Tandem Queueing System with Different Types of Customers (Queueing Theory) (Analytical and Stochastic Modeling) Part 2

Numerical Results

In the numerical experiment, we solve the problem of optimal choice of the number N of servers at Station 1 and the number R of servers at Station 2.

We consider the following cost criterion (an average gain per unit time under the steady-state operation of the system) of the system operation:

where a is the cost of utilization of a server at Station 2 per unit time (maintenance cost), cr is a gain for successful service of each customer of type r, r = 1, 2.

Our aim is to find numerically the optimal number R of servers at Station 2 that provides the maximal value to the cost criterion for different number N of servers at Station 1.

The input flow to the tandem is described by the MMAP defined by the following matrices:

Intensities of arrival of type-r customers are equal to 8.33357, respectively. Coefficient of correlation of two successive inter-arrival times is equal to 0.2. Coefficients of correlation of two successive intervals between arrivals of type-1 and type-2 customers are equal to 0.0425364 and 0.178071, respectively.

The service rates of customers at Station 1 are defined by The service rate of customers at Station 2 is

Let us first illustrate dependence of probabilityof successful service of type-1 customers at Station-1, probability ___of successful service of type-1 customers at Station-2, and probabilityof successful service of type-1 customers at both stations.

Table 1 contains the values of probabilityon the number N of servers at Station 1. Evidently, this probability does not depend on the number R of servers at Station 2.

Table 1. Dependence of probabilityon the number N of servers at Station 1

number N of servers 5		10	15	20	35
	0.317301	0.599657	0.823246	0.9532157	0.9999932

Table 2 contains the values of probabilityon the number N of servers at Station 1 and the number R of servers at Station 2.

	R= 1	R = 3	R = 5	R = 8	i? = 10	R= 13	R= 17	R = 20
N = 5	0.155	0.439	0.675	0.902	0.969	0.997	0.998	0.999
N= 10	0.088	0.258	0.418	0.633	0.753	0.887	0.976	0.995
N= 15	0.065	0.193	0.317	0.491	0.598	0.739	0.882	0.948
N = 20	0.057	0.169	0.277	0.432	0.529	0.662	0.810	0.893
N = 35	0.054	0.161	0.265	0.413	0.507	0.636	0.783	0.869

Table 3 contains the values of probability Psucc,i on the number N of servers at Station 1 and the number R of servers at Station 2.

Table 3. Dependence of probability Psucc,1 of successful service of type-1 customers both stations on N and R

	R= 1	R = 3	R = 5	R = 8	i? = 10	R = 13	R= 17	R = 20
N = 5	0.049	0.139	0.214	0.286	0.307	0.316	0.317	0.317
N= 10	0.053	0.154	0.251	0.379	0.451	0.532	0.585	0.596
N= 15	0.054	0.159	0.261	0.404	0.492	0.609	0.726	0.781
N = 20	0.055	0.161	0.264	0.412	0.504	0.631	0.772	0.851
N = 35	0.055	0.161	0.265	0.413	0.507	0.636	0.783	0.869

Let now consider optimization problem. Let the values of the cost coefficients c1, c2, a are assumed to be as follows: c1 = 15, c2 = 1, a =1.

The value of criterion I as a function of the numbers R and N of servers at Station 2 and Station 1 is presented in Figure 2.

Fig. 2. The cost criterion as a function of the number of servers at Station 1 and Station 2

Fig. 3. The cost criterion as a function of the number of servers at Station 2 under different number of servers at Station 1

Fig. 4. The cost criterion as a function of the number of servers at Station 1 under different number of servers at Station 2

For better understanding the behavior of cost criterion we present the values of the criterion as function of R under different but fixed values of N in Figure 3 and the values of the criterion as function of N under different but fixed values of R in Figure 4.

Table 4 contains the values of cost criterion I for some values of R and N. The optimal values of I as a function of R for different values of N are shown by bold numbers.

Table 4. Dependence of cost criterion I on R and N

	i? = 1	R = 3	i? = 5	R = 8	R = 10	R= 13	R= 17	R = 20
N = 5	2.877	3.133	3.001	1.804	0.332	-2.447	-6.423	-9.423
N= 10	5.319	5.871	6.274	6.495	6.294	5.308	2.634	-0.080
N=15	7.212	7.851	8.393	8.979	9.178	9.091	8.028	6.3898
N = 20	8.306	8.974	9.558	10.249	10.559	10.736	10.268	9.243
N = 35	8.699	9.373	9.967	10.681	11.016	11.251	10.922	10.066

Table 5 contains the relative profit provided by the optimal value R* of a parameter R comparing to other values of a parameter. This relative profit is calculated by formula

Table 5. Dependence of the relative profit on R and N

	i? = 1	R = 3	R = 5	R = 8	R= 10	R = 13	R= 17	R = 20
N = 5	8.167	0.00	4.193	42.409	89.386	178.10	305.04	400.79
N= 10	18.103	9.604	3.399	0.00	3.089	18.268	59.447	101.23
N=15	21.412	14.453	8.551	2.165	0.00	0.943	12.524	30.385
N = 20	22.631	16.415	10.976	4.539	1.651	0.00	4.368	13.903
N = 35	22.681	16.686	11.407	5.059	2.086	0.00	2.919	10.530

It is seen from Figure 4 and Table 4 that, for each R, the value I(N) of the gain increases in the beginning and then becomes constant. It is explained by the fact that, for large value of N, all entering customers get service at Station 1 so that the gain from the successful service at this station and the rate of input flow at Station 2 become constant. Since the number R of servers at Station 2 is constant, the gain from the successful service at Station 2 becomes constant too. Thus, the value of total gain does not vary for large N.

To illustrate effect of correlation in the arrival process (and justify consideration of the MMAP instead of much simpler stationary Poisson arrival processes, we consider, in addition to the MMAP introduced above and having coefficient of correlation 0.2, two other MMAPs. These MMAPs have the same intensities of customers arrival, but different coefficient of correlation of successive inter-arrival times. One MMAP is a superposition of two independent stationary Poisson arrival processes with intensities equal to Ai = 1.66671 and A2 = 8.33357, respectively. This MMAP has correlation equal to zero.

The second MMAP defined by the following matrices:

Fig. 5. Dependence of probabilityof successful service of type-1 customers at Station-1 on the average arrival rate A

Fig. 6. Dependence of probabilityof successful service of type-1 customers at Station-2 on the average arrival rate A

Coefficient of correlation of two successive inter-arrival times is equal to 0.1. Coefficients of correlation of two successive intervals between arrivals of type-1 and type-2 customers are equal to 0.0154306, 0.0853564, respectively.

Figures 5 and 6 show dependence of probabilitiesof successful service of type-1 customers at Station-1 andof successful service of type-1 customers at Station-2 on the average arrival rate A. This rate is varied by means of multiplying the matrices Dk, k = 0,1, 2, by a scalar.

It is evidently seen from Figures 5 and 6, that the value of probabilities andessentially depend on correlation in arrival process. It is interesting to observe that probabilityof a customer loss at Station-1 increases when correlation in the arrival process grows, while probabilitydecreases. Explanation of this fact is as follows. Higher correlation in arrival process implies more bursty arrival of customers. There are time intervals when arrivals occur frequently and loss probability is high. So, higher correlation implies bigger value of probability. This implies that the arrival flow to Station 2 has smaller intensity, so loss probability at this Station is smaller.

Conclusion

We have analyzed the tandem queue with two stations defined by the multi-server queues without buffers where a part of arriving customers are served only at Station 1 while the others have to be served at both stations. Optimization problem is considered and solved. Cost criterion includes the gain of the system obtained from the service of customers and the maintenance cost of the servers at Station 2.

The obtained results can be applied for optimization of the structure of an office computer network with partial access of officers to the external network.

Results can be extended to the case of more general so called PH – phase type distribution of the service times, see, e.g., [5]. However computer realization in this case becomes much more time consuming due to the essential increase of the state space of the Markov chain under study that will additionally include components indicating the current phases of the service at each busy server or the number of servers at each phase of service of two types of customers at Station 1 and service at Station 2.

The results can be also extended to the case when not only the number of servers at both stations should be optimized, but additionally the share of customers, which are permitted to go to Station 2 after the service at Station 1, is a subject of optimization.