Question

Question 2 Individual customers arrive at a gas station randomly. The time of each arrival Tn has the following probability density function: fTa (t) There are c pumps. The time it takes to fill a gas tank at a particular pump is exponentially distributed with mean џ. Pumping times are independent Find the stationary distribution of the number of customers at the gas station (waiting for a pump, or pumping). Assume λ. Simplify the result as much as possible (no infinite sums!

Answer 1

This is M/M/c queue

In MIMIc queues, the arrival rate remains same as MIM1 queues but the service rate will depend on the number of servers. The service rate will be nu for n<-c. As soon as the number of customers exceeds c, the service rate becomes uc as shown in Figure 9.8. 0 C+1 Figure 9.8: Transition of states in MIMIc Queuing Model . The serv ice rate, Me in this case will be (n11) n < c for n 1,2, c (cu) n> c for n-c, c+1, The probability of having n customers in the service system can be written in similar way as we wrote for MIM|I model but with revised service rate.

Or λη 1 (2 n!(μ n=