A markov-modulated M/M/1 queue with group arrivals

We study anM/M/1 group arrival queue in which the arrival rate, service time distributions and the size of each group arrival depend on the state of an underlying finite-state Markov chain. Using Laplace transforms and matrix analysis, we derive the results for the queue length process, its limit distribution and the departure process. In some special cases, explicit results are obtained which are analogous to known classic results.     

?/M/1: On the equilibrium distribution of customer arrivals

Each day a facility commences service at time zero. All customers arriving prior to time T are served during that day. The queuing discipline is First-Come First-Served. Each day, each person in the population chooses whether or not to visit the facility that day. If he decides to visit, he arrives at an instant of time such that his expected waiting time in the queue is minimal. We investigate the arrival rate of customers in equilibrium, where each customer is fully aware of the characteristics of the system. We show that the arrival rate is constant before opening time, but that in general it is not constant between opening and closing time. For the case of exponential distribution of service time, we develop a set of equations from which the equilibrium queue size distribution and expected waiting time can be numerically computed as functions of time.     

Explicit solutions for the steady state distributions in M/PH/1 queues with workload dependent balking

We consider an M/PH/1 queue with balking based on the workload. An arriving customer joins the queue and stays until served only if the system workload is below a fixed level at the time of arrival. The steady state workload distribution in such a system satisfies an integral equation. We derive a differential equation for Phase type service time distribution and we solve it explicitly, with Erlang, Hyper-exponential and Exponential distributions as special cases. We illustrate the results with numerical examples.     

Steady state results for the M/M(a,b)/c batch-service system

The transportation system considered in this paper has a number of vehicles with capacity constraint, which take passengers from a source terminal to various destinations and return to the terminal. The trip times are considered to be independent and identically distributed random variables with a common exponential distribution. Passengers arrive at the terminal in accordance with a Poisson process. The system is operated under the following policy: when a vehicle is available and there are at least ‘a’ passengers waiting for service, then a vehicle is dispatched immediately. A recursive algorithm is derived to obtain the steady-state probability P(m, j) that there are m idle vehicles and j waiting passengers in the queue. Analytical expressions have been derived for passenger queue length distribution, average passenger queue length, the r-th moment of passenger waiting time in the queue, service batch size distribution and the average service batch size, all in terms of P(0,0).     

An explicit solution to a tandem queueing model

We consider two queues in tandem, each with an exponential server, and with deterministic arrivals to the first queue. We obtain an explicit solution for the steady state distribution of the process (N₁(t), N₂(t), Y(t)), where N_j(t) is the queue length in the jth queue and Y(t) measures the time elapsed since the last arrival. Then we obtain the marginal distributions of (N₁(t), N₂(t)) and of N₂(t). We also evaluate the solution in various limiting cases, such as heavy traffic. This revised version was published online in June 2006 with corrections to the Cover Date.     

The GI/M/1 queue with start-up period and single working vacation and Bernoulli vacation interruption

Consider a GI/M/1 queue with start-up period and single working vacation. When the system is in a closed state, an arriving customer leading to a start-up period, after the start-up period, the system becomes a normal service state. And during the working vacation period, if there are customers at a service completion instant, the vacation can be interrupted and the server will come back to the normal working level with probability p (0 ? p ? 1) or continue the vacation with probability 1 − p. Meanwhile, if there is no customer when a vacation ends, the system is closed. Using the matrix-analytic method, we obtain the steady-state distributions for the queue length at both arrival epochs and arbitrary epochs, the waiting time and sojourn time.     

The last departure time from an M t /G/∞ queue with a terminating arrival process

This paper studies the last departure time from a queue with a terminating arrival process. This problem is motivated by a model of two-stage inspection in which finitely many items come to a first stage for screening. Items failing first-stage inspection go to a second stage to be examined further. Assuming that arrivals at the second stage can be regarded as an independent thinning of the departures from the first stage, the arrival process at the second stage is approximately a terminating Poisson process. If the failure probabilities are not constant, then this Poisson process will be nonhomogeneous. The last departure time from an M _t /G/∞ queue with a terminating arrival process serves as a remarkably tractable approximation, which is appropriate when there are ample inspection resources at the second stage. For this model, the last departure time is a Poisson random maximum, so that it is possible to give exact expressions and develop useful approximations based on extreme-value theory.      

The shorter queue problem: A numerical study using the matrix-geometric solution

We consider a service system with two similar servers in which a customer, on arrival, joins the shorter queue. The state of the system is described by a pair (i,j), j?i, where i and j are the number of customers in the shorter and the longer queue, respectively. The stationary probability vector and several performance characteristics are obtained using the matrix-geometric solution technique.     

Analysis of the queue-length distribution for the discrete-time batch-service Geo/G/1/K queue

In this paper, we consider a discrete-time finite-capacity queue with Bernoulli arrivals and batch services. In this queue, the single server has a variable service capacity and serves the customers only when the number of customers in system is at least a certain threshold value. For this queue, we first obtain the queue-length distribution just after a service completion, using the embedded Markov chain technique. Then we establish a relationship between the queue-length distribution just after a service completion and that at a random epoch, using elementary ‘rate-in = rate-out’ arguments. Based on this relationship, we obtain the queue-length distribution at a random (as well as at an arrival) epoch, from which important performance measures of practical interest, such as the mean queue length, the mean waiting time, and the loss probability, are also obtained. Sample numerical examples are presented at the end.     

A Recursive Algorithm for State Dependent GI/M/1/N Queue with Bernoulli-Schedule Vacation

In this paper, we study a renewal input working vacations queue with state dependent services and Bernoulli-schedule vacations. The model is analyzed with single and multiple working vacations. The server goes for exponential working vacation whenever the queue is empty and the vacation rate is state dependent. At the instant of a service completion, the vacation is interrupted and the server resumes a regular busy period with probability 1???q (if there are customers in the queue), or continues the vacation with probability q (0?≤?q?≤?1). We provide a recursive algorithm using the supplementary variable technique to numerically compute the stationary queue length distribution of the system. Finally, using some numerical results, we present the parameter effect on the various performance measures.     

Controlling the GI/M/1 queue by conditional acceptance of customers

In this paper we consider the problem of controlling the arrival of customers into a GI/M/1 service station. It is known that when the decisions controlling the system are made only at arrival epochs, the optimal acceptance strategy is of a control-limit type, i.e., an arrival is accepted if and only if fewer than n customers are present in the system. The question is whether exercising conditional acceptance can further increase the expected long run average profit of a firm which operates the system. To reveal the relevance of conditional acceptance we consider an extension of the control-limit rule in which the nth customer is conditionally admitted to the queue. This customer may later be rejected if neither service completion nor arrival has occurred within a given time period since the last arrival epoch. We model the system as a semi-Markov decision process, and develop conditions under which such a policy is preferable to the simple control-limit rule.     

Performance analysis of GI/M/1 queue with working vacations and vacation interruption

In this paper, we analyze a single-server vacation queue with a general arrival process. Two policies, working vacation and vacation interruption, are connected to model some practical problems. The GI/M/1 queue with such two policies is described and by the matrix analysis method, we obtain various performance measures such as mean queue length and waiting time. Finally, using some numerical examples, we present the parameter effect on the performance measures and establish the cost and profit functions to analyze the optimal service rate η during the vacation period.     

Ergodicity and analysis of the process describing the system state in polling systems with two queues

Consider a polling system of two queues served by a single server that visits the queues in cyclic order. The polling discipline in each queue is of exhaustive-type, and zero-switchover times are considered. We assume that the arrival times in each queue form a Poisson process and that the service times form sequences of independent and identically distributed random variables, except for the service distribution of the first customer who is served at each polling instant (the time in which the server moves from one queue to the other one). The sufficient and necessary conditions for the ergodicity of such polling system are established as well as the stationary distribution for the continuous-time process describing the state of the system. The proofs rely on the combination of three embedded processes that were previously used in the literature. An important result is that ρ=1 can imply ergodicity in one specific case, where ρ is the typical traffic intensity for polling systems, and ρ<1 is the classical non-saturation condition.     

The Periodic BMAP/PH/c Queue

In queueing theory, most models are based on time-homogeneous arrival processes and service time distributions. However, in communication networks arrival rates and/or the service capacity usually vary periodically in time. In order to reflect this property accurately, one needs to examine periodic rather than homogeneous queues. In the present paper, the periodic BMAP/PH/c queue is analyzed. This queue has a periodic BMAP arrival process, which is defined in this paper, and phase-type service time distributions. As a Markovian queue, it can be analysed like an (inhomogeneous) Markov jump process. The transient distribution is derived by solving the Kolmogorov forward equations. Furthermore, a stability condition in terms of arrival and service rates is proven and for the case of stability, the asymptotic distribution is given explicitly. This turns out to be a periodic family of probability distributions. It is sketched how to analyze the periodic BMAP/M _t /c queue with periodically varying service rates by the same method.     

An M/G/1 queue with two phases of service subject to the server breakdown and delayed repair

This paper deals with the steady-state behaviour of an M/G/1 queue with an additional second phase of optional service subject to breakdowns occurring randomly at any instant while serving the customers and delayed repair. This model generalizes both the classical M/G/1 queue subject to random breakdown and delayed repair as well as M/G/1 queue with second optional service and server breakdowns. For this model, we first derive the joint distributions of state of the server and queue size, which is one of chief objectives of the paper. Secondly, we derive the probability generating function of the stationary queue size distribution at a departure epoch as a classical generalization of Pollaczek–Khinchin formula. 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.     

The GI/M/1 queue with phase-type working vacations and vacation interruption

Consider a GI/M/1 queue with phase-type working vacations and vacation interruption where the vacation time follows a phase-type distribution. The server takes the original work at the lower rate during the vacation period. And, the server can come back to the normal working level at a service completion instant if there are customers at this instant, and not accomplish a complete vacation. From the PH renewal process theory, we obtain the transition probability matrix. Using the matrix-analytic method, we obtain the steady-state distributions for the queue length at arrival epochs, and waiting time of an arbitrary customer. Meanwhile, we obtain the stochastic decomposition structures of the queue length and waiting time. Two numerical examples are presented lastly.     

A G/M/1 retrial queue with constant retrial rate

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.     

An M/G/1 queue with second optional service

We study an M/G/1 queue with second optional service. Poisson arrivals with mean arrival rate (>0) all demand the first essential service, whereas only some of them demand the second optional service. The service times of the first essential service are assumed to follow a general (arbitrary) distribution with distribution function B(v) and that of the second optional service are exponential with mean service time 1/₂ (₂>0). The time-dependent probability generating functions have been obtained in terms of their Laplace transforms and the corresponding steady state results have been derived explicitly. Also the mean queue length and the mean waiting time have been found explicitly. The well-known Pollaczec–Khinchine formula and some other known results including M/D/1, M/E_k/1 and M/M/1 have been derived as particular cases.