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

[WMDhamnekar]

Homework Statement

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 $μ_i, i=1,2,3,$ What proportion of time is server i idle?

Relevant Equations

Not required.

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 ( S_i) represent the state where server ( i ) is idle.
- Since there are three servers, we have states ( S₁, S₂, )and ( S₃).

2. Transition Rates:
- The service times are exponential with rates $( \mu_1, \mu_2, )$ and $( \mu_3 ).$
- 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 $( \mu_i ).$

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:
$[ P_1 (\mu_2 + \mu_3) = \mu_2 P_2 + \mu_3 P_3 ] [ P_2 (\mu_1 + \mu_3) = \mu_1 P_1 + \mu_3 P_3 ] [ P_3 (\mu_1 + \mu_2) = \mu_1 P_1 + \mu_2 P_2 ]$

4. Normalization Condition:
- The sum of the proportions must equal 1:
[P₁+ P₂ + P₃= 1]

5. Solve the System of Equations:
- Solve the above system of linear equations to find ( P₁, P₂ ) and ( P₃ ).

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

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

$[P_2 (\mu_1 + \mu_3) = \mu_1 P_1 + \mu_3 P_3]$

$[P_3 (\mu_1 + \mu_2) = \mu_1 P_1 + \mu_2 P_2]$

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

By solving these equations, you can find the exact proportions $( P_1, P_2, ) and ( P_3 ).$