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In this paper we consider a discrete-time G e o / G / 1 queue with N -policy and setup time, where the customers' input rate varies according to the server's status: idle, setup and busy states. By using the total probability decomposition technique, we study the transient and equilibrium properties of the queue length from the beginning of the arbitrary initial state, and obtain both the recursion expressions of the z -transformation of the transient queue length distribution and the recursion expressions of the steady state queue length distribution at arbitrary time epoch n + . The results obtained in this paper indicate that the queue length distribution in equilibrium no longer follows the stochastic decomposition discipline. The important relations between the steady state queue length distributions at different time epochs ( n - , n , n + ) are discovered. Finally, by numerical examples, we discuss the sensitivity of the steady state queue length distribution towards system parameters, and illustrate the application of the expressions for the steady state queue length distribution in the system capacity design.

This paper considers an infinite buffer M / M / c queueing system in which servers follow a multi-threshold vacation policy. With such a policy, at a service completion instant, if the number of customers in the system is less than a prefixed threshold value, part of servers together take a single vacation (or leave for a random amount of time doing other secondary job). At the vacation completion instant, they return to the system for serving the customers. Some practical production and inventory systems or call centers could be modeled as this Markovian queue with a multi-threshold vacation policy. Using the Markovian process model, we obtain the exact closed-form expression of rate matrix and the stationary distribution of the number of customers in the system. A cost model is developed to search the joint optimal values of the thresholds of vacation policy and service rate of each server, which minimizes the long-term average cost. Some numerical results are presented to illustrate the optimization procedures.

We consider the M/M/s/K retrial queues in which a customer who is blocked to enter the service facility may leave the system with a probability that depends on the number of attempts of the customer to enter the service facility. Approximation formulae for the distributions of the number of customers in service facility, waiting time in the system and the number of retrials made by a customer during its waiting time are derived. Approximation results are compared with the simulation.

In this paper, we model a discrete-time infinite buffer renewal input single server queue with changeover time under (a, c, b) policy. The service and changeover time are geometrically distributed. The server begins service if there are at least c customers in the queue and the services are performed in batches of minimum size a and maximum size b (a ≤ c ≤ b). At the instant of service completion, if the queue size is less than c but not less than a secondary limit a, the server continues to serve and it will be in the changeover period if the queue size is a − 1. Employing the supplementary variable and recursive techniques, we have derived the steady state queue length distributions at pre-arrival and arbitrary epochs. Based on the queue length distributions, some performance measures and cost functions have been discussed. Using a genetic algorithm, the optimum value of the service rate at minimum cost has been evaluated. Numerical results showing the effect of model parameters on the key performance m...

In this paper, asymptotic properties of the loss probability are considered for an M/G/1/N queue with server vacations and exhaustive service discipline, denoted by an M/G/1/N-(V, E)-queue. Exact asymptotic rates of the loss probability are obtained for the cases in which the traffic intensity is smaller than, equal to and greater than one, respectively. When the vacation time is zero, the model considered degenerates to the standard M/G/1/N queue. For this standard queueing model, our analysis provides new or extended asymptotic results for the loss probability. In terms of the duality relationship between the M/G/1/N and GI/M/1/N queues, we also provide asymptotic properties for the standard GI/M/1/N model.

We analyze queueing model with priority scheduling by sup- plementary variable method. Customers are classified into two types ( type-1 and type-2 ) according to their characteristics. Customers of each type arrive by independent Poisson processes, and all customers regardless of type have same general service time. The service order of each type is determined by the queue length of type-1 buffer. If the queue length of type-1 customer exceeds a threshold L, the service priority is given to the type-1 customer. Otherwise, the service priority is given to type-2 customer. Method of supplementary variable by remaining service time gives us information for queue length of two buffers. That is, we derive the differential difference equations for our queueing system. We obtain joint probability generating function for two queue lengths and the remaining service time. Also, the mean queue length of each buffer is derived.

A single server Markovian queue is considered. The arriving units are served in batches by using Accessibility and NonAccessibility rules with varying service rates. The expressions for the steady state probabilities when the server is busy as well as idle are derived. The mean and variance for the number of units in the queue are obtained. The expected waiting time of units is also attained. Numerical results for number of units in the queue are computed for various values of  and exhibited the corresponding graphs when the remaining parameters are fixed.

We consider an M/M/1 queueing system with inventory under the $$(r,Q)$$ policy and with lost sales, in which demands occur according to a Poisson process and service times are exponentially distributed. All arriving customers during stockout are lost. We derive the stationary distributions of the joint queue length (number of customers in the system) and on-hand inventory when lead times are random variables and can take various distributions. The derived stationary distributions are used to formulate long-run average performance measures and cost functions in some numerical examples.

We consider a stationary multi-class FIFO M/G/1 queue with exponential working vacations, where a server works at two different processing rates. There are K classes of customers, and the arrival rates and the distributions of the amount of service requirements of arriving customers depend on both their customer classes and the server state. When the system becomes empty, the server takes a working vacation, during which customers are served at processing rate ( >

We consider a repair facility consisting of one repairman and two arrival streams of failed items, from bases 1 and 2. The arrival processes are independent Poisson processes, and the repair times are independent and identically exponentially distributed. The item types are exchangeable, and a failed item from base 1 could just as well be returned to base 2, and vice versa. The rule according to which backorders are satisfied by repaired items is the longest queue rule: At the completion of a service (repair), the repaired item is delivered to the base that has the largest number of failed items. We point out a direct relation between our model and the classical longer queue model. We obtain simple expressions for several probabilities of interest, and show how all two-dimensional queue length probabilities may be obtained. Finally, we derive the sojourn time distributions.

This paper considers the M/M/Nrepairable queuing system. The customers' arrival is a Poisson process. The servers are subject to breakdown according to Poisson processes with different rates in idle time and busy time, respectively. The breakdown servers are repaired by repairmen, and the repair time is an exponential distribution. Using probability generating function and transform method, we obtain the steady-state probabilities of the system states, the steady-state availability of the servers, and the mean queueing length of the model.

The discrete-time G/D/1 queues with serially correlated batch arrivals and unit service times have wide applications in modern telecommunication systems. Despite the rich literature in their performance analysis, no simple formula on the relation between system size and sojourn time is known. We show that for this specific type of queues, the Little's law can be extended to higher moments. The benefit of this generalised result is that once the moments of either performance measure are available, those of the other will be obtained simultaneously. This result is applied to a particular example of OO-G/D/1 system, where the mean, variance, and skewness of the sojourn delay are derived in closed-form. Numerical examples are given to examine how the correlation influences these performance measures.

An APPL extension that computes the exact distribution of the nth customer’s sojourn time in an M∕M∕s queue with k customers initially present is derived in this chapter. Algorithms for computing the covariance between sojourn times for an M∕M∕1 queue with k customers present at time zero are also developed. Maple computer code is developed to implement the transient queue analysis for many system measures of performance without regard to traffic intensity (i.e., the system may be unstable with traffic intensity greater than one).

We consider an M/G/1 Markovian feedback queue with multiclass customers. We derive functional equations for the stationary distribution of the queue size and the total response time. A system of linear equations is also derived for the moments of the queue size and the total response time distributions. The mean and the variance of the queue size and the total response time can be computed by solving the system of linear equations.

This paper considers a BMAP/M/∞ queue with a batch Markovian arrival process (BMAP) and an exponential service time distribution. We first prove that the BMAP/M/∞ queue is stable if and only if the expectation of the logarithm of the batch-size distribution is finite. Using this result, we also present the stability condition for an infinite-server queue with a multiclass batch Markovian arrival process and class-dependent exponential service times.

We study the single server B/D/1 FIFO queue with Bernoulli inter-arrival times t (t{0,1}) and arbitrary deterministic service time s (s

The subject of this paper is the problem of estimating the service time distribution of the M/G/∞ queue from incomplete data on the queue. The goal is to estimate G from observations of the queue-length process at the points of the regular grid on a fixed time interval. We propose an estimator and analyze its accuracy over a family of target service time distributions. An upper bound on the maximal risk is derived. The problem of estimating the arrival rate is considered as well.

We consider a horizontal traffic queue (HTQ) on a periodic road segment, where vehicles arrive according to a spatio-temporal Poisson process, and depart after traveling a distance that is sampled independently and identically from a spatial distribution. When inside the queue, the motion of vehicles is governed by a car following model. We consider safe first and second order car following models. In the first order models, the speed of a vehicle is proportional to a power m > 0 of the distance to the vehicle in front. For the first order model, we show monotonicity of the service rate in between arrivals and departures. We extend the busy period calculations for M/G/1 queue to our setting, including for non-empty initial conditions in order to derive a probabilistic lower bound on the throughput for the m > 1 case. For the m < 1 case, we study throughput under a release control policy, where the additional expected waiting time caused by the release control policy is interpreted as the magnitude of the perturbation to the arrival process. We derive a lower bound on throughput for a given combination of maximum allowable perturbation and m < 1. For the second order model, we design a safe release control policy which guarantees a provable lower bound on the throughput. Illustrative simulation results are also presented.

Recently, Ammar [1] has discussed the transient behavior of a multiple vacations queue with impatient customers. In this paper, a similar technique is used to derive a new elegant explicit solution for an M/M/1 vacation queue with impatient customers and a waiting server, where the server is allowed to take a vacation whenever the system is empty after waiting for a random period of time. If the server does not return from the vacation before the expiry of the customer impatience time, the customer abandons the system forever. Moreover, the formulas of mean and variance expressed in terms of the obtained possibilities for this model.

We study the age of information, which is a recently introduced metric for measuring the freshness of a continually updated piece of information as observed at a remote monitor. The age of information metric has been studied for a variety of different queuing systems. In this work, we introduce a packet deadline as a control mechanism and study its impact on the average age of information for an M/M/1/2 queuing system. We analyze the system for a fixed deadline and derive a mathematical expression for the average age. We numerically evaluate the expression and show the relationship of the age performance to that of the M/M/1/1 and M/M/1/2 systems. We show that the system with a deadline constraint can outperform both the M/M/1/1 and M/M/1/2 without such a deadline.

We consider a two-queue polling model in which customers upon arrival join the shorter of two queues. Customers arrive according to a Poisson process and the service times in both queues are independent and identically distributed random variables having the exponential distribution. The two-dimensional process of the numbers of customers at the queue where the server is and at the other queue is a two-dimensional Markov process. We derive its equilibrium distribution using two methodologies: the compensation approach and a reduction to a boundary value problem. Keywords: Polling models; Join the shorter queue; Compensation approach; Boundary value problem

Nowadays, Local Area Networks (LAN) are one of the most popular networks, and the LAN performance is very important for operators. The LAN method has been applied as an essential infrastructure of numerous companies and organizations for a long time. This study aims to evaluate the M/M/1 queuing model in LAN Based on Poisson and Exponential Distribution and compare the traffic behavior of these Distributions in terms of some essential parameters. Moreover, it also aims to design and implement a model to perform the M/M/1 queuing model with different metrics and finally analyze the results to evaluate traffic behavior of M/M/1 queuing model for Poisson and Exponential Distribution in LAN.

Queueing systems with feedback are well suited for the description of message transmission and manufacturing processes where a repeated service is required. In the present paper we investigate a rather general single server queue with a Markovian Arrival Process (MAP), Phase-type (PH) service-time distribution, a finite buffer and feedback which operates in a random environment. A finite state Markovian random environment affects the parameters of the input and service processes and the feedback probability. The stationary distribution of the queue and of the sojourn times as well as the loss probability are calculated. Moreover, Little’s law is derived.

This chapter is concerned with the analysis of a single server retrial queue with two types of service, Bernoulli vacation and feedback. The server provides two types of service i.e., type 1 service with probability??1 and type 2 service with probability ??2. We assume that the arriving customer who finds the server busy upon arrival leaves the service area and are queued in the orbit in accordance with an FCFS discipline and repeats its request for service after some random time. After completion of type 1 or type 2 service the unsatisfied customer can feedback and joins the tail of the retrial queue with probability f or else may depart from the system with probability 1–f. Further the server takes vacation under Bernoulli schedule mechanism, i.e., after each service completion the server takes a vacation with probability q or with probability p waits to serve the next customer. For this queueing model, the steady state distributions of the server state and the number of customers in the orbit are obtained using supplementary variable technique. Finally the average number of customers in the system and average number of customers in the orbit are also obtained.

Around 50 years ago, P. Naor has derived the optimal social and individual admission rules to an M/M/1 queue. In both cases, the optimal policies were identified to be of a pure threshold type: admit if and only if the number queued upon arrival is below some threshold. The value of the threshold in the individual optimal case was shown to be larger than the one for the social optimal criterion. We make the observation that admitting according to a threshold policy requires only the information of whether the queue is above or below a threshold. We call these “red” and “green” light, respectively, associated with a threshold, say L. The question that we pose in this paper is: what happens if one restricts to the above information pattern but let the threshold level L be chosen by the system which signals to arrivals whether the queue is above or below the threshold. Can one find a choice of a threshold that will induce an equilibrium that performs better than in the case that full information is available? We also examine the question of what is the threshold that maximizes the revenue for the queue. We show that the choice of threshold that maximizes the system’s performance at equilibrium is the same as under the full information case if the service in the queue follows the FIFO discipline.

This paper, analyze the parametric performance measures in Poisson arrival and Erlang service time with k phase queuing system the arrival rate and service rate being Triangular fuzzy numbers. Using Zaden Extension principle to estimate the uncertainty associated with the input parameters and triangular membership function. This function has been used to statistical applications to obtained the error performance of the waiting time in the Queue and in the system. The basic idea of M/E k /1/∞/ FIFO is to transform to a fuzzy queue by applying the — cut approach, since the system performance measures are expressed by membership function rather than by crisp values. A numerical example is included.

We present a detail theoretical analysis of the response time and waiting time in the M/M/m FCFS preemptive-resume priority queueing system in the steady state by scrutinizing and extending the previous studies by Brosh (1969), Segal (1970), Buzen and Bondi (1983), Tatashev (1984), and Zeltyn et al. (2009). In particular, we analyze the durations of intermittent waiting times and service times during the response time of a tagged customer of each priority class that is preempted by the arrivals of higher-priority class customers. Numerical examples are shown in order to demonstrate the computation of theoretical formulas.

The present investigation deals with the transient analysis of Markovian queue model with multi stage service. Here, current analysis deals with M/M/1 queueing model with finite capacity. The arrival of the customer in the system follows Poisson process and the customers are served following exponential distribution. The brokendown server is immediately send for repair process so as to become as good as earlier. Transient analysis of finite Markovian queueing model has been studied using Runge-Kutta method. Various performance measures like queue length of the system, waiting time, throughput, service state probabilities have also been obtained.

We consider a discrete-time GI/G/1 queue in which the server takes exactly one vacation each time the system becomes empty. The interarrival times of arriving customers, the service times, and the vacation times are all generic discrete random variables. Under our study, we derive an exact transform-free expression for the stationary system size distribution through the modified supplementary variable technique. Utilizing obtained results, we introduce a simple two-moment approximation for the system size distribution. From this, approximations for the mean system size along with the system size distribution could be obtained. Finally, some numerical examples are given to validate the proposed approximation method.

The analysis of priority queues where both the arrival and the service processes are correlated does not have a long literature. Only a few results are known, that attack the problem with the matrix geometric machinery. Unfortunately, these results have some restrictions that limit their usability significantly, for example they require the calculation of infinite series of matrices and infinite summations, that can be implemented only by truncation. The method presented in this paper calculates only the queue length moments, but without relying on infinite series of matrices and provides procedures to calculate the arising infinite sums accurately in an efficient way by means of linear equations, matrix-quadratic equations and a coupled matrix-quadratic equation. The numerical examples demonstrate that the presented method is several orders of magnitudes faster than the existing ones. From the large number of queue length moments it is possible to obtain lower and upper bounds for the queue length distribution by using existing moment based distribution estimation results.

This paper analyzes the steady-state behavior of a discrete-time single-server queueing system with correlated service times and server vacations. The vacation times of the server are independent and geometrically distributed, and their durations are integral multiples of slot duration. The customers are served one at a time under discrete-time Markovian service process. The new service process starts with the initial phase distribution independent of the path followed by the previous service process when the server returns from a vacation and finds at least one waiting customer. The matrix-geometric method is used to obtain the probability distribution of system-length at prearrival epoch. With the help of Markov renewal theory approach, we also derive the system-length distribution at an arbitrary epoch. The analysis of actual-waiting-time distribution in the queue measured in slots has also been carried out. In addition, computational experiences with a variety of numerical results are discussed to display the effect of the system parameters on the performance measures.

This paper deals with a single server M/G/1 queue with two phases of heterogeneous service and unreliable server. We assume that customers arrive to the system according to a Poisson process with rate λ. After completion of two successive phases of service the server either goes for a vacation with probability p(0 ⩽ p ⩽ 1) or may continue to serve the next unit, if any, with probability q(=1 − p). Otherwise it remains in the system until a customer arrives. While the server is working with any phase of service, it may breakdown at any instant and the service channel will fail for a short interval of time. For this model, we first derive the joint distribution of state of the server and queue size, which is one of the chief objectives of the paper. Secondly, we derive the probability generating function of the stationary queue size distribution at a departure epoch. Next, we derive Laplace Stieltjes transform of busy period distribution and waiting time distribution. Finally we obtain some important performance measures and reliability indices of this model.

We consider the GI/G/1 queue where customers are served in random order and the service time distribution has a finite exponential moment. We derive the large deviations result for the waiting time distribution by showing that the asymptotic decay rate of the waiting time distribution is the same as that of the busy period distribution.

When the service time distribution has a finite exponential moment, we present a large deviations result for the busy period distribution in the M/G/1 queue without the assumption of Abate and Whitt (1997) and Kyprianou (1971).

In this paper, we are concerned with the analytical treatment of an GI/M/1 retrial queue with constant retrial rate. Constant retrial rate is typical for some real world systems where the intensity of individual retrials is inversely proportional to the number of customers in the orbit or only one customer from the orbit is allowed to make the retrials. In our model, a customer who finds the server busy joins the queue in the orbit in accordance with the FCFS (first-come-first-out) discipline and only the oldest customer in the queue is allowed to make the repeated attempts to reach the server. A distinguishing feature of the considered system is an arbitrary distribution of inter-arrival times, while the overwhelming majority of the papers is devoted to the retrial systems with the stationary Poisson arrival process. We carry out an extensive analytical analysis of the queue in steady state using the well-known matrix analytic technique. The ergodicity condition and simple expressions for the stationary distributions of the system states at pre-arrival, post-arrival and arbitrary times are derived. The important and difficult problem of finding the stationary distribution of the sojourn time is solved in terms of the Laplace–Stieltjes transform. Little’s formula is proved. Numerical illustrations are presented.

In the case of M / M / 1 queue with constrained workload (finite dam), we consider two strategies for investing overflow of the dam. Expected value of the present value of each strategy is computed using restricted Laplace transforms and properties of M / M / 1 queues.

We consider the M/M/m retrial queue with nonpersistent customers. Liu et al. (2011) [12] provided the asymptotic lower and upper bounds for the stationary distribution of the number of customers in the orbit. In this paper we strengthen Liu, Wang and Zhao’s result by finding the exact tail asymptotic formula.

We present a priority queue that supports insert in worst-case constant time, and delete-min, access-min, delete, and decrease of an element x in worst-case O(log(min{wx,qx})) time, where wx (respectively, qx) is the number of elements that were accessed after (respectively, before) the last access to x and are still in the priority queue at the time when the corresponding operation is performed. (An access to an element is accounted for by any priority-queue operation that involves this element.) Our priority queue then has both the working-set and the queueish properties; and, more strongly, it satisfies these properties in the worst-case sense. From the results in Iacono (2001) [11] and Elmasry et al. (2011) [7], our priority queue also satisfies the static-finger, static-optimality, and unified bounds. Moreover, we modify our priority queue to realize a new unifying property — the time-finger property — which encapsulates both the working-set and the queueish properties.

Working vacation queue models are well applied in the modeling and analysis of the router in optical networks. The GI/Geo/c queue with working vacations is studied in this paper. Through establishing two-dimensional Markov chain and using matrix-geometric solution method, the stability condition is derived. Adopting UL-type RG-factorization of irreducible Markov chain, the stationary distribution is given. Based on these, the probability distribution of queue-length and PGF of waiting time are obtained in the end.

M/G/1 queue with server vacations period depending on the previous vacation and customer’s arrival is considered. Most existing studies on M/G/1 queue with server vacations assume that server vacations are independent of customers’ arrival. However, some vacations are terminated by some class o f customers’ arrival in certain queueing systems. In this paper, therefore, we investigate M/G/1 queue with server vacations where each vacation period has different distribution and vacation length is influenced by customers’ arrival. Laplace-Stieltjes transform of the waiting time distribution and the distribution of number of customers waiting for each class of customers are respectively derived. As performance measures, mean waiting time and average number of customers waiting for each class of customers are also derived.

In this paper, a multi phase M/G/1queueing system with Bernoulli feedback where the server takes multiple vacation is considered. All the poisson arrivals with mean arrival rate will demand any of the multi essential services.. The service times of the first essential service are assumed to follow a general distribution Bi(v). After the completion of any of the n services, if the customer is dissatisfied he can join the tail of the queue for receiving another regular service with probability p. Otherwise the customer may depart from the system with the probability q=1-p. If there is no customer in the queue, then the server can go for vacation and vacation periods are exponentially distributed with mean vacation time .On returning from vacation , if the server again founds no customer waiting in the queue, then it again goes for vacation. The server continues to go for vacation until he finds at least one customer in the system. We find the time dependent probability generating function in terms of Laplace transforms and derive explicitly the corresponding steady state results.

In this paper we introduce the adaptive MMAP[K] arrival process and analyze the adaptive MMAP[K]/PH[K]/1 queue. In such a queueing system, customers of K different types with Markovian inter-arrival times and possibly correlated customer types, are fed to a single server queue that makes use of r thresholds. Service times are phase-type and depend on the type of customer in service. Type k customers are accepted with some probability a(l,k) if the current workload is between threshold i - 1 and i. The manner in which the arrival process changes its state after generating a type k customer also depends on whether the customer is accepted or rejected. The solution method exists in reducing the joint workload and arrival process to a fluid queue with r thresholds, the steady state of which is expressed using matrix analytic methods. The time and memory complexity of this approach is also shown to be linear in the number of thresholds, allowing us to study systems with thousands of thresholds. Markovian multi-type queues with customer impatience form a subclass of the queues considered in this paper. A numerical method to determine the probability of abandonment and the waiting time distribution is provided if the patience distributions have finite support, while for general customer impatience numerical examples show that accurate approximate results can be obtained using a step-function approach. Numerical examples with adaptive sources that model certain types of admission and congestion control are also included. (C) 2012 Elsevier B.V. All rights reserved.

An MX/M/1 queueing system with threshold switching of service regimes at the instant of a change in the number of customers and the same system with threshold blocking of the flow of customers are considered. For these systems, an algorithm is proposed for determination of the stationary distribution of the number of customers and stationary characteristics (the mean queue length, the mean time of waiting in a queue, the variance of the queue length, the probability of the customer service for a system with blocking). For the case when the minimal number of incoming customers in a group is comparable with threshold value h, the stationary characteristics are found in an explicit form. For the system with threshold switching of service regimes (without blocking), two problems of optimal synthesis are solved. In these problems, the optimal values of threshold h and of the service intensity in the post-threshold regime are determined. The obtained results are verified with the help of simulation models developed with the help of the GPSS World software.

This letter investigates a single-server discrete-time queuing system with single-slot service times. For a broad class of arrival processes, a closed-form expression for the mean queue content in steady state is obtained. Apart from being stationary-ergodic, the arrival process adheres to a regeneration property when there are no arrivals in a slot. Well-studied arrival processes such as autoregressive arrival processes and M / G / ∞ -input or train arrival processes adhere to this property.

This paper analyzes a multi-server queueing system with multiple vacations in which the vacation duration of each server is exponentially distributed. When all the servers are on vacation, customers are impatient if their waiting times exceed a constant value. To derive the stationary distribution of the system, we employ the matrix analytic method. Specifically, our queueing model is represented as an M / G / 1 -type Markov chain, and the stationary distribution is simply obtained by using its transition structures. Furthermore, we explicitly derive the tail decay rate of the stationary distribution, and demonstrate that the decay rate is not always equal to that for the queueing model without vacations.

A recent article on bulk-service queues gives a generating function for the steady-state probabilities of the embedded Markov chain for a single-server infinite-space system in which, the customers arrive according to a Poisson process and are served in batches with quorum a and capacity b, and the service time follows a general distribution dependent on batch size. This system is equivalent to the bulk queueing model M/Gj(a,b)/1, whose general solution requires finding the roots of the denominator of the underlying generating function. The article claims that the use of roots may result in numerical inaccuracies, especially for large values of b. Hence it only solves for the finite-space model M/Gj(a,b)/1/(B + b) using (B + 1) simultaneous linear equations. We present a simple way to obtain the probability distribution of queue length at post-departure epochs for the infinite-space model M/Gj(a,b)/1 using roots, then an alternative method to solve the finite-space queue M/Gj(a,b)/1/(B + b). We d...

This paper presents a new approach to the functional approximation of the M/G/1/N built on a Taylor series approach. Specifically, we establish an approximative expression for the remainder term of the Taylor series that can be computed in an efficient manner. As we will illustrate with numerical examples, the resulting Taylor series approximation turns out to be of practical value. © 2012 The Author(s).

This paper considers a discrete-time queue with N-policy and LAS-DA (late arrival system with delayed access) discipline. By using renewal process theory and probability decomposition techniques, the authors derive the recursive expressions of the queue-length distributions at epochs n −, n +, and n. Furthermore, the authors obtain the stochastic decomposition of the queue length and the relations between the equilibrium distributions of the queue length at different epochs (n −, n +, n and departure epoch D n ).

We present a discrete time single-server two-level mixed service polling systems with two queue types, one center queue and N normal queues. Two-level means the center queue will be successive served after each normal queue. In the first level, server visits between the center queue and the normal queue. In the second level, normal queues are polled by a cyclic order. Mixed service means the service discipline are exhaustive for center queue, and parallel 1-limited for normal queues. We propose an imbedded Markov chain framework to drive the closed-form expressions for the mean cycle time, mean queue length, and mean waiting time. Numerical examples demonstrate that theoretical and simulation results are identical the new system efficiently differentiates priorities.