New results on the single server queue with a batch markovian arrival process

We consider a single server infinite capacity queueing system, where the arrival process is a batch Markovian arrival process (BMAP). Particular BMAPs are the batch Poisson arrival process, the Markovian arrival process (MAP), many batch arrival processes with correlated interarrival times and batch sizes, and superpositions of these processes. We note that the MAP includes phase-type (PH) renewal processes and non-renewal processes such as the Markov modulated Poisson process (MMPP).The server applies Kella's vacation scheme, i.e., a vacation policy where the decision of whether to take a new vacation or not, when the system is empty, depends on the number of vacations already taken in the current inactive phase. This exhaustive service discipline includes the single vacation T-policy, T(SV), and the multiple vacation T-policy, T(MV). The service times are i.i.d. random variables, independent of the interarrival times and the vacation durations. Some important performance measures such as the distribution functions and means of the virtual and the actual waiting times are given. Finally, a numerical example is presented.

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A single server queue with finite-geometric-cluster arrival process

ICC 91 International Conference on Communications Conference Record ◽

10.1109/icc.1991.162227 ◽

2002 ◽

Author(s):

Z. Zhang

Keyword(s):

Arrival Process ◽

Single Server ◽

Single Server Queue ◽

Server Queue

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A monotonicity result for a single-server queue subject to a Markov-modulated Poisson process

Journal of Applied Probability ◽

10.2307/3215223 ◽

1995 ◽

Vol 32 (4) ◽

pp. 1103-1111 ◽

Cited By ~ 6

Author(s):

Qing Du

Keyword(s):

Poisson Process ◽

Arrival Rate ◽

Loss Probability ◽

Arrival Process ◽

Single Server ◽

Single Server Queue ◽

Markov Modulated Poisson Process ◽

Server Queue ◽

Monotonicity Result ◽

Markov Modulated

Consider a single-server queue with zero buffer. The arrival process is a three-level Markov modulated Poisson process with an arbitrary transition matrix. The time the system remains at level i (i = 1, 2, 3) is exponentially distributed with rate cα i. The arrival rate at level i is λ i and the service time is exponentially distributed with rate μ i. In this paper we first derive an explicit formula for the loss probability and then prove that it is decreasing in the parameter c. This proves a conjecture of Ross and Rolski's for a single-server queue with zero buffer.

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A single-server queue with server vacations and a class of non-renewal arrival processes

Advances in Applied Probability ◽

10.2307/1427464 ◽

1990 ◽

Vol 22 (3) ◽

pp. 676-705 ◽

Cited By ~ 307

Author(s):

David M. Lucantoni ◽

Kathleen S. Meier-Hellstern ◽

Marcel F. Neuts

Keyword(s):

Waiting Times ◽

Arrival Process ◽

Single Server ◽

Single Server Queue ◽

Arbitrary Time ◽

Markov Modulated Poisson Process ◽

Special Cases ◽

Server Queue ◽

Arrival Processes ◽

Markov Modulated

We study a single-server queue in which the server takes a vacation whenever the system becomes empty. The service and vacation times and the arrival process are all assumed to be mutually independent. The successive service times and the vacation times each form independent, identically distributed sequences with general distributions. A new class of non-renewal arrival processes is introduced. As special cases, it includes the Markov-modulated Poisson process and the superposition of phase-type renewal processes.Algorithmically tractable equations for the distributions of the waiting times at an arbitrary time and at arrivals, as well as for the queue length at an arbitrary time, at arrivals, and at departures are established. Some factorizations, which are known for the case of renewal input, are generalized to this new framework and new factorizations are obtained. The algorithmic implementation of these results is discussed.

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A monotonicity result for a single-server queue subject to a Markov-modulated Poisson process

Journal of Applied Probability ◽

10.1017/s0021900200103572 ◽

1995 ◽

Vol 32 (04) ◽

pp. 1103-1111 ◽

Cited By ~ 1

Author(s):

Qing Du

Keyword(s):

Poisson Process ◽

Arrival Rate ◽

Loss Probability ◽

Arrival Process ◽

Single Server ◽

Single Server Queue ◽

Markov Modulated Poisson Process ◽

Server Queue ◽

Monotonicity Result ◽

Markov Modulated

Consider a single-server queue with zero buffer. The arrival process is a three-level Markov modulated Poisson process with an arbitrary transition matrix. The time the system remains at level i (i = 1, 2, 3) is exponentially distributed with rate cα i . The arrival rate at level i is λ i and the service time is exponentially distributed with rate μ i . In this paper we first derive an explicit formula for the loss probability and then prove that it is decreasing in the parameter c. This proves a conjecture of Ross and Rolski's for a single-server queue with zero buffer.

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The total waiting time in a busy period of a stable single-server queue, II

Journal of Applied Probability ◽

10.2307/3212102 ◽

1969 ◽

Vol 6 (3) ◽

pp. 565-572 ◽

Cited By ~ 15

Author(s):

D. J. Daley ◽

D. R. Jacobs

Keyword(s):

Waiting Time ◽

Busy Period ◽

Bessel Functions ◽

Arrival Process ◽

Single Server ◽

Single Server Queue ◽

Common Distribution ◽

Poisson Arrival ◽

Server Queue ◽

Common Distribution Function

This paper is a continuation of Daley (1969), referred to as (I), whose notation and numbering is continued here. We shall indicate various approaches to the study of the total waiting time in a busy period2 of a stable single-server queue with a Poisson arrival process at rate λ, and service times independently distributed with common distribution function (d.f.) B(·). Let X'i denote3 the total waiting time in a busy period which starts at an epoch when there are i (≧ 1) customers in the system (to be precise, the service of one customer is just starting and the remaining i − 1 customers are waiting for service). We shall find the first two moments of X'i, prove its asymptotic normality for i → ∞ when B(·) has finite second moment, and exhibit the Laplace-Stieltjes transform of X'i in M/M/1 as the ratio of two Bessel functions.

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