M/M/2 with Heterogeneous Servers

Consider a variant of the M/M/2 queue where the service rates of the two processors are not identical. Denote the service rate of the first processor by 1 andtheservicerateofthesecondprocessorby2,where1 >2.Inthecase of heterogeneous servers, the rule is that when both servers are idle, the faster server is scheduled for service before the slower one. Define the utilization, , for this system to be = .

1 +2

Set up a CTMC and determine the mean number of jobs in the system and the

mean response time. You should get

E[N] = 1/A(1 )2

where

14.6 exercises 267

A=12(1+2)/(+2) +1/1

14.5 M/M/2 with Heterogeneous Servers Consider a variant of the M/M/2 queue where the service rates of the two processors are not identical. Denote the service rate of the first processor by H1 and the service rate of the second processor by M2, where p > H2. In the case of heterogeneous servers, the rule is that when both servers are idle, the faster server is scheduled for service before the slower one. Define the utilization, p, for this system to be p= mathe Set up a CTMC and determine the mean number of jobs in the system and the mean response time. You should get 1 E[N] = (14.14) A(1-P) 14.6 EXERCISES 267 where 1 A= Mi Mg(1+2p) + X(X + H2) (14.15) 14.5 M/M/2 with Heterogeneous Servers Consider a variant of the M/M/2 queue where the service rates of the two processors are not identical. Denote the service rate of the first processor by H1 and the service rate of the second processor by M2, where p > H2. In the case of heterogeneous servers, the rule is that when both servers are idle, the faster server is scheduled for service before the slower one. Define the utilization, p, for this system to be p= mathe Set up a CTMC and determine the mean number of jobs in the system and the mean response time. You should get 1 E[N] = (14.14) A(1-P) 14.6 EXERCISES 267 where 1 A= Mi Mg(1+2p) + X(X + H2) (14.15)