What is the idle time of each server in the three server system?

[Dhamnekar Win,odd]

 

Question:

Two customers move about among three servers. Upon completion of service at a server, the customer leaves that server and enters service at whichever of the other two servers is free. If the service times at server i are exponential with rate [latex]\mu_i, i= 1,2,3 [/latex], What proportion of time is server i idle?

 

My solution:

 

To determine the proportion of time each server is idle in this system, we can use the concept of Markov chains and queueing theory. Here’s a step-by-step outline of the approach:

1. Define the States:

   • Let [latex]( S_i ) [/latex]represent the state where server ( i ) is idle.
   • Since there are three servers, we have states [latex]( S_1, S_2, )[/latex] and [latex]( S_3 ).[/latex]

2. Transition Rates:

   • The service times are exponential with rates [latex]( \mu_1, \mu_2, ) and ( \mu_3 ).[/latex]
   • When a customer finishes service at server ( i ), they move to one of the other two servers. The transition rate from server ( i ) to server ( j ) is [latex]( \mu_i ).[/latex]

3. Balance Equations:

   • For each server ( i ), the proportion of time it is idle, denoted by ( P_i), can be found by solving the balance equations.
   • The balance equations for the idle times are:[latex] [ P_1 (\mu_2 + \mu_3) = \mu_2 P_2 + \mu_3 P_3 ] [ P_2 (\mu_1 + \mu_3) = \mu_1 P_1 + \mu_3 P_3 ] [ P_3 (\mu_1 + \mu_2) = \mu_1 P_1 + \mu_2 P_2 ][/latex]

4. Normalization Condition:

   • The sum of the proportions must equal 1: [latex][ P_1 + P_2 + P_3 = 1 ][/latex]

5. Solve the System of Equations:

   • Solve the above system of linear equations to find [latex]( P_1, P_2, ) and ( P_3 ).[/latex]

Let’s solve these equations step-by-step:

1. From the balance equations:[latex] [ P_1 (\mu_2 + \mu_3) = \mu_2 P_2 + \mu_3 P_3 ] [ P_2 (\mu_1 + \mu_3) = \mu_1 P_1 + \mu_3 P_3 ] [ P_3 (\mu_1 + \mu_2) = \mu_1 P_1 + \mu_2 P_2 ][/latex]

2. Using the normalization condition: [ P₁ + P₂ + P₃ = 1 ]

By solving these equations, you can find the exact proportions ( P₁, P₂, ) and ( P₃ ). 

Edited October 20 by Dhamnekar Win,odd