Week 10 - Queueing theory

Question 1

Arrival & service rates INDEPENDENT of the state - single server

1. Prob. that server is IDLE
2. Proportion of time the server is BUSY
3. Expected # of customers being served, S
4. Expected # of customers in the SYSTEM, Ls
5. Expected # of customers in the QUEUE, Lq
6. Expected queuing time, Wq
7. Expected time in the system, Ws
8. CONDITIONAL expected queueing time, wq

Answer 1

1. Prob. that server is IDLE
   P0 = 1-ρ
2. Proportion of time the server is BUSY
   = ρ
3. Expected # of customers being served, S
   = ρ
4. Expected # of customers in the SYSTEM, Ls
   = ρ/1-ρ
5. Expected # of customers in the QUEUE, Lq
   = Ls - S
6. Expected queuing time, Wq
   = Lq/lambda (Little’s law)
7. Expected time in the system, Ws
   = Ls/lambda
   = Wq + 1/µ
8. CONDITIONAL expected queueing time, wq
   = Wq/1-P0

Question 2

Arrival & service rates INDEPENDENT of the state - multiple servers

1. Expected proportion of IDLE TIME PER SERVER
2. Expected # of customers in the QUEUE, Lq
3. Expected waiting time in the QUEUE, Wq
4. Expected time in the SYSTEM, Ws
5. Expected # of customers in the SYSTEM, Ls
6. Expected # of customers being SERVED, S

Answer 2

1. Expected proportion of IDLE TIME PER SERVER
   = 1-ρ
2. Expected # of customers in the QUEUE, Lq
   = P0 (sρ)^s/s! x ρ/(1-ρ)^2
3. Expected waiting time in the QUEUE, Wq
   = Lq/lambda
4. Expected time in the SYSTEM, Ws
   = Wq + 1/µ
5. Expected # of customers in the SYSTEM, Ls
   = lambda Ws (Little’s law)
6. Expected # of customers being SERVED, S
   = Ls - Lq
   = sρ

Question 3

Arrival & service rates INDEPENDENT of the state - single server
4 assumptions made

Answer 3

1. Poisson arrival process
2. Arrival rate is constant
3. Exponentially distributed service times (memoryless)
4. System is always in STEADY STATE