The Simulation of a Queueing System Consist of Two Parallel Heterogeneous Channels with no Waiting Line
Vedat Sağlam^{1}^{, *}, Murat Sağır^{1}, Erdinç Yücesoy^{1}, Müjgan Zobu^{2}
^{1}Department of Statistics, Ondokuz Mayıs University, Samsun, Turkey
^{2}Department of Statistics, Amasya University, Amasya, Turkey
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Vedat Sağlam, Murat Sağır, Erdinç Yücesoy, Müjgan Zobu. The Simulation of a Queueing System Consist of Two Parallel Heterogeneous Channels with no Waiting Line.Pure and Applied Mathematics Journal.Vol.4, No. 5, 2015, pp. 216218. doi: 10.11648/j.pamj.20150405.12
Abstract: A queueing system with two parallel heterogenous channels without waiting is considered. In this queueing system customer arrivals are Poisson distributed with rate. Each customer has exponentially distributed service time with parameter at channel. When a customer arrives this system if both the service channels are available, the customer has service with or probabilities at first and second service channels respectively. If one of the service channels is available, the customer has service at this service channel or leaves the system without being served if both of the service channels are busy. We have obtained mean waiting time and mean number of customers of the system and a simulation of this system is performed.
Keywords: Heterogeneous Queueing Systems, Loss Probability, Optimization, Simulation, Markov Chain, Kolmogorov Equation, Transition Rates
1. Introduction
Stochastic queueing models and their applications have important roles in fields such as transportation, computer networks, communication, production lines and etc. There are a lot of studies regarding queueing systems with homogeneous Poisson arrivals. In fact in real life practical applications the arrivals to queueing systems are nonhomogeneous. The mean customer number and the distribution of waiting time in a tandem queueing system with Poisson arrivals are obtained in [1]. The response time of a tandem Markovian queue with blocking is studied in [2]. A heterogeneous stochastic queueing system with two channel in which no waiting is allowed is analyzed by [3]. Performance measures of a tandem queueing system with two stages are obtained and these performance measures are given in [4]. A queueing model with nonhomogeneous bulk arrivals under parallel and series configuration is developed and analyzed in [5]. A number of different models for Markovmodulated queueing systems surveyed. Then a model in which the workload process and the secondary process together constitute a Markov compound Poisson process is analyzed by [6]. In [7] a tandem queueing system that can be formulated as a continuoustime Markov chain is considered and this system is investigated how to maximize the throughput when the queue is considered and several monotonicity properties of optimal policies for such a system are proved in [8]. The algorithms for the MultiServer Queue is had in [12]. A queueing system with two parallel heterogeneous channels without waiting is considered in [9], in this system the arrivals are Poisson distributed with rate. Each customer has exponentially distributed service time with parameter at channel. When a customer arrives this system if both the service channels are available, the customer has service with or probabilities at first and second service channels respectively. If one of the service channels is available, the customer has service at this service channel or leaves the system without being served if both of the service channels are busy so loss occurs. This loss probability optimized in [9]. In this paper we obtain mean waiting time and mean number of customers of the system studied in [9]. In addition, the mean waiting time of this system is optimized under customized conditions. Finally a 100.000 step simulation is done for this system and it is seen that the simulation results tend to exact results.
2. Definition of the Queueing System
The customers arrive this queueing system according to Poisson distribution with parameter. In service, each customer has exponentially distributed service time with service parameter. When a customer arrives this system if both the services are available the customer has service with and probabilities at first and second service channels respectively. If one of the service channel is available the customer has service at this service channel or leaves the system without being served if both of the service channels are busy so loss occurs. The system is shown in Figure 1 below.
3. Kolmogorov Equations
Transient Solutions in Markovian queueing systems are obtained in [13]. Two dimensional Markov chain defines the probability that there are customers at first service channel and there are customers at second service channel at a given time . Where . The Kolmogorov differential equations of these transient probabilities are obtained as below:
(1)
where
(2)
The transition rates matrices is:
(3)
in which and and
Transient probabilities of the system above are found as:
(4)
(5)
(6)
(7)
Erlang B formula is given in [10, 11, 14]. The loss probability in queuing systems with heterogeneous servers is shown by [15, 16]. Where, is the probability that the system is empty and is the probability that first and second service channels are busy also this is the loss probability. The optimization of loss probability is as follows:
When and then loss probability is minimum [9].
When under condition the loss probability takes its minimum value for .
(8)
as a result homogeneous system is better than heterogeneous system for criteria [9].
4. Performance Measures of System
Performance measure of since there is no waiting in this queueing system, the mean customer number and the mean waiting time in the system is found.
Mean customer number in the system
(9)
Mean waiting number in the system and its optimization
(10)
Similarly, optimizations for mean waiting time in the system can be done. The optimizations done for loss probability above are also valid for .
When and , the loss probability is minimum.
(11)
For and under condition , the loss probability has its minimum value for .
(12)
To make minimum, product must be maximum. Under condition , product is maximum for .
(13)
5. Simulation
Table 1. Mean waiting time in the system and loss Probability for .
Iteration Numbers  Simulation Value  Exact Value  



 
500  0.0620  0.6579  0,07770  0,68493 
1000  0.0540  0.6790  0,07770  0,68493 
5000  0.0606  0.7034  0,07770  0,68493 
10000  0.0590  0.6885  0,07770  0,68493 
Table 2. Mean waiting time in the system and loss Probability for .
Iteration Numbers  Simulation Value  Exact Value  



 
500  0.0820  0.8219  0,08236  0,79860 
1000  0.0810  0.8012  0,08236  0,79860 
5000  0.0836  0.7942  0,08236  0,79860 
10000  0.0718  0.7955  0,08236  0,79860 
Table 3. Mean waiting time in the system and loss Probability for .
Iteration Numbers  Simulation Value  Exact Value  



 
500  0.0860  0.8571  0,08529  0,87438 
1000  0.0850  0.8564  0,08529  0,87438 
5000  0.0824  0.8504  0,08529  0,87438 
10000  0.0858  0.8422  0,08529  0,87438 
Table 4. Mean waiting time in the system and loss probability for .
Iteration Numbers  Simulation Value  Exact Value  



 
500  0.0820  0.8289  0,08288  0,83333 
1000  0.0820  0.8217  0,08288  0,83333 
5000  0.0828  0.8383  0,08288  0,83333 
10000  0.0824  0.8304  0,08288  0,83333 
6. Conclusion
Performance measures, the mean waiting time and the mean number of customers are obtained in this paper. In addition the optimization of mean waiting time is done under conditions given as , , the loss probability and for and under condition .
Loss probability and mean waiting time in this system is obtained by 500, 1000, 5000 and 10000 iteration steps and these results are given in Table1, Table2, Table3 and Table4 respectively. Then it is seen that these simulation results tend to exact values. The number of service channels in the system can be adjusted or a waiting line can be added to this system for further studies.
References