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

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 ).