Regeneration in tandem queues

1981 ◽  
Vol 13 (1) ◽  
pp. 221-230 ◽  
Author(s):  
E. Nummelin
Keyword(s):  
Markov Chain ◽  
Waiting Times ◽  
Tandem Queues ◽  
Tandem Queue ◽  
Input Process ◽  
Service Times ◽  
Interarrival Times ◽  
Regeneration Times

Consider a tandem queue with renewal input process and i.i.d. service times (at each server). This paper is concerned with the construction of regeneration times for the multivariate Markov chain formed by the interarrival times, waiting times and service times of the customers.

Regeneration in tandem queues

1981 ◽  
Vol 13 (01) ◽  
pp. 221-230 ◽  
Author(s):  
E. Nummelin
Keyword(s):  
Markov Chain ◽  
Waiting Times ◽  
Tandem Queues ◽  
Tandem Queue ◽  
Input Process ◽  
Service Times ◽  
Interarrival Times ◽  
Regeneration Times

Consider a tandem queue with renewal input process and i.i.d. service times (at each server). This paper is concerned with the construction of regeneration times for the multivariate Markov chain formed by the interarrival times, waiting times and service times of the customers.

Regeneration in tandem queues with multiserver stations

1988 ◽  
Vol 25 (02) ◽  
pp. 391-403 ◽  
Author(s):  
Karl Sigman
Keyword(s):  
Markov Chain ◽  
Renewal Process ◽  
Single Server ◽  
Tandem Queues ◽  
Random Assignment ◽  
Tandem Queue ◽  
Ergodic Markov Chain ◽  
Fifo Queue ◽  
Harris Chain ◽  
Server System

A tandem queue with a FIFO multiserver system at each stage, i.i.d. service times and a renewal process of external arrivals is shown to be regenerative by modeling it as a Harris-ergodic Markov chain. In addition, some explicit regeneration points are found. This generalizes the results of Nummelin (1981) in which a single server system is at each stage and the result of Charlot et al. (1978) in which the FIFO GI/GI/c queue is modeled as a Harris chain. In preparing for our result, we study the random assignment queue and use it to give a new proof of Harris ergodicity of the FIFO queue.

A conservation property for general GI/G/1 queues with an application to tandem queues

1979 ◽  
Vol 11 (03) ◽  
pp. 660-672 ◽  
Author(s):  
E. Nummelin
Keyword(s):  
Limit Theorem ◽  
Total Variation ◽  
Renewal Process ◽  
Markov Renewal Process ◽  
Output Process ◽  
Tandem Queues ◽  
Tandem Queue ◽  
Input Process ◽  
Conservation Property ◽  
Positive Recurrent

We show that, if the input process of a generalGI/G/1 queue is a positive recurrent Markov renewal process then the output process, too, is a positive recurrent Markov renewal process (the conservation property). As an application we consider a general tandem queue and prove a total variation limit theorem for the associated waiting and service times.

On the input-output map of a G/G/1 queue

1994 ◽  
Vol 31 (04) ◽  
pp. 1128-1133 ◽  
Author(s):  
Cheng-Shang Chang
Keyword(s):  
Direct Consequence ◽  
Input Output ◽  
Input Process ◽  
Departure Process ◽  
Service Times ◽  
Interarrival Times ◽  
Independent And Identically Distributed

In this note, we consider G/G/1 queues with stationary and ergodic inputs. We show that if the service times are independent and identically distributed with unbounded supports, then for a given mean of interarrival times, the number of sequences (distributions) of interarrival times that induce identical distributions on interdeparture times is at most 1. As a direct consequence, among all the G/M/1 queues with stationary and ergodic inputs, the M/M/1 queue is the only queue whose departure process is identically distributed as the input process.

scholarly journals The stationary local sojourn time in single server tandem queues with renewal input

1999 ◽  
Vol 12 (4) ◽  
pp. 417-428
Author(s):  
Pierre Le Gall
Keyword(s):  
Sojourn Time ◽  
Stationary Regime ◽  
General Distribution ◽  
Single Server ◽  
Tandem Queues ◽  
Tandem Queue ◽  
Service Times

We start from an earlier paper evaluating the overall sojourn time to derive the local sojourn time in stationary regime, in a single server tandem queue of (m+1) stages with renewal input. The successive service times of a customer may or may not be mutually dependent, and are governed by a general distribution which may be different at each sage.

A conservation property for general GI/G/1 queues with an application to tandem queues

1979 ◽  
Vol 11 (3) ◽  
pp. 660-672 ◽  
Author(s):  
E. Nummelin
Keyword(s):  
Limit Theorem ◽  
Total Variation ◽  
Renewal Process ◽  
Markov Renewal Process ◽  
Output Process ◽  
Tandem Queues ◽  
Tandem Queue ◽  
Input Process ◽  
Conservation Property ◽  
Positive Recurrent

We show that, if the input process of a general GI/G/1 queue is a positive recurrent Markov renewal process then the output process, too, is a positive recurrent Markov renewal process (the conservation property). As an application we consider a general tandem queue and prove a total variation limit theorem for the associated waiting and service times.

On the input-output map of a G/G/1 queue

1994 ◽  
Vol 31 (4) ◽  
pp. 1128-1133 ◽  
Author(s):  
Cheng-Shang Chang
Keyword(s):  
Direct Consequence ◽  
Input Output ◽  
Input Process ◽  
Departure Process ◽  
Service Times ◽  
Interarrival Times ◽  
Independent And Identically Distributed

In this note, we consider G/G/1 queues with stationary and ergodic inputs. We show that if the service times are independent and identically distributed with unbounded supports, then for a given mean of interarrival times, the number of sequences (distributions) of interarrival times that induce identical distributions on interdeparture times is at most 1. As a direct consequence, among all the G/M/1 queues with stationary and ergodic inputs, the M/M/1 queue is the only queue whose departure process is identically distributed as the input process.

scholarly journals Single server queueing networks with varying service times and renewal input

2000 ◽  
Vol 13 (4) ◽  
pp. 429-450 ◽  
Author(s):  
Pierre Le Gall
Keyword(s):  
Network Management ◽  
Queueing Networks ◽  
Queueing Network ◽  
Single Server ◽  
Simulation Methods ◽  
Tandem Queues ◽  
Tandem Queue ◽  
Queueing Delay ◽  
New Concepts ◽  
Service Times

Using recent results in tandem queues and queueing networks with renewal input, when successive service times of the same customer are varying (and when the busy periods are frequently not broken up in large networks), the local queueing delay of a single server queueing network is evaluated utilizing new concepts of virtual and actual delays (respectively). It appears that because of an important property, due to the underlying tandem queue effect, the usual queueing standards (related to long queues) cannot protect against significant overloads in the buffers due to some possible “agglutination phenomenon” (related to short queues). Usual network management methods and traffic simulation methods should be revised, and should monitor the partial traffic streams loads (and not only the server load).

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