We study a finite buffer
In many real world queueing systems, the server may be unavailable for a random period of time when there is no customer in the waiting line at a service completion instant. This random period of server absence is often called server vacation; see Doshi [
Servi and Finn [
For some controllable queueing systems with vacations, it is usually assumed that the server is available or unavailable completely depending upon the number of customers present in the system. Whenever the system is empty, the server goes on vacation. In the instant at which the server returns back from a vacation and finds at least
State dependent queues are dependent on the queue size, arrivals, and their service times. This is applicable in many areas like cellular manufacturing cells, routers, and switches that regulate the transmission of information packages having finite buffer capacity. In many of these applications, the arrival and service rates depend on the state of the queue. These queues have applications on arrivals and service rates which yield less waiting time in the system. Chao and Rahman [
The present paper is an extension of the work of [
Motivated by the above observations, this paper aims to contribute to the theory of BS-VI models with
The paper is structured as follows. Next section presents the description and analysis of the model. Some special cases are derived in Section
In this paper, we consider a
The system length distributions at prearrival epoch are obtained by developing differential-difference equations at steady state. Treating the remaining interarrival time as supplementary variable, we write the equations as
Solving the set of equations from (
Substituting
Let
Setting
Some models available in the literature are deduced as special cases of our model by taking specific values of the parameters
If
If
If
In this section, some operating characteristics such as the average number of customers in the queue
In this subsection, we formulate an expected cost model, in which mean service rate during vacation
The total expected cost function per unit time is given by
Given a 3-point pattern, we can fit a quadratic function through corresponding functional values that has a unique minimum,
To validate the results obtained earlier, some numerical computations have been done and some of them are presented in the form of tables and graphs. The parameters of the system are taken as
Table
Performance characteristics of
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1.222528 | 1.222661 | 1.222806 | 1.223020 |
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2.156477 | 2.156633 | 2.156803 | 2.157055 |
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0.000027 | 0.000027 | 0.000027 | 0.000027 |
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1.511926 | 1.512090 | 1.512269 | 1.512534 |
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2.666959 | 2.667152 | 2.667362 | 2.667674 |
Figure
Impact of
Figure
Effect on
A similar observation of above figure can be made from Figure
Impact of
The effect of
Search for optimum service rate during working vacation period (
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2.50000 | 2.60000 | 2.70000 | 124.3275 | 124.2732 | 124.3748 | 2.58482 | 124.2712 |
2.50000 | 2.58482 | 2.60000 | 124.3275 | 124.2712 | 124.2732 | 2.58409 | 124.2712 |
2.50000 | 2.58409 | 2.58482 | 124.3275 | 124.2712 | 124.2712 | 2.58394 | 124.2712 |
2.50000 | 2.58394 | 2.58409 | 124.3275 | 124.2712 | 124.2712 | 2.58398 | 124.2712 |
2.58394 | 2.58398 | 2.58409 | 124.2712 | 124.2712 | 124.2712 | 2.58398 | 124.2712 |
2.58394 | 2.58398 | 2.58398 | 124.2712 | 124.2712 | 124.2712 | — | — |
Effect on
Table
Sensitivity analysis of MWV and MWV-VI models.
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MWV | MWV-VI | ||||
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1 | 1.1 | 0.138792 | 089.547341 | 1.034160 | 0.417693 | 086.290059 | 0.758794 |
1.3 | 0.391248 | 099.693814 | 1.043741 | 0.560349 | 099.666960 | 1.122521 | |
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3 | 1.1 | 1.165482 | 101.761264 | 0.827047 | 0.947490 | 098.967713 | 0.823598 |
1.3 | 1.455129 | 110.737740 | 0.951105 | 1.110389 | 107.311209 | 1.043741 | |
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5 | 1.1 | 1.843442 | 108.197096 | 0.633725 | 1.832134 | 107.507853 | 0.618745 |
1.3 | 2.242303 | 116.015649 | 0.654932 | 2.147315 | 115.435968 | 0.692579 |
In this paper, we have carried out an analysis of a renewal input state dependent
The authors declare that there is no conflict of interests regarding the publication of this paper.