Stationary distribution of queue length in G / M / 1 queue with two-stage service policy

We consider a G / M / 1 queue with two-stage service policy. The server starts to serve with rate of μ₁ customers per unit time until the number of customers in the system reaches λ. At this moment, the service rate is changed to that of μ₂ customers per unit time and this rate continues until the system is empty. We obtain the stationary distribution of the number of customers in the system.     

A batch arrival retrial queue with two phases of service and Bernoulli vacation schedule

We consider an M X /G/1 queueing system with two phases of heterogeneous service and Bernoulli vacation schedule which operate under a linear retrial policy. In addition, each individual customer is subject to a control admission policy upon the arrival. This model generalizes both the classical M/G/1 retrial queue with arrivals in batches and a two phase batch arrival queue with a single vacation under Bernoulli vacation schedule. We will carry out an extensive stationary analysis of the system , including existence of the stationary regime, embedded Markov chain, steady state distribution of the server state and number of customer in the retrial group, stochastic decomposition and calculation of the first moment.     

The randomized vacation policy for a batch arrival queue

This paper examines an M^[x]/G/1${\text{M}}^{[\text{x}]}/\text{G}/1$ queueing system with a randomized vacation policy and at most J vacations. Whenever the system is empty, the server immediately takes a vacation. If there is at least one customer found waiting in the queue upon returning from a vacation, the server will be immediately activated for service. Otherwise, if no customers are waiting for service at the end of a vacation, the server either remains idle with probability p   or leaves for another vacation with probability 1-p$1-p$. This pattern continues until the number of vacations taken reaches J. If the system is empty by the end of the J  th vacation, the server is dormant idly in the system. If there is one or more customers arrive at server idle state, the server immediately starts his services for the arrivals. For such a system, we derive the distributions of important characteristics, such as system size distribution at a random epoch and at a departure epoch, system size distribution at busy period initiation epoch, idle period and busy period, etc. Finally, a cost model is developed to determine the joint suitable parameters (p^∗,J^∗)$(p^{\ast }\text{,}{J}^{\ast })$ at a minimum cost, and some numerical examples are presented for illustrative purpose.     

The infinite-buffer single server queue with a variant of multiple vacation policy and batch Markovian arrival process

We consider an infinite-buffer single server queue where arrivals occur according to a batch Markovian arrival process (BMAP). The server serves until system emptied and after that server takes a vacation. The server will take a maximum number H of vacations until either he finds at least one customer in the queue or the server has exhaustively taken all the vacations. We obtain queue length distributions at various epochs such as, service completion/vacation termination, pre-arrival, arbitrary, departure, etc. Some important performance measures, like mean queue lengths and mean waiting times, etc. have been obtained. Several other vacation queueing models like, single and multiple vacation model, queues with exceptional first vacation time, etc. can be considered as special cases of our model.     

Equilibrium arrival times to a queue with order penalties

Suppose customers need to choose when to arrive to a congested queue with some desired service at the end, provided by a single server that operates only during a certain time interval. We study a model where the customers incur not only congestion (waiting) costs but also penalties for their index of arrival. Arriving before other customers is desirable when the value of service decreases with every admitted customer. This may be the case for example when arriving at a concert or a bus with unmarked seats or going to lunch in a busy cafeteria. We provide game theoretic analysis of such queueing systems with a given number of customers, specifically we characterize the arrival process which constitutes a symmetric Nash equilibrium.     

Periodic solution to the time-inhomogeneous multi-server Poisson queue

We derive the periodic family of asymptotic distributions and the periodic moments for number in the queue for the multi-server queue with Poisson arrivals and exponential service for time-varying periodic arrival and departure rates, and time-varying periodic number of servers. The method is a straight-forward application of generating functions.     

Optimal multithreshold control for a BMAP/G/1 queue with N service modes

This paper deals with the problem of the optimal service rate control in the system with BMAP (Batch Markovian Arrival Process) arrival stream. An algorithm for the computation of the embedded stationary queue length distribution is developed. The procedure for the cost criteria calculation is elaborated for any fixed parameters of the multithreshold control policy. This revised version was published online in June 2006 with corrections to the Cover Date.     

On the loss process in a batch arrival queue

In this paper, we investigate the loss process in a finite-buffer queue with batch arrivals and total rejection discipline. In such a model, if the buffer has insufficient capacity to accept all the customers included in an arriving batch, the whole batch is blocked and lost. This scheme is especially useful in performance evaluation of buffering processes in IP (internet protocol) networks. The main result of this paper is a closed-form formula for the joint distribution of the length of the first lost series of batches and the time of the first loss. Moreover, the limiting distribution (as the buffer size grows to infinity) is shown.     

Discrete-time queue with Bernoulli bursty source arrival and generally distributed service times

In this contribution, a discrete-time single-server infinite-capacity queue with correlated arrivals and general service times is investigated. Arrivals of cells are modelled as an on/off source process with geometrically distributed on-periods and off-periods, which is called Bernoulli bursty source. Based on the probability generating function technique, closed-form expression of some performance measures of system, such as average buffer content, unfinished work, cell delay and so on, are obtained. Finally, the effects of system parameters on performance measures are illustrated by some numerical examples.     

Asymptotic analysis of the M/G/1 queue with a time‐dependent arrival rate

We consider the M/G/1 queue with an arrival rate λ that depends weakly upon time, as λ = λ(εt) where ε is a small parameter. In the asymptotic limit ε → 0, we construct approximations to the probability p _n(t)that η customers are present at time t. We show that the asymptotics are different for several ranges of the (slow) time scale Τ= εt. We employ singular perturbation techniques and relate the various time scales by asymptotic matching. This revised version was published online in June 2006 with corrections to the Cover Date.     

Finite buffer vacation models under E-limited with limit variation service and Markovian arrival process

We consider a finite-buffer single server queue with single (multiple) vacation(s) and Markovian arrival process. The service discipline is E-limited with limit variation (ELV). Several other service disciplines like, Bernoulli scheduling, nonexhaustive and E-limited service can be treated as special cases of the ELV service.     

An optimal repair policy for systems with a limited number of repairs

Traditionally in reliability literature, the repair facilities are always available. This work considers a more general case in which the repair facilities are not always available, but are available only until a fixed number of repairs have been completed. Different assumptions are made to analytically determine an optimal repair policy maximizing the expected reward.     

An optimal maintenance policy for a server with decreasing arrival rate

We study an optimal maintenance policy for the server in a queueing system. Customers arrive at the server in a Poisson stream and are served by an exponential server, which is subject to multiple states indicating levels of popularity. The server state transitions are governed by a Markov process. The arrival rate depends on the server state and it decreases as the server loses popularity. By maintenance the server state recovers completely, though the customers in the system are lost at the beginning of maintenance. The customers who arrive during maintenance are also lost. In this paper, two kinds of such systems are considered. The first system receives a unit reward when a customer arrives at the system and pays a unit cost for each lost customer at the start of maintenance. The second system receives a unit reward at departure, and pays nothing for lost customers at the beginning of maintenance. Our objective is to maximize the total expected discounted profit over an infinite time horizon. We use a semi-Markov decision process to formulate the problem and are able to establish some properties for the optimal maintenance policy under certain conditions.     

Asymptotically optimal dividend policy for regime-switching compound Poisson models

This work develops asymptotically optimal dividend policies to maximize the expected present value of dividends until ruin.Compound Poisson processes with regime switching are used to model the surplus and the switching（a continuous-time controlled Markov chain） represents random environment and other economic conditions.Assuming the switching to be fast varying together with suitable conditions,it is shown that the system has a limit that is an average with respect to the invariant measure of a related Markov chain.Under simple conditions,the optimal policy of the limit dividend strategy is a threshold policy.Using the optimal policy of the limit system as a guide,feedback control for the original surplus is then developed.It is demonstrated that the constructed dividend policy is asymptotically optimal.     

A BMAP/SM/1 queue with service times depending on the arrival process

We study a BMAP/>SM/1 queue with batch Markov arrival process input and semi‐Markov service. Service times may depend on arrival phase states, that is, there are many types of arrivals which have different service time distributions. The service process is a heterogeneous Markov renewal process, and so our model necessarily includes known models. At first, we consider the first passage time from level {κ+1} (the set of the states that the number of customers in the system is κ+1) to level {κ} when a batch arrival occurs at time 0 and then a customer service included in that batch simultaneously starts. The service descipline is considered as a LIFO (Last‐In First‐Out) with preemption. This discipline has the fundamental role for the analysis of the first passage time. Using this first passage time distribution, the busy period length distribution can be obtained. The busy period remains unaltered in any service disciplines if they are work‐conserving. Next, we analyze the stationary workload distribution (the stationary virtual waiting time distribution). The workload as well as the busy period remain unaltered in any service disciplines if they are work‐conserving. Based on this fact, we derive the Laplace–Stieltjes transform for the stationary distribution of the actual waiting time under a FIFO discipline. In addition, we refer to the Laplace–Stieltjes transforms for the distributions of the actual waiting times of the individual types of customers. Using the relationship between the stationary waiting time distribution and the stationary distribution of the number of customers in the system at departure epochs, we derive the generating function for the stationary joint distribution of the numbers of different types of customers at departures. This revised version was published online in June 2006 with corrections to the Cover Date.     

Analysis of a single server queue with semi-Markovian service interruption

In this paper we analyze a discrete-time single server queue where the service time equals one slot. The numbers of arrivals in each slot are assumed to be independent and identically distributed random variables. The service process is interrupted by a semi-Markov process, namely in certain states the server is available for service while the server is not available in other states. We analyze both the transient and steady-state models. We study the generating function of the joint probability of queue length, the state and the residual sojourn time of the semi-Markov process. We derive a system of Hilbert boundary value problems for the generating functions. The system of Hilbert boundary value problems is converted to a system of Fredholm integral equations. We show that the system of Fredholm integral equations has a unique solution. This revised version was published online in June 2006 with corrections to the Cover Date.     

An approximate analysis of a cyclic server queue with limited service and reservations

In this paper, we examine a queueing problem motivated by the pipeline polling protocol in satellite communications. The model is an extension of the cyclic queueing system withM-limited service. In this service mechanism, each queue, after receiving service on cyclej, makes a reservation for its service requirement in cyclej + 1. The main contribution to queueing theory is that we propose an approximation for the queue length and sojourn-time distributions for this discipline. Most approximate studies on cyclic queues, which have been considered before, examine the means only. Our method is an iterative one, which we prove to be convergent by using stochastic dominance arguments. We examine the performance of our algorithm by comparing it to simulations and show that the results are very good.     

Decay rate for a PH/M/2 queue with shortest queue discipline

In this paper, we consider a PH/M/2 queue in which each server has its own queue and arriving customers join the shortest queue. For this model, it has been conjectured that the decay rate of the tail probabilities for the shortest queue length in the steady state is equal to the square of the decay rate for the queue length in the corresponding PH/M/2 model with a single queue. We prove this fact in the sense that the tail probabilities are asymptotically geometric when the difference of the queue sizes and the arrival phase are fixed. Our proof is based on the matrix analytic approach pioneered by Neuts and recent results on the decay rates. AMS subject classifications: 60K25 · 60K20 · 60F10 · 90B22     

An M/G/1-type queuing model with service times depending on queue length

A study is made of an M/G/1-type queuing model in which customers receive one type of service until such time as, at the end of a service, the queue size is found to exceed a given value N, N ≥ 1. Then a second type of service is put into effect and remains in use until the queue size is reduced to a fixed value K, 0 ≤ K ≤ N. Equations are derived for the stationary probabilities both at departure times and at general times. An algorithm is developed that allows the rapid computation of the mean queue length and some important probabilities.