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

1982 ◽  
Vol 19 (1) ◽  
pp. 245-249 ◽  
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.

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

1982 ◽  
Vol 19 (01) ◽  
pp. 245-249 ◽  
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.

Approximations for delays using asymptotic estimates

1984 ◽  
Vol 16 (01) ◽  
pp. 6
Author(s):  
David Y. Burman ◽  
Donald R. Smith
Keyword(s):  
Markov Process ◽  
Poisson Process ◽  
Heavy Traffic ◽  
Single Server ◽  
Asymptotic Estimates ◽  
Single Server Queue ◽  
Average Delay ◽  
Multiserver Queues ◽  
Server Queue

Consider a general single-server queue where the customers arrive according to a Poisson process whose rate is modulated according to an independent Markov process. The authors have previously reported on limits for the average delay in light and heavy traffic. In this paper we review these results, extend them to multiserver queues, and describe some approximations obtained from them for general delays.

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

1995 ◽  
Vol 32 (4) ◽  
pp. 1103-1111 ◽  
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.

Average delay in queues with non-stationary Poisson arrivals

1978 ◽  
Vol 15 (03) ◽  
pp. 602-609 ◽  
Author(s):  
Sheldon M. Ross
Keyword(s):  
Poisson Process ◽  
Intensity Function ◽  
Arrival Process ◽  
Single Server ◽  
Actual Process ◽  
Average Delay ◽  
Average Value ◽  
Poisson Arrival ◽  
Poisson Arrivals ◽  
Customer Delay

One of the major difficulties in attempting to apply known queueing theory results to real problems is that almost always these results assume a time-stationary Poisson arrival process, whereas in practice the actual process is almost invariably non-stationary. In this paper we consider single-server infinite-capacity queueing models in which the arrival process is a non-stationary process with an intensity function ∧(t), t ≧ 0, which is itself a random process. We suppose that the average value of the intensity function exists and is equal to some constant, call it λ, with probability 1. We make a conjecture to the effect that ‘the closer {∧(t), t ≧ 0} is to the stationary Poisson process with rate λ ' then the smaller is the average customer delay, and then we verify the conjecture in the special case where the arrival process is an interrupted Poisson process.

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

1995 ◽  
Vol 32 (04) ◽  
pp. 1103-1111 ◽  
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.

A monotonicity result for the workload in Markov-modulated queues

1998 ◽  
Vol 35 (03) ◽  
pp. 741-747 ◽  
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.

The continuity of the single server queue

1972 ◽  
Vol 9 (02) ◽  
pp. 370-381 ◽  
Author(s):  
Douglas P. Kennedy
Keyword(s):  
Metric Spaces ◽  
Arrival Process ◽  
Single Server ◽  
Implicit Assumption ◽  
Single Server Queue ◽  
Virtual Waiting Time ◽  
Convergence Of Probability Measures ◽  
Server Queue ◽  
Poisson Arrivals ◽  
Service Times

In many applications of queueing theory assumptions of either Poisson arrivals or exponential service times are made. The implicit assumption is that if the actual arrival process approximates a Poisson process and the service times are close to exponential, then the quantities of interest in the real queueing system (viz. the virtual waiting time, queue length, idle times, etc.), will approximate those of the idealized model. The continuity of the single server queue acting as functionals of the arrival and service processes is established. The proof involves an application of the theory of weak convergence of probability measures on metric spaces.

Approximations for delays using asymptotic estimates

1984 ◽  
Vol 16 (1) ◽  
pp. 6-6
Author(s):  
David Y. Burman ◽  
Donald R. Smith
Keyword(s):  
Markov Process ◽  
Poisson Process ◽  
Heavy Traffic ◽  
Single Server ◽  
Asymptotic Estimates ◽  
Single Server Queue ◽  
Average Delay ◽  
Multiserver Queues ◽  
Server Queue

Consider a general single-server queue where the customers arrive according to a Poisson process whose rate is modulated according to an independent Markov process. The authors have previously reported on limits for the average delay in light and heavy traffic. In this paper we review these results, extend them to multiserver queues, and describe some approximations obtained from them for general delays.

A monotonicity result for the workload in Markov-modulated queues

1998 ◽  
Vol 35 (3) ◽  
pp. 741-747 ◽  
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.

The continuity of the single server queue

1972 ◽  
Vol 9 (2) ◽  
pp. 370-381 ◽  
Author(s):  
Douglas P. Kennedy
Keyword(s):  
Metric Spaces ◽  
Arrival Process ◽  
Single Server ◽  
Implicit Assumption ◽  
Single Server Queue ◽  
Virtual Waiting Time ◽  
Convergence Of Probability Measures ◽  
Server Queue ◽  
Poisson Arrivals ◽  
Service Times

In many applications of queueing theory assumptions of either Poisson arrivals or exponential service times are made. The implicit assumption is that if the actual arrival process approximates a Poisson process and the service times are close to exponential, then the quantities of interest in the real queueing system (viz. the virtual waiting time, queue length, idle times, etc.), will approximate those of the idealized model. The continuity of the single server queue acting as functionals of the arrival and service processes is established. The proof involves an application of the theory of weak convergence of probability measures on metric spaces.

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