This paper studies a discrete-time N-policy Geo/G/1 queueing system with feedback and repairable server. With a probabilistic analysis method and renewal process theory, the steady-state system size distribution is derived. Further, the steady-state system size distribution derived in this work is extremely suitable for numerical calculations. Numerical example illustrates the important application of steady-state system size distribution in system capacity design for a network access proxy system.
Over the past years, the analysis of discrete-time queueing systems has received more attention in the literatures. This is because the discrete-time queues are more appropriate than their continuous-time counterparts in their applicability for many computer and communication systems in which time is divided into fixed-length time intervals (for details, e.g., see [
It is well known that the N-policy introduced by Yadin and Naor [
Feedback was introduced by Takács [
A remarkable phenomenon in a queueing system is its server breakdown. For instance, in computer systems, the machine may be subject to scheduled backups and unforeseeable failures. Queues with server breakdowns can also be called repairable queues. For only the studies on discrete-time repairable queues, the reader may refer to Wang and Zhang [
However, only very few works in the literature concerned with N-policy queues with feedback and repairable server have been done. Particularly, the researches of the discrete-time N-policy queues with feedback and repairable server are not found. Further, in most works of the N-policy queue, the probability generating function (PGF) of steady-state system size distribution rather than steady-state system size distribution was obtained. In this paper, we investigate a discrete-time N-policy Geo/G/1 queueing system with feedback and repairable server. With a probability analysis method and renewal process theory, we derive the steady-state system size distribution. It is noted that the steady-state system size distribution derived in this paper is quite suitable for numerical calculations. What is more, numerical examples illustrate the applications of steady-state system size distribution in system capacity design for a network access proxy system.
The rest of the paper is organized as follows. The next section presents the model assumptions and some preliminaries. In Section
We consider the following discrete-time N-policy Geo/
(1) Let the time axis be slotted into intervals of equal length with the length of a slot being one unit. To be more specific, let the time axis be marked by
Various time epochs in a late arrival system with delayed access.
(2) When there are
(3) The server may fail when and only when it is serving a customer. The failed server will be repaired immediately. After repair, the server is as good as new and continues to serve the customer whose service has not been finished yet. We assume that the service time for a customer is cumulative.
(4) The lifetime of server has a geometric distribution with parameter
(5) All random variables are mutually independent. At the initial time
Throughout this paper, we adopt the following notations.
The “generalized service time of a customer”
The PGF and the mean of
For customer
Let
The “server busy period” denotes the time interval from the time when the server begins to serve a customer until the system becomes empty, which contains some possible repair times of server due to its failures in the process of service.
Let
The server busy period initiated with
The “system idle period" denotes the time interval from the time the system becomes empty until the first customer arrives and enters the system.
The “server idle period” denotes the time interval from the time the system becomes empty until the server begins to serve the first customer.
Let
(i) If
(ii) If
If
If
In this section, above all, we will discuss the system size distribution at any epoch
Let
For
Denote by
Let
In the light of the above service order, if the epoch
Let
For
Noting that the system is empty at time epoch
For
For
For
For
For
Letting for for where
when
Applying
Also, for
Let
Using
Equation (
Let
Equation (
If
Our study has a potential application in a network access proxy system. In such a system, Channel requests, grants, data transmissions, and receptions all proceed in fixed time intervals. That is, service request sending, preprocessing, and processing are done in a discrete-time manner. Service requests sent by the users can be modelled as a Bernoulli process with rate
Numerical results in Table
The steady-state service request number distribution for a network access proxy system (
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0.14661 | 0.17341 | 0.18040 | 0.18265 | 0.10049 | 0.03816 | 0.01227 |
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0.00483 | 0.00456 | 0.00452 | 0.00427 | 0.00388 | 0.00336 | 0.00276 |
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0.00162 | 0.00127 | 0.00100 | 0.00079 | 0.00063 | 0.00050 | 0.00040 |
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0.00026 | 0.00021 | 0.00016 | 0.00013 | 0.00010 | 0.00008 | 0.00007 |
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0.00004 | 0.00003 | 0.00002 | 0.00001 | 0.00000 | 0.00000 | 0.00000 |
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With Matlab 7.0, the data here are accurate for five places of decimals.
Denote by
The above three probability values are greater than 23
Let
The optimal system capacity of a network access proxy system for different values of controllable parameter
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34 | 31 | 24 | 18 | 13 | 7 |
In this paper, we consider a discrete-time N-policy Geo/G/1 queueing system with feedback and repairable server. With a probability analysis and renewal process theory, we discuss the transient system size distribution and derive the steady-state system size distribution and its PGF. It should be noted that the steady-state system size distribution derived by this paper is quite suitable for numerical calculation. Numerical examples show the application of steady-state system size distribution in system capacity design for a network access proxy system. In the future, further study, such as the random N-policy Ge
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors would like to thank the referees and the editor for their valuable comments and suggestions. This work is supported by the Basic and Frontier Research Foundation of Chongqing of China (cstc2013jcyjA00008) and the Scientific Research Starting Foundation for Doctors of Chongqing University of Technology (2012ZD48).