Vacation policies in an M/G/1 type queueing system with finite capacity

This paper deals with a queueing system with finite capacity in which the server passes from the active state to the inactive state each time a service terminates withv customers left in the system. During the active (inactive) phases, the arrival process is Poisson with parameter (₀). Denoting byu _n the duration of thenth inactive phase and byx _n the number of customers present at the end of thenth inactive phase, we assume that the bivariate random vectors {(v _n ,x _n ),n 1} are i.i.d. withx _n v+l a.s. The stationary queue length distributions immediately after a departure and at an arbitrary instant are related to the corresponding distributions in the classical model.     

An M/M/1 retrial queue with unreliable server

We analyze an unreliable M/M/1 retrial queue with infinite-capacity orbit and normal queue. Retrial customers do not rejoin the normal queue but repeatedly attempt to access the server at i.i.d. intervals until it is found functioning and idle. We provide stability conditions as well as several stochastic decomposability results.     

AnM/M/1 retrial queue with control policy and general retrial times

We consider anM/M/1 retrial queueing system in which the retrial time has a general distribution and only the customer at the head of the queue is allowed to retry for service. We find a necessary and sufficient condition for ergodicity and, when this is satisfied, the generating function of the distribution of the number of customers in the queue and the Laplace transform of the waiting time distribution under steady-state conditions. The results agree with known results for special cases.Supported by KOSEF 90-08-00-02.     

On the M/G/1 retrial queueing system with linear control policy

Summary This paper is concerned with the study of a newM/G/1 retrial queueing system in which the delays between retrials are exponentially distributed random variables with linear intensityg(n)=α+nμ, when there aren≥1 customers in the retrial group. This new retrial discipline will be calledlinear control policy. We carry out an extensive analysis of the model, including existence of stationary regime, stationary distribution of the embedded Markov chain at epochs of service completions, joint distribution of the orbit size and the server state in steady state and busy period. The results agree with known results for special cases.     

The M/G/1 retrial queue with bernoulli schedule

We consider anM/G/1 retrial queue with infinite waiting space in which arriving customers who find the server busy join either (a) the retrial group with probabilityp in order to seek service again after a random amount of time, or (b) the infinite waiting space with probabilityq(=1–p) where they wait to be served. The joint generating function of the numbers of customers in the two groups is derived by using the supplementary variable method. It is shown that our results are consistent with known results whenp=0 orp=1.     

Stability and irreducibility of queueing systems with finite capacity

In this paper, we obtain a readily verifiable condition of stability for GI/G/1 queueing systems with finite capacity. A necessary and sufficient condition of irreducibility of the queueing size process is involved. Under this assumption, we derive general conditions of recurrence (positive recurrence) for the general process describing the state of the system. The conditions of irreducibility and recurrence are based on restrictions over the supports of the interarrival and the service distributions, which are easy to check in practice. The positive recurrence is also connected to the first moments of both distributions. This revised version was published online in June 2006 with corrections to the Cover Date.     

A maximum entropy approach for the busy period of the M/G /1 retrial queue

This paper concerns the busy period of a single server queueing model with exponentially distributed repeated attempts. Several authors have analyzed the structure of the busy period in terms of the Laplace transform but, the information about the density function is limited to first and second order moments. We use the maximum entropy principle to find the least biased density function subject to several mean value constraints. We perform results for three different service time distributions: 3-stage Erlang, hyperexponential and exponential. Also a numerical comparative analysis between the exact Laplace transform and the corresponding maximum entropy density is presented. AMS subject classification: 90B05 90B22     

A recursive method to the optimal control of an M/G/1 queueing system with finite capacity and infinite capacity

We study a single removable server in an infinite and a finite queueing systems with Poisson arrivals and general distribution service times. The server may be turned on at arrival epochs or off at service completion epochs. We present a recursive method, using the supplementary variable technique and treating the supplementary variable as the remaining service time, to obtain the steady state probability distribution of the number of customers in a finite system. The method is illustrated analytically for three different service time distributions: exponential, 3-stage Erlang, and deterministic. Cost models for infinite and finite queueing systems are respectively developed to determine the optimal operating policy at minimum cost.     

对一类等待空间有限的抢占优先权排队的分析

讨论M/M/1抢占优先权排队模型, 且假设低优先权顾客的等待空间有限. 该模型可以用有限位相拟生灭过程来描述. 由矩阵解析方法, 对该拟生灭过程进行了分析, 并得到排队模型平稳队长的计算公式, 最后还用数值 结果说明了方法的有效性.     

The M/G/1/K blocking formula and its generalizations to state-dependent vacation systems and priority systems

The formula for the blocking probability for the finite capacity M/G/1/K in terms of the steady state occupancy probability distribution of M/G/1 and the system utilization is known [Keilson, J. Royal Statistical Soc. Serie B, 28 (1966) 190–201]. The validity of this relationship is demonstrated for a broad class of state dependent M/G/1 vacation systems and priority systems. New methods are employed which may also be of interest in their own right.This research was conducted while J. Keilson was a Senior Staff Scientist at GTE Laboratories Incorporated.     

Approximation of M/M/c retrial queue with PH-retrial times

We consider the M/M/c retrial queues with PH-retrial times. Approximation formulae for the distribution of the number of customers in service facility and the mean number of customers in orbit are presented. Some numerical results are presented.     

Markov-modulatedPH/G/1 queueing systems

We study a PH/G/1 queue in which the arrival process and the service times depend on the state of an underlying Markov chain J(t) on a countable state spaceE. We derive the busy period process, waiting time and idle time of this queueing system. We also study the Markov modulated E_K/G/1 queueing system as a special case.     

M/(G1, G2)/1 retrial queue under Bernoulli vacation schedules with general repeated attempts and starting failures

This paper investigates a batch arrival retrial queue with general retrial times, where the server is subject to starting failures and provides two phases of heterogeneous service to all customers under Bernoulli vacation schedules. Any arriving batch finding the server busy, breakdown or on vacation enters an orbit. Otherwise one customer from the arriving batch enters a service immediately while the rest join the orbit. After the completion of two phases of service, the server either goes for a vacation with probability p or may wait for serving the next customer with probability (1 − p). We construct the mathematical model and derive the steady-state distribution of the server state and the number of customers in the system/orbit. Such a model has potential application in transfer model of e-mail system.     

Steady-state analysis of M/M/c/c-type retrial queueing systems with constant retrial rate

The paper deals with a research of bivariate Markov process \(\{X(t), t\ge 0\}\) whose state space is a lattice semistrip \(S(X)=\{0,1,{\ldots },c\} \times Z_{+}\). The process \(\{X(t), t\ge 0\}\) describes the service policy of a multi-server retrial queue in which the rate of repeated flow does not depend on the number of sources of retrial calls. In this class of queues, a vector–matrix representation of steady-state distribution was obtained. This representation allows to write down the stationary probabilities through the model parameters in closed form and to propose the closed formulas of its main performance measures. The investigative techniques use an approximation of the initial model by means of the truncated one and the direct passage to the limit.     

两类具有N-策略和单重休假的M/G/1排队系统的最优控制策略

本文考虑两类具有N-策略和服务员单重休假的M/G/1排队系统,其中一类是休假不可中断,另一类是休假可中断。利用系统稳态队长的随机分解特性导出稳态队长的概率母函数,并讨论了系统空闲率与附加平均队长对系统一些参数的敏感性。进一步,在建立费用结构的基础上,应用更新报酬过程理论导出了系统长期运行单位时间内所产生的成本期望费用的显示表达式,同时通过数值计算实例确定了使得系统在长期运行单位时间内所产生的成本期望费用最小的控制策略N^*,以及当休假时间为定长T时的二维最优控制策略（N^*,T^*）。     

Approximation of M/M/s/K retrial queue with nonpersistent customers

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.     

The BMAP/PH/N retrial queue with Markovian flow of breakdowns

We consider a multi-server retrial queue with the Batch Markovian Arrival Process (BMAP). The servers are identical and independent of each other. The service time distribution of a customer by a server is of the phase (PH) type. If a group of primary calls meets idle servers the primary calls occupy the corresponding number of servers. If the number of idle servers is insufficient the rest of calls go to the orbit of unlimited size and repeat their attempts to get service after exponential amount of time independently of each other. Busy servers are subject to breakdowns and repairs. The common flow of breakdowns is the MAP. An event of this flow causes a failure of any busy server with equal probability. When a server fails the repair period starts immediately. This period has PH type distribution and does not depend on the repair time of other broken-down servers and the service time of customers occupying the working servers. A customer whose service was interrupted goes to the orbit with some probability and leaves the system with the supplementary probability. We derive the ergodicity condition and calculate the stationary distribution and the main performance characteristics of the system. Illustrative numerical examples are presented.     

Optimal prices for finite capacity queueing systems

We prove a lower bound on the optimal price for a fairly large class of blocking systems with general arrival and service processes, determine optimal price expressions for M/M/1/m and M/GI/s/s systems, and investigate how optimal prices change with changes in the size of the waiting room and service capacity.     

A recursive method for the N policy G/M/1 queueing system with finite capacity

This paper studies a single removable server in a G/M/1 queueing system with finite capacity operating under the N policy. We provide a recursive method, using the supplementary variable technique and treating the supplementary variable as the remaining interarrival time, to develop the steady-state probability distributions of the number of customers in the system. The method is illustrated analytically for exponential interarrival time distribution. Numerical results for various system performance measures are presented for four different interarrival time distributions such as exponential, 2-stage hyperexponential, 4-stage Erlang, and deterministic.