- #1
WMDhamnekar
MHB
- 376
- 28
- 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 ( Si) 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 ( Pi), 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 ).##
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 ( Si) 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 ( Pi), 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 ).##
