consider a two server system which a customer is served first by server 1, then by server 2 and then departs. the server times at server i are exponential random variables with rate ui i = 1,2.
a)When you arrive you find server 1 free and you find server one is free and one customers at server 2 .find the expected time that you spend in the system?
b)a)When you arrive you find server 1 free and you find server one is free and two customers at server 2 .find the expected time that you spend in the system?
To calculate the expected time spent in the system in both scenarios, we need to consider the service times at each server and the arrival rate of customers.
a) When you arrive and find server 1 free and one customer at server 2:
In this case, the customer at server 2 needs to be served first before you can begin service. The service time at server 1 is exponentially distributed with rate u1 = 1, and the service time at server 2 is exponentially distributed with rate u2 = 2.
To calculate the expected time spent in the system, we need to sum the time spent at each server:
Expected time spent at server 1 = (1/u1) = 1
Expected time spent at server 2 = (1/u2) = 1/2
Expected time spent in the system = (Expected time spent at server 1) + (Expected time spent at server 2) = 1 + 1/2 = 3/2 = 1.5 units of time.
b) When you arrive and find server 1 free and two customers at server 2:
In this case, both customers at server 2 need to be served before you can begin service. The service time at server 1 is exponentially distributed with rate u1 = 1, and the service time at server 2 is exponentially distributed with rate u2 = 2.
To calculate the expected time spent in the system, we need to sum the time spent at each server:
Expected time spent at server 1 = (1/u1) = 1
Expected time spent at server 2 = (1/u2) + (1/u2) = 1/2 + 1/2 = 1
Expected time spent in the system = (Expected time spent at server 1) + (Expected time spent at server 2) = 1 + 1 = 2 units of time.
So, in scenario (a), the expected time spent in the system is 1.5 units of time, and in scenario (b), the expected time spent in the system is 2 units of time.