BROWNIAN MOTION MINUS THE INDEPENDENT INCREMENTS: REPRESENTATION AND QUEUING APPLICATION

2020 ◽  
pp. 1-25
Author(s):  
Kerry Fendick
Keyword(s):  
Brownian Motion ◽  
Maximum Likelihood ◽  
Markov Process ◽  
Maximum Likelihood Estimators ◽  
Single Server ◽  
Single Server Queue ◽  
Stationary Increments ◽  
Transient Distribution ◽  
Independent Increments ◽  
Server Queue

This paper relaxes assumptions defining multivariate Brownian motion (BM) to construct processes with dependent increments as tractable models for problems in engineering and management science. We show that any Gaussian Markov process starting at zero and possessing stationary increments and a symmetric smooth kernel has a parametric kernel of a particular form, and we derive the unique unbiased, jointly sufficient, maximum-likelihood estimators of those parameters. As an application, we model a single-server queue driven by such a process and derive its transient distribution conditional on its history.

The single-server queue with random service output

1975 ◽  
Vol 12 (04) ◽  
pp. 763-778 ◽  
Author(s):  
O. J. Boxma
Keyword(s):  
Service Process ◽  
Service Rate ◽  
Single Server ◽  
Queueing Model ◽  
Single Server Queue ◽  
Independent Increments ◽  
Server Queue ◽  
Work Done

In this paper a problem arising in queueing and dam theory is studied. We shall consider a G/G*/1 queueing model, i.e., a G/G/1 queueing model of which the service process is a separable centered process with stationary independent increments. This is a generalisation of the well-known G/G/1 model with constant service rate. Several results concerning the amount of work done by the server, the busy cycles etc., are derived, mainly using the well-known method of Pollaczek. Emphasis is laid on the similarities and dissimilarities between the results of the ‘classical’ G/G/1 model and the G/G*/1 model.

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.

The single-server queue with random service output

1975 ◽  
Vol 12 (4) ◽  
pp. 763-778 ◽  
Author(s):  
O. J. Boxma
Keyword(s):  
Service Process ◽  
Service Rate ◽  
Single Server ◽  
Queueing Model ◽  
Single Server Queue ◽  
Independent Increments ◽  
Server Queue ◽  
Work Done

In this paper a problem arising in queueing and dam theory is studied. We shall consider a G/G*/1 queueing model, i.e., a G/G/1 queueing model of which the service process is a separable centered process with stationary independent increments. This is a generalisation of the well-known G/G/1 model with constant service rate.Several results concerning the amount of work done by the server, the busy cycles etc., are derived, mainly using the well-known method of Pollaczek. Emphasis is laid on the similarities and dissimilarities between the results of the ‘classical’ G/G/1 model and the G/G*/1 model.

The single server queue with Poisson input and semi-Markov service times

1966 ◽  
Vol 3 (1) ◽  
pp. 202-230 ◽  
Author(s):  
Marcel F. Neuts
Keyword(s):  
Markov Process ◽  
Time Dependence ◽  
Stationary Solutions ◽  
Single Server ◽  
Single Server Queue ◽  
Poisson Input ◽  
Server Queue ◽  
Semi Markov Process ◽  
Poisson Arrivals ◽  
Service Times

We assume that the successive service times in a single server queue with Poisson arrivals form an m-state semi-Markov process.The results for the M/G/1 queue are extended to this case. Both the time-dependence and the stationary solutions are discussed.

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.

The single server queue with Poisson input and semi-Markov service times

1966 ◽  
Vol 3 (01) ◽  
pp. 202-230 ◽  
Author(s):  
Marcel F. Neuts
Keyword(s):  
Markov Process ◽  
Time Dependence ◽  
Stationary Solutions ◽  
Single Server ◽  
Single Server Queue ◽  
Poisson Input ◽  
Server Queue ◽  
Semi Markov Process ◽  
Poisson Arrivals ◽  
Service Times

We assume that the successive service times in a single server queue with Poisson arrivals form an m-state semi-Markov process. The results for the M/G/1 queue are extended to this case. Both the time-dependence and the stationary solutions are discussed.

Sign in / Sign up

Export Citation Format

Share Document