# Tandem queue - response time distribution

In tandem queue with two queuing system, each server has exp(mu0) and exp(mu1) service time distribution and arrival rate is poisson(lambda). Scheduling policy is FCFS. What would be the response time distribution in the system?

Using laplace transform, I know how to calculate for single queue (M/M/1) but how to calculate for 2 queues in tandem network. What would be the conditional laplace transform of response time given n jobs n the system?

Thanks.

It is known that such a model has a product-form solution. Indeed, the response time for a tandem queue was the subject of a paper by Edgar Reich in 1957 (open access).

Reich, Edgar. Waiting times when queues are in tandem. The Annals of Mathematical Statistics 28.3 (1957): 768-773.

Therefore, as the solution by bkd.online notes, you compute the response time distribution by summing the independent response times for each of the nodes.

With jackson's result, each subqueue of the tandem queue would behave as if its a separate M/M/1 queue. The resultant response time of the system is hence summation of response times of all the components.

$R = \displaystyle\sum_{i}R_i$

With this reasoning, distribution of the response time of the whole system should be hypoexponential. If response time distributions of all the 'n' queues are same (with same parameter), then the resultant distribution would be n-stage erlang.

• The summation of response times holds if the queues are multi-server (M/M/c)? – Guillermo Guardastagno Oct 16 '17 at 20:12