M/G/1 Type Vacation Models: Nonexhaustive Service

2006 ◽  
pp. 77-127
Author(s):  
Naishuo Tian ◽  
Zhe George Zhang
Keyword(s):  
Vacation Models

Time-dependent process of M/G/1 vacation models with exhaustive service

1992 ◽  
Vol 29 (02) ◽  
pp. 418-429 ◽  
Author(s):  
Hideaki Takagi
Keyword(s):  
Steady State ◽  
Initial Conditions ◽  
Setup Time ◽  
Time Dependent ◽  
Time Model ◽  
Vacation Models ◽  
Single Vacation ◽  
Multiple Vacation ◽  
Virtual Waiting Time ◽  
Vacation Model

Generalized M/G/1 vacation systems with exhaustive service include multiple and single vacation models and a setup time model possibly combined with an N-policy. In these models with given initial conditions, the time-dependent joint distribution of the server's state, the queue size, and the remaining vacation or service time is known (Takagi (1990)). In this paper, capitalizing on the above results, we obtain the Laplace transforms (with respect to time) for the distributions of the virtual waiting time, the unfinished work (backlog), and the depletion time. The steady-state limits of those transforms are also derived. An erroneous expression for the steady-state distribution of the depletion time in a multiple vacation model given by Keilson and Ramaswamy (1988) is corrected.

Time-dependent process of M/G/1 vacation models with exhaustive service

1992 ◽  
Vol 29 (2) ◽  
pp. 418-429 ◽  
Author(s):  
Hideaki Takagi
Keyword(s):  
Steady State ◽  
Initial Conditions ◽  
Setup Time ◽  
Time Dependent ◽  
Time Model ◽  
Vacation Models ◽  
Single Vacation ◽  
Multiple Vacation ◽  
Virtual Waiting Time ◽  
Vacation Model

Generalized M/G/1 vacation systems with exhaustive service include multiple and single vacation models and a setup time model possibly combined with an N-policy. In these models with given initial conditions, the time-dependent joint distribution of the server's state, the queue size, and the remaining vacation or service time is known (Takagi (1990)). In this paper, capitalizing on the above results, we obtain the Laplace transforms (with respect to time) for the distributions of the virtual waiting time, the unfinished work (backlog), and the depletion time. The steady-state limits of those transforms are also derived. An erroneous expression for the steady-state distribution of the depletion time in a multiple vacation model given by Keilson and Ramaswamy (1988) is corrected.

scholarly journals Some reflections on the Renewal-theory paradox in queueing theory

1998 ◽  
Vol 11 (3) ◽  
pp. 355-368 ◽  
Author(s):  
Robert B. Cooper ◽  
Shun-Chen Niu ◽  
Mandyam M. Srinivasan
Keyword(s):  
Waiting Time ◽  
Queueing Theory ◽  
Waiting Times ◽  
Renewal Theory ◽  
Vacation Models ◽  
Polling Models ◽  
Mean Waiting Time ◽  
The Mean ◽  
Expository Paper ◽  
Set Up

The classical renewal-theory (waiting time, or inspection) paradox states that the length of the renewal interval that covers a randomly-selected time epoch tends to be longer than an ordinary renewal interval. This paradox manifests itself in numerous interesting ways in queueing theory, a prime example being the celebrated Pollaczek-Khintchine formula for the mean waiting time in the M/G/1 queue. In this expository paper, we give intuitive arguments that “explain” why the renewal-theory paradox is ubiquitous in queueing theory, and why it sometimes produces anomalous results. In particular, we use these intuitive arguments to explain decomposition in vacation models, and to derive formulas that describe some recently-discovered counterintuitive results for polling models, such as the reduction of waiting times as a consequence of forcing the server to set up even when no work is waiting.

scholarly journals Entropy maximization and the busy period of some single-server vacation models

2004 ◽  
Vol 38 (3) ◽  
pp. 195-213 ◽  
Author(s):  
Jesus R. Artalejo ◽  
Maria J. Lopez-Herrero
Keyword(s):  
Busy Period ◽  
Single Server ◽  
Server Vacation ◽  
Entropy Maximization ◽  
Vacation Models

Oscillating random walk models for GI/G/1 vacation systems with Bernoulli schedules

1986 ◽  
Vol 23 (03) ◽  
pp. 790-802 ◽  
Author(s):  
J. Keilson ◽  
L. D. Servi
Keyword(s):  
Random Walk ◽  
State Space ◽  
Complex Plane ◽  
General Class ◽  
Stochastic Behavior ◽  
Vacation Models ◽  
Server Vacations ◽  
Fixed Probability ◽  
Oscillating Random Walk

Processors handling multi-class traffic typically alternate between serving a particular class of traffic and performing other tasks, e.g., secondary service tasks or routine maintenance. The stochastic behavior of such systems is modeled by a newly introduced class of Bernoulli GI/G/1 vacation models. For this model, when a vacation is completed and customers are present, a customer is served. When a customer has just been served and other customers are present, the server accepts a customer with fixed probability p or commences a vacation of prespecified random duration with probability 1 – p. Whenever no customers are present, a vacation is taken. When p = 0 or p = 1 this schedule reduces to the previously introduced single service schedule and the exhaustive service schedule, respectively. An analysis of all three schedules on a state space incorporating server vacations is presented using simple methods in the complex plane. It is shown that the recent decomposition results for exhaustive service extend to the more general class of Bernoulli schedules.

scholarly journals ON A GENERIC CLASS OF LÉVY-DRIVEN VACATION MODELS

2009 ◽  
Vol 24 (1) ◽  
pp. 1-12 ◽  
Author(s):  
Onno Boxma ◽  
Offer Kella ◽  
Michel Mandjes
Keyword(s):  
Steady State ◽  
Content Distribution ◽  
Levy Process ◽  
Active Period ◽  
Queuing Systems ◽  
Stieltjes Transform ◽  
Server Vacation ◽  
Vacation Models ◽  
Buffer Content ◽  
Generic Class

This article analyzes a generic class of queuing systems with server vacation. The special feature of the models considered is that the duration of the vacations explicitly depends on the buffer content evolution during the previous active period (i.e., the time elapsed since the previous vacation). During both active periods and vacations, the buffer content evolves as a Lévy process. For two specific classes of models, the Laplace–Stieltjes transform of the buffer content distribution at switching epochs between successive vacations and active periods, and in steady state, is derived.

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