A simulation model to implement multiple client class server-client ...

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Average waiting time Class 1(W 1 ) Table 1: A comparison based on Littles law Average waiting time Class 2(W 2 ) Total number of customs class 1(N 1 ) Total number of customs class 2(N 2 ) Measured average number of customers in the Calculation of littles law W 1 × N 1 + W 2 × N 1 system 54.53763 51.49038 93 104 0.20854 0.20854 46.25325 49.35146 999 993 1.90426 1.90426 28.23506 29.22124 1238 1243 1.42554 1.42554 14.12709 14.39264 1676 1658 0.9508 0.9508 17.09113 17.26543 2030 1944 1.36518 1.36518 14.42547 14.40802 2536 2468 1.44284 1.44284 units for each client class in this validation. We used 18 combinations of stochastic arrival rate and service rates from exponential distribution to simulate workloads and processing times of both client classes. All these combinations were selected to maintain the system in the steady state. The workload scripts generated from arrival rate were given to the ClientClassWorkload- Generator instance of each client class and the processing times of ResourceUnits were generated from service rate from exponential distribution in each experiment. A experiment was conducted for 50,000 ticks. The StatisticCalulator instance was used to compute the final statistics of the experiment including, the average response time, average arrival rates and average number of customers in the system. In these calculations the system was considered as two sub systems, each providing services to a corresponding client class. The comparison of the statics were done as the total number of clients in these two sub systems as equal to total measured number of clients in the systems when both sub systems considered together. These statics were an exact match for all of these experiments. Some of the selected experimental results are summarized in Table 1 indicates that measured number in the system is precisely equal to the calculations of the Little’s law. The same results were observed for the experiments conducted in deterministic arrival and service rates. Hence, the multi-client class simulations implemented from the DES model described in Section 3, precisely conform to the Little’s law. This result also indicates that all the request input to the system leave the system. Further, implementation of the DES model including the statistical calculations is correct. 4.2. Conformant to single-server queuing system (M/M/1) In this section a single-server queuing system is developed and simulated, and then the measurements are compared to theoretic results of (M/M/1) queuing system from literature. The simulation was implemented with a single resource unit and queue. The workload script of a single client is generated according to Poisson arrival process. The resource reservation time (processing time) is generated according to the exponential distribution. All the components available from the DES model are used in this implementation as well. The 18 experiments were conducted with the same arrival and processing time combinations utilized in Section 4.1. Due to the measured results are compared with probabilistic theoretical values, each experiment was run for 200,000 ticks. As the basis of validation, we compared the measured average number of customers in the system from the simulations with the expected number of customers calculated 6
Table 2: A comparison based on single-server queuing system (M/M/1) λ µ L theoretical L measured 0.02 0.1 0.247 0.244 0.025 0.067 0.423 0.418 0.04 0.056 2.622 2.591 0.022 0.03 2.799 2.756 0.02 0.027 2.838 2.801 0.02 0.04 0.979 0.968 0.033 0.067 0.976 0.96 from queuing theoretic results. Let us say λ and µ represents the mean arrival rate and mean service time respectively. Theoretically the expected customers in the system is calculated as follows: L theoretical = λ µ − λ So that, given the simulated λ and µ, the L theoretical calculated from equation (2) should approximately equal to L measured from the simulation. In order to quantify the statistical significance of the difference, we also conducted a Kolmogorov-Smirnov test using the data of the 18 simulations conducted under different λ and µ. The compassion of results of some of the experiments are summarized in Table 2. The results of Kolmogorov-Smirnov test producedx D statistics of 0.11 and P statistic of 1. In nutshell if the P value is less than 0.05 there is significant difference between the data sets. However, since for this case P = 1 concludes that the data set of L measured and L theoretical has no significant difference. As a consequence, the single-server queuing system (M/M/1) constructed from the DES model conform to the queuing theoretic results. This confirms the M/M/1 queuing system implemented using the DES model constructs which includes scheduling of a single queue and resource is correct. 4.3. Conformant to multi-server queuing system (M/M/c) In this section we construct a multi-server queuing system serving a single queue. For this system, the same assumptions used in Section 4.2 are maintained. However, (c=) 5 resource units are used to represent 5 servers in the system. 18 experiments were conducted under same settings as in Section 4.2 in order to gather measurement data. The same measurement of the average number of customers in the system was used for the comparison. The theoretical calculation is done as follows: r c ∑c−1 p 0 = ( c!(1 − ρ) + (c − 1) r n n! )−1 (3) L theoretical = r + n=0 (2) r c ρ c!(1 − ρ) 2 p 0, (4) Where r = λ µ , ρ = r c , c = number of servers (5 for this experiment). The results are summarized in Table 3. 7
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