonment time distributions. The time-varying performance measures of interest are: (1) the expected number of customers in queue, (2) the variance of the number of customers in queue, (3) the expected number of abandonments, and (4) the virtual waiting time distribution of a customer arriving at an arbitrary moment in time. We refer to our model as G-RAND since it analyzes a general queue using the randomization method. A computational experiment shows that our model allows the accurate analysis of small- to medium-sized problem instances.

An operation of a single-machine manufacturing system is modeled by an unreliable finite-buffer-type queuing system with Poisson arrivals, in which service times, failure-free times and times of repairs are totally independent and exponentially distributed random variables. Applying the idea of embedded Markov chain and the formula of total probability a system of integral equations for the transient conditional queue-size distributions of jobs present in the system at fixed time t is built. The solution of the corresponding system written for Laplace transforms is obtained in a compact form using the potential technique.

We study the paper Zhernovyi and Zhernovyi [Zhernovyi, K.Y. and Y.V. Zhernovyi, “An M θ /G/1/m system with two-threshold hysteresis strategy of service intensity switching,” Journal of Communications and Electronics, Vol.12, No.2(2012), pp.127-140]. In the paper, authors used the Korolyuk potential method to obtain the stationary queue length distribution. Instead, our note makes an attempt to apply the most frequently used methods : the embedded Markov chain and the supplementary variable method. We derive the queue length distribution at a customer"s departure epoch and then at an arbitrary epoch.

Often in a real world system with a fairly large population, members are not individually traceable for various reasons. As a result, the relationship between a member’s behavior and the system’s behavior is quite hard to understand. In this paper, we focus on a fundamental problem in such a system: the relationship between its population size and the life-time distribution for its members. We answer two questions: (A) If the life-time distribution is known and the times when members join are observable, how do we best estimate the population size? (B) If the population size can be observed accurately, how do we estimate the unknown life-time distribution for members? In the paper we focus on (B), using the results of (A) as a basis. We model the system as a G / G I / ∞ queue with incomplete information, where jobs, once entering the queue, are no longer tracked. With this model, the population size is the number of jobs in the queue and the life times of members are the service times of the jobs. The problem (A) is to estimate the number of jobs in the queue, with known arrival times and a known service-time distribution. We show that, in terms of mean square error, the best deterministic estimator for the (stochastic) number of jobs in the system can be constructed using the survival function of the service-time distribution. The problem (B) is to estimate the unknown service-time distribution in a G / G I / ∞ queue where the number of jobs are observable. We demonstrate that the service-time distribution can be inferred indirectly from continuous observations of the number of jobs in the queue, and then propose a few easy-to-implement algorithms. Using only a limited amount of memory, these on-line, streaming algorithms continuously refine their results which, over time, converge to the true service-time distribution.

Many real-life systems can be modeled using the classical M/G/c/N queue. A frequently-used approach is to replace the general service time distribution by a phase-type distribution since the M/Ph/c/N queue can be described by familiar balance equations. The downside of this approach is that the size of the resulting state space suffers from the ''dimensionality curse'', i.e., exhibits combinatorial growth as the number of servers and/or phases increases. To circumvent this complexity issue, we propose to use a reduced state description in which the state of only one server is represented explicitly, while the other servers are accounted for through their rate of completions. The accuracy of the resulting approximation is generally good and, moreover, tends to improve as the number of servers in the system increases. Its computational complexity in terms of the number of states grows only linearly in the number of servers and phases.

Point-queue model is widely used in the dynamic user equilibrium (DUE) analysis in discrete-time or continuous-time form. In this paper, a continuous time point queue is proposed. In the former studies, the negativity of the queue length of the original point-queue model is shown and some improvement has been made. Based on the observation that the original point-queue model is actually a queuing model with a server and a buffer with infinite capacity, a fluid approximation (FA) model is proposed to interpret the original point-queue model. Three essential components are a flow balance function, an exit flow function and a time-dependent capacity utilization ratio function, which are all in continuous form. During the analysis, the theoretical proof and numerical study of the non-negativity of queue length are accomplished. With the first-order Taylor expansion, this paper applies the Gronwall's inequality to prove the non-negativity of queue-length. Through numerical testing different specific FA models in the solution scheme, the authors can show that the negativity of the queue length in the FA model is overcome and some differences between the FA and former studies are discussed. Based on the testing, the capability of the authors' model in approximating the point- queue model is demonstrated.

The M/G/1 model is the fundamental basis of the queueing system in many network systems. Usually, the study of the M/G/1 is limited by the assumption of single queue and infinite capacity. In practice, however, these postulations may not be valid, particularly when dealing with many real-world problems. In this paper, a two-stage state-space approach is devoted to solving the state probabilities for the multi-queue finite-capacity M/G/1 model, i.e. q-M/G/1/Ki with Ki buffers in the ith queue. The state probabilities at departure instants are determined by solving a set of state transition equations. Afterward, an embedded Markov chain analysis is applied to derive the state probabilities with another set of state balance equations at arbitrary time instants. The closed forms of the state probabilities are also presented with theorems for reference. Applications of Little's theorem further present the corresponding results for queue lengths and average waiting times. Simulation experiments have demonstrated the correctness of the proposed approaches.

A multichannel queueing system with an unlimited queue length is considered. In the system, the service time and time intervals between arrivals of customer batches are characterized by exponential distributions and a hysteretic mechanism is applied to control the input flow intensity. The system receives two independent types of flows of customers one of which is blocked in the overload mode. An algorithm for determination of the stationary distribution of the number of customers and the stationary characteristics (the mean queue length, the mean time of waiting in the queue, the probability of a loss of customers) is proposed. The obtained results are verified with the help of a simulation model developed with the help of the GPSS World tools.

Mθ/G/1/m and Mθ/G/1 queues are considered in the case when the service time and input flow parameters depend on the queue length and are determined at the instants of completion of the service of customers. With the help of an approach based on the idea of Korolyuk’s potential method, the Laplace transforms are found for the distribution of the number of customers in a system within the busy period and for the distribution function of the busy period. The mean duration of the busy period and the stationary distribution of the number of customers in a system are determined. An Mθ/G/1 system with one threshold of functioning mode switching is considered as a particular case. The obtained results are verified with the help of simulation models developed with the use of the GPSS World tools.

In this paper, we study a T-IPH/Geo/1 queue model, where T-IPH denotes the discrete-time phase type distribution defined on a birth and death process with countably many states. The queue model can be described by a quasi-birth-and-death process with countably phases. Using operator-geometric solution method, we first give the expression of the operator and the joint stationary distribution. Then we obtain the steady-state distributions for the number of customers in the system, and waiting time for an arbitrary customer.

In this paper, we have obtained explicit expressions for the time dependent probabilities of the M/M/1queue with server vacations under a multiple vacation scheme. The corresponding steady state probabilities have been obtained. We also obtain the time dependent performance measures of the systems

In this paper, two new formulations of the steady-state probability distribution of the number of customers in an M / E r / 1 queue system are presented. On one hand, a new recurrent method which is numerically stable and computationally more efficient than the existing ones, in terms of memory and time, is presented. On the other hand, a new analytic formula, simpler than the current ones, which additionally has associated a probabilistic interpretation of its terms, that gives a new insight about this queue and its behavior, is also presented.

We are concerned with the problem of characterizing the distribution of the maximum number Z(t(0)) of customers during a fixed time interval [0, t(0)] in the M/M/c retrial queue, which is shown to have a matrix exponential form. We present a simple condition on the service and retrial rates for the matrix exponential solution to be explicit or algorithmically tractable. Our methodology is based on splitting methods and the use of eigen-values and eigenvectors. A particularly appealing feature of our solution is that it allows us to obtain global error control. Specifically, we derive an approximating solution p(x; t(0)) = p(x; t(0); epsilon) verifying [P(Z(t(0)) = i + j, for any epsilon > 0 and initial numbers i of busy servers and j of customers in orbit.

In this paper, we consider a batch arrival MX/M/1 queue model with working breakdown. The server may be subject to a service breakdown when it is busy, rather than completely stoping service, it will decrease its service rate. For this model, we analyze a two-dimensional Markov chain and give its quasi upper triangle transition probability matrix. Under the system stability condition, we derive the probability generating function (PGF) of the stationary queue length, and then obtain its stochastic decomposition, which shows the relationship with that of the classical MX/M/ 1 queue without vacation or breakdown. Besides, we also give the Laplace-Stieltjes transform (LST) of the stationary waiting time distribution of an arbitrary customer in a batch. Finally, some numerical examples are given to illustrate the effect of the parameters on the system performance measures.

This paper studies a control problem for optimal switching on and off a cloud computing services modeled by an M=M=1 queue with holding, running and switching costs. The main result is that an average-optimal policy either always runs the system or is an (M; N)- policy defined by two thresholds M and N, such that the system is switched on upon an arrival epoch when the system size accumulates to N and it is switched off upon a departure epoch when the system size decreases to M. We compare the optimal (M; N)-policy with the classical (0; N)-policy and show the non-optimality of it.

In this paper, a closed-form time-dependent busy period queue length probability is obtained for the M / E k / 1 queue. This probability is frequently needed when we compare the length of the busy period and the maximum amount of service that can be rendered to the existing customers. The transient probability is given in terms of the generalized modified Bessel function of the second type of Griffiths et al. (2006a). The queue length probability for the M / M / 1 queue is also presented as a special case.

This paper presents a queue-length analysis of Geoi¾?Gi¾?1 queue with r,N-policy and different input rate. Using a different method, the recursive expressions of queue-length distribution at different epochs are obtained. Furthermore, some performance measures are also investigated. Finally, the Tabu search algorithm is used to search the joint optimum value of r,N, which minimizes the state-dependent operating cost. Copyright © 2014 John Wiley & Sons, Ltd.

In this article, the M/M/k/N/N queue is modeled as a continuous-time homogeneous Markov system with finite state size capacity (HMS/cs). In order to examine the behavior of the queue a continuous-time homogeneous Markov system (HMS) constituted of two states is used. The first state of this HMS corresponds to the source and the second one to the state with the servers. The second state has a finite capacity which corresponds to the number of servers. The members of the system which can not enter the second state, due to its finite capacity, enter the buffer state which represents the system's queue. In order to examine the variability of the state sizes formulae for their factorial and mixed factorial moments are derived in matrix form. As a consequence, the pmf of each state size can be evaluated for any t ∈ ℝ+. The theoretical results are illustrated by a numerical example.

This paper considers a multi-server batch arrival M^X/M/c queue with impatient customers. We give an exact expression for the loss probability, which can be expressed in terms of the waiting time distribution in the standard M^X/M/c queue with no impatience. This solves the problem conjectured by Boots and Tijms (1999).