An M/G/1 queue with cyclic service times

In this paper we consider a single server queue with Poisson arrivals and general service distributions in which the service distributions are changed cyclically according to customer sequence number. This model extends a previous study that used cyclic exponential service times to the treatment of general service distributions. First, the stationary probability generating function and the average number of customers in the system are found. Then, a single vacation queueing system with aN-limited service policy, in which the server goes on vacation after servingN consecutive customers is analyzed as a particular case of our model. Also, to increase the flexibility of using theM/G/1 model with cyclic service times in optimization problems, an approximation approach is introduced in order to obtain the average number of customers in the system. Finally, using this approximation, the optimalN-limited service policy for a single vacation queueing system is obtained.On leave from the Department of Industrial Engineering, Iran University of Science and Technology, Narmak, Tehran 16844, Iran.     

Approximations for the conditional waiting times in the GI/G/c queue

This note presents a two-moment approximation for the conditional average waiting time in the standard multi-server queue and an approximation for the tail probabilities of the conditional waiting time distribution in the standard single-server queue. These approximations have been tested by extensive numerical experiments.     

Flow time distributions in aK classM/G/1 priority feedback queue

In this paper, aK classM/G/1 queueing system with feedback is examined. Each arrival requires at least one, and possibly up toK service phases. A customer is said to be in classk if it is waiting for or receiving itskth phase of service. When a customer finishes its phasek ≤K service, it either leaves the system with probabilityp _k, or it instantaneously reenters the system as a classk + 1 customer with probability (1 −p _k). It is assumed thatp _k = 1. Service is non-preemptive and FCFS within a specified priority ordering of the customer classes. Level crossing analysis of queues and delay cycle results are used to derive the Laplace-Stieltjes Transform (LST) for the PDF of the sojourn time in classes 1,…,k;k ≤K.     

Time-dependent analysis of M/G/1 vacation models with exhaustive service

We analyze the time-dependent process in severalM/G/1 vacation models, and explicitly obtain the Laplace transform (with respect to an arbitrary point in time) of the joint distribution of server state, queue size, and elapsed time in that state. Exhaustive-serviceM/G/1 systems with multiple vacations, single vacations, an exceptional service time for the first customer in each busy period, and a combination ofN-policy and setup times are considered. The decomposition property in the steady-state joint distribution of the queue size and the remaining service time is demonstrated.     

TheM/G/1 retrial queue with the server subject to starting failures

In this paper, we study a retrial queueing model with the server subject to starting failures. We first present the necessary and sufficient condition for the system to be stable and derive analytical results for the queue length distribution as well as some performance measures of the system in steady state. We show that the general stochastic decomposition law forM/G/1 vacation models also holds for the present system. Finally, we demonstrate that a few well known queueing models are special cases of the present model and discuss various interpretations of the stochastic decomposition law when applied to each of these special cases.Partially supported by the Natural Sciences and Engineering Research Council of Canada, grant OGP0046415.Partially supported by internal research grant of Mount Saint Vincent University.     

Discrete-time GI/G/1 retrial queues with time-controlled vacation policies

A discrete-time GI/G/1 retrial queue with Bernoulli retrials and time-controlled vacation policies is investigated in this paper. By representing the inter-arrival, service and vacation tlmes using a Markov-based approach, we are able to analyze this model as a level-dependent quasi-birth-and-death （LDQBD） process which makes the model algorithmically tractable. Several performance measures such as the stationary probability distribution and the expected number of customers in the orbit have been discussed with two different policies： deterministic time-controlled system and random time-controlled system. To give a comparison with the known vacation policy in the literature, we present the exhaustive vacation policy as a contrast between these policies under the early arrival system （EAS） and the late arrival system with delayed access （LAS-DA）. Significant difference between EAS and LAS-DA is illustrated by some numerical examples.     

A simple method for computing state probabilities of theM/G/1 andGI/M/1 finite waiting space queues

This paper presents a simple method for computing steady state probabilities at arbitrary and departure epochs of theM/G/1/K queue. The method is recursive and works efficiently for all service time distributions. The only input required for exact evaluation of state probabilities is the Laplace transform of the probability density function of service time. Results for theGI/M/1/K –1 queue have also been obtained from those ofM/G/1/K queue.     

Tail probabilities of low-priority waiting times and queue lengths in MAP/GI/1 queues

We consider the problem of estimating tail probabilities of waiting times in statistical multiplexing systems with two classes of sources – one with high priority and the other with low priority. The priority discipline is assumed to be nonpreemptive. Exact expressions for the transforms of these quantities are derived assuming that packet or cell streams are generated by Markovian Arrival Processes (MAPs). Then a numerical investigation of the large-buffer asymptotic behavior of the the waiting-time distribution for low-priority sources shows that these asymptotics are often non-exponential.     

Application of the method of collective marks to someM/G/1 vacation models with exhaustive service

We use the Method of Collective Marks to analyze some time-dependent processes in theM/G/1 queue with single and multiple vacations. With the server state specified at a fixed timet>0, the Laplace transforms with respect tot of mixed transforms for the joint distribution of the number of departures by timet, the queue length, the virtual waiting time, the elapsed and remaining service/vacation times at timet are derived by means of probabilistic interpretations. The Laplace-Stieltjes transform of the virtual waiting time at timet is also given. Some well known results are special cases.This research was supported by the University of Amsterdam.     

Transient solution for a finite capacity M/G a,b /1 queueing system with vacations to the server

In this paper we consider a single server queueing system with Poisson input, general service and a waiting room that allows only a maximum of b customers to wait at any time. A minimum of a customers are required to start a service and the server goes for a vacation whenever he finds less than a customers in the waiting room after a service. If the server returns from a vacation to find less than a customers waiting, he begins another vacation immediately. Using the theory of regenerative processes we derive expressions for the time dependent system size probabilities at arbitrary epochs.     

Settling time bounds forM/G/1 queues

This paper addresses the question of how long it takes for anM/G/1 queue, starting empty, to approach steady state. A coupling technique is used to derive bounds on the variation distance between the distribution of number in the system at timet and its stationary distribution. The bounds are valid for allt. This research was supported in part by a grant from the AT&T Foundation and NSF grant DCR-8351757.     

An exact FCFS waiting time analysis for a general class of G/G/s queueing systems

A closed form expression for the waiting time distribution under FCFS is derived for the queueing system MGE_k/MGE_m/s, where MGE_n is the class of mixed generalized Erlang probability density functions (pdfs) of ordern, which is a subset of the Coxian pdfs that have rational Laplace transform. Using the calculus of difference equations and based on previous results of the author, it is proved that the waiting time distribution is of the form 1- , under the assumption that the rootsU _j are distinct, i.e. belongs to the Coxian class of distributions of order . The present approach offers qualitative insight by providing exact and asymptotic expressions, generalizes and unifies the well known theories developed for the G/G/1,G/M/s systems and leads to an algorithm, which is polynomial if only one of the parameterss orm varies, and is exponential if both parameters vary. As an example, numerical results for the waiting time distribution of the MGE₂/MGE₂/s queueing system are presented.     

Conditional and unconditional distributions forM/G/1 type queues with server vacations

M/G/1 queues with server vacations have been studied extensively over the last two decades. Recent surveys by Boxma [3], Doshi [5] and Teghem [14] provide extensive summary of literature on this subject. More recently, Shanthikumar [11] has generalized some of the results toM/G/1 type queues in which the arrival pattern during the vacations may be different from that during the time the server is actually working. In particular, the queue length at the departure epoch is shown to decompose into two independent random variables, one of which is the queue length at the departure epoch (arrival epoch, steady state) in the correspondingM/G/1 queue without vacations. Such generalizations are important in the analysis of situations involving reneging, balking and finite buffer cyclic server queues. In this paper we consider models similar to the one in Shanthikumar [11] but use the work in the system as the starting point of our investigation. We analyze the busy and idle periods separately and get conditional distributions of work in the system, queue length and, in some cases, waiting time. We then remove the conditioning to get the steady state distributions. Besides deriving the new steady state results and conditional waiting time and queue length distributions, we demonstrate that the results of Boxma and Groenendijk [2] follow as special cases. We also provide an alternative approach to deriving Shanthikumar's [11] results for queue length at departure epochs.     

Geo/G/1 queues with disasters and general repair times

This paper discusses discrete-time single server Geo/G/1 queues that are subject to failure due to a disaster arrival. Upon a disaster arrival, all present customers leave the system. At a failure epoch, the server is turned off and the repair period immediately begins. The repair times are commonly distributed random variables. We derive the probability generating functions of the queue length distribution and the FCFS sojourn time distribution. Finally, some numerical examples are given.     

Analysis of M/G/1-Queues with Setup Times and Vacations under Six Different Service Disciplines

Single server M/G/1-queues with an infinite buffer are studied; these permit inclusion of server vacations and setup times. A service discipline determines the numbers of customers served in one cycle, that is, the time span between two vacation endings. Six service disciplines are investigated: the gated, limited, binomial, exhaustive, decrementing, and Bernoulli service disciplines. The performance of the system depends on three essential measures: the customer waiting time, the queue length, and the cycle duration. For each of the six service disciplines the distribution as well as the first and second moment of these three performance measures are computed. The results permit a detailed discussion of how the expected value of the performance measures depends on the arrival rate, the customer service time, the vacation time, and the setup time. Moreover, the six service disciplines are compared with respect to the first moments of the performance measures.     

观察时间服从Erlang(n)分布的对偶模型红利支付

本文研究了观察时间服从Erlang（n）分布的对偶模型红利支付问题.在收益额的拉普拉斯变换是有理拉普拉斯变换的情况下,获得了破产之前总贴现红利V（u;b）的求解方法.该结果推广了文献[8]的相应结论.     

On a BMAP/G/1 G-queue with setup times and multiple vacations

In this paper, we consider a BMAP/G/1 G-queue with setup times and multiple vacations. Arrivals of positive customers and negative customers follow a batch Markovian arrival process (BMAP) and Markovian arrival process (MAP) respectively. The arrival of a negative customer removes all the customers in the system when the server is working. The server leaves for a vacation as soon as the system empties and is allowed to take repeated (multiple) vacations. By using the supplementary variables method and the censoring technique, we obtain the queue length distributions. We also obtain the mean of the busy period based on the renewal theory.     

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.     

具有不同到达率的带有启动时间的多级适应性休假Mξ/G/1排队模型

本文研究具有不同到达率的带有启动时间的多级适应性休假M~ξ／G／1排队模型,应用嵌入马尔可夫链方法推导出了稳态队长和等待时间(先到先服务规则)分布,并验证了稳态队长和稳态等待时间具有随机分解性,而且给出了忙期分布．许多关于M~ξ／G／1的排队模型都可以看作是此模型的特例．     

具有不同到达率的带有启动时间的多级适应性休假M^ξ／G／1排队模型

本文研究具有不同到达率的带有启动时间的多级适应性休假M^ξ/G/1排队模型，应用嵌入马尔可夫链方法推导出了稳态队长和等待时间（先到先服务规则）分布，并验证了稳态队长和稳态等待时间具有随机分解性，而且给出了忙期分布．许多关于M^ξ/G/1的排队模型都可以看作是此模型的特例．