Regeneration in tandem queues with multiserver stations

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.

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Markov chains with exponential return times are finitary

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2020.100 ◽

2020 ◽

pp. 1-9

Author(s):

OMER ANGEL ◽

YINON SPINKA

Keyword(s):

Markov Chain ◽

Markov Chains ◽

Renewal Process ◽

Proper Subgroup ◽

Ergodic Markov Chain ◽

Countable State Space ◽

Return Times ◽

Exponential Tails ◽

Stationary Version ◽

Countable State

Abstract Consider an ergodic Markov chain on a countable state space for which the return times have exponential tails. We show that the stationary version of any such chain is a finitary factor of an independent and identically distributed (i.i.d.) process. A key step is to show that any stationary renewal process whose jump distribution has exponential tails and is not supported on a proper subgroup of ℤ is a finitary factor of an i.i.d. process.

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

Advances in Applied Probability ◽

10.1017/s0001867800032869 ◽

1979 ◽

Vol 11 (03) ◽

pp. 660-672 ◽

Cited By ~ 5

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.

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On the multiserver queue with finite waiting room and controlled input

Advances in Applied Probability ◽

10.2307/1427148 ◽

1985 ◽

Vol 17 (2) ◽

pp. 408-423 ◽

Cited By ~ 11

Author(s):

Jewgeni Dshalalow

Keyword(s):

Markov Chain ◽

Queue Length ◽

Limiting Distribution ◽

Embedded Markov Chain ◽

Single Server ◽

Simple Relationship ◽

Waiting Room ◽

Original Process ◽

State Dependent ◽

Server System

In this paper we study a multi-channel queueing model of type with N waiting places and a non-recurrent input flow dependent on queue length at the time of each arrival. The queue length is treated as a basic process. We first determine explicitly the limit distribution of the embedded Markov chain. Then, by introducing an auxiliary Markov process, we find a simple relationship between the limiting distribution of the Markov chain and the limiting distribution of the original process with continuous time parameter. Here we simultaneously combine two methods: solving the corresponding Kolmogorov system of the differential equations, and using an approach based on the theory of semi-regenerative processes. Among various applications of multi-channel queues with state-dependent input stream, we consider a closed single-server system with reserve replacement and state-dependent service, which turns out to be dual (in a certain sense) in relation to our model; an optimization problem is also solved, and an interpretation by means of tandem systems is discussed.

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Regeneration in tandem queues

Advances in Applied Probability ◽

10.2307/1426476 ◽

1981 ◽

Vol 13 (1) ◽

pp. 221-230 ◽

Cited By ~ 16

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.

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

Journal of Applied Mathematics and Stochastic Analysis ◽

10.1155/s1048953399000349 ◽

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.

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

Advances in Applied Probability ◽

10.2307/1426960 ◽

1979 ◽

Vol 11 (3) ◽

pp. 660-672 ◽

Cited By ~ 8

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.

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Single server queueing networks with varying service times and renewal input

Journal of Applied Mathematics and Stochastic Analysis ◽

10.1155/s1048953300000368 ◽

2000 ◽

Vol 13 (4) ◽

pp. 429-450 ◽

Cited By ~ 3

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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User-Optimal State-Dependent Routeing in Parallel Tandem Queues with Loss

Journal of Applied Probability ◽

10.1239/jap/1143936259 ◽

2006 ◽

Vol 43 (1) ◽

pp. 274-281 ◽

Cited By ~ 10

Author(s):

Scott Spicer ◽

Ilze Ziedins

Keyword(s):

Loss Probability ◽

Single Server ◽

Tandem Queues ◽

Tandem Queue ◽

Optimal State ◽

Final Destination ◽

State Dependent ◽

In Series ◽

Queues In Series ◽

Residual Service

We consider a system of parallel, finite tandem queues with loss. Each tandem queue consists of two single-server queues in series, with capacities C1 and C2 and exponential service times with rates μ1 and μ2 for the first and second queues, respectively. Customers that arrive at a queue that is full are lost. Customers arriving at the system can choose which tandem queue to enter. We show that, for customers choosing a queue to maximise the probability of their reaching the destination (or minimise their individual loss probability), it will sometimes be optimal to choose queues with more customers already present and/or with greater residual service requirements (where preceding customers are further from their final destination).

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User-Optimal State-Dependent Routeing in Parallel Tandem Queues with Loss

Journal of Applied Probability ◽

10.1017/s0021900200001522 ◽

2006 ◽

Vol 43 (01) ◽

pp. 274-281 ◽

Cited By ~ 1

Author(s):

Scott Spicer ◽

Ilze Ziedins

Keyword(s):

Loss Probability ◽

Single Server ◽

Tandem Queues ◽

Tandem Queue ◽

Optimal State ◽

Final Destination ◽

State Dependent ◽

In Series ◽

Queues In Series ◽

Residual Service

We consider a system of parallel, finite tandem queues with loss. Each tandem queue consists of two single-server queues in series, with capacities C 1 and C 2 and exponential service times with rates μ1 and μ2 for the first and second queues, respectively. Customers that arrive at a queue that is full are lost. Customers arriving at the system can choose which tandem queue to enter. We show that, for customers choosing a queue to maximise the probability of their reaching the destination (or minimise their individual loss probability), it will sometimes be optimal to choose queues with more customers already present and/or with greater residual service requirements (where preceding customers are further from their final destination).

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