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

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 monotonicity result for the workload in Markov-modulated queues

Journal of Applied Probability ◽

10.1017/s0021900200016387 ◽

1998 ◽

Vol 35 (03) ◽

pp. 741-747 ◽

Cited By ~ 1

Author(s):

Nicole Bäuerle ◽

Tomasz Rolski

Keyword(s):

Poisson Process ◽

Arrival Process ◽

Single Server ◽

Single Server Queue ◽

Convex Ordering ◽

Markov Modulated Poisson Process ◽

Server Queue ◽

Monotonicity Result ◽

Markov Modulated ◽

Stationary Workload

We consider a single server queue where the arrival process is a Markov-modulated Poisson process and service times are independent and identically distributed and independent from arrivals. The underlying intensity process is assumed ergodic with generator cQ, c > 0. We prove under some monotonicity assumptions on Q that the stationary workload W(c) is decreasing in c with respect to the increasing convex ordering.

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A monotonicity result for the workload in Markov-modulated queues

Journal of Applied Probability ◽

10.1239/jap/1032265221 ◽

1998 ◽

Vol 35 (3) ◽

pp. 741-747 ◽

Cited By ~ 23

Author(s):

Nicole Bäuerle ◽

Tomasz Rolski

Keyword(s):

Poisson Process ◽

Arrival Process ◽

Single Server ◽

Single Server Queue ◽

Convex Ordering ◽

Markov Modulated Poisson Process ◽

Server Queue ◽

Monotonicity Result ◽

Markov Modulated ◽

Stationary Workload

We consider a single server queue where the arrival process is a Markov-modulated Poisson process and service times are independent and identically distributed and independent from arrivals. The underlying intensity process is assumed ergodic with generator cQ, c > 0. We prove under some monotonicity assumptions on Q that the stationary workload W(c) is decreasing in c with respect to the increasing convex ordering.

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

Advances in Applied Probability ◽

10.1017/s0001867800019947 ◽

1990 ◽

Vol 22 (03) ◽

pp. 676-705 ◽

Cited By ~ 45

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 Convexity Property of a Markov-Modulated Queueing Loss System

Probability in the Engineering and Informational Sciences ◽

10.1017/s0269964800003788 ◽

1995 ◽

Vol 9 (2) ◽

pp. 193-199 ◽

Cited By ~ 1

Author(s):

Charles Du ◽

Michael Pinedo

Keyword(s):

Transition Probability ◽

Loss Probability ◽

Arrival Process ◽

Service Rate ◽

Single Server ◽

Convexity Property ◽

Loss System ◽

Markov Modulated Poisson Process ◽

Speed Parameter ◽

Markov Modulated

In this note we consider a single-server queueing loss system with zero buffer. The arrival process is a nonstationary Markov-modulated Poisson process. The arrival process in state i is Poisson with rate λi. The process remains in state i for a time that is exponentially distributed with rate Cαi, with c being a control or speed parameter. The service rate in state i is exponentially distributed with rate μi. The process moves from state i to state j with transition probability qij. We are interested in the loss probability as a function of c. In this note we show that, under certain conditions, the loss probability decreases when the c increases. As such, this result generalizes a result obtained earlier by Fond and Ross.

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On ross's conjectures about queues with non-stationary poisson arrivals

Journal of Applied Probability ◽

10.1017/s0021900200028485 ◽

1982 ◽

Vol 19 (01) ◽

pp. 245-249 ◽

Cited By ~ 6

Author(s):

D. P. Heyman

Keyword(s):

Poisson Process ◽

Arrival Process ◽

Single Server ◽

Single Server Queue ◽

Average Delay ◽

Server Queue ◽

Poisson Arrivals

Ross (1978) conjectured that the average delay in a single-server queue is larger when the arrival process is a non-stationary Poisson process than when it is a stationary Poisson process with the same rate. We present an example where equality obtains. When the number of waiting-positions is finite, Ross conjectured that the proportion of lost customers is greater in the nonstationary case. We present a counterexample to this conjecture.

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On ross's conjectures about queues with non-stationary poisson arrivals

Journal of Applied Probability ◽

10.2307/3213936 ◽

1982 ◽

Vol 19 (1) ◽

pp. 245-249 ◽

Cited By ~ 18

Author(s):

D. P. Heyman

Keyword(s):

Poisson Process ◽

Arrival Process ◽

Single Server ◽

Single Server Queue ◽

Average Delay ◽

Server Queue ◽

Poisson Arrivals

Ross (1978) conjectured that the average delay in a single-server queue is larger when the arrival process is a non-stationary Poisson process than when it is a stationary Poisson process with the same rate. We present an example where equality obtains. When the number of waiting-positions is finite, Ross conjectured that the proportion of lost customers is greater in the nonstationary case. We present a counterexample to this conjecture.

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Loss Probability of a Burst Arrival Finite Queue with Synchronized Service

Probability in the Engineering and Informational Sciences ◽

10.1017/s026996480000245x ◽

1992 ◽

Vol 6 (2) ◽

pp. 201-216 ◽

Cited By ~ 8

Author(s):

Masakiyo Miyazawa

Keyword(s):

Markov Chain ◽

Loss Probability ◽

Buffer Size ◽

Embedded Markov Chain ◽

Main Concern ◽

Single Server ◽

Finite Buffer ◽

Single Server Queue ◽

Server Queue ◽

Computation Algorithm

We are concerned with a burst arrival single-server queue, where arrivals of cells in a burst are synchronized with a constant service time. The main concern is with the loss probability of cells for the queue with a finite buffer. We analyze an embedded Markov chain at departure instants of cells and get a kind of lumpability for its state space. Based on these results, this paper proposes a computation algorithm for its stationary distribution and the loss probability. Closed formulas are obtained for the first two moments of the numbers of cells and active bursts when the buffer size is infinite.

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Recurrent emptiness in a finite queue with increasing arrival rate

Journal of Applied Probability ◽

10.1017/s0021900200094559 ◽

1976 ◽

Vol 13 (02) ◽

pp. 423-426

Author(s):

Stig I. Rosenlund

Keyword(s):

Sufficient Conditions ◽

Arrival Rate ◽

Necessary And Sufficient Conditions ◽

Single Server ◽

Single Server Queue ◽

Server Queue ◽

Necessary And Sufficient

For a single-server queue with one waiting place and increasing arrival rate some necessary and sufficient conditions for infinitely many returns to emptiness with probability one are given.

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