15. Queueing Theory (Management

1. Queuing System components

• Queueing discipline

• Arrival process

• Service process

• Number of servers (workers, staff members, etc.)

2. Queueing discipline

First come first serve (FCFS)

Last come first serve (LCFS)

Service in random order (SIRO)

Shortest Processing Time first (SPT)

Priority queue

Preemptive queue (we can interrupt a job even after the job started)

Non-preemptitve queue (we cannot interrupt a job once started)

3. Arrival process

• Poisson arrival (M)

(M stands for Markovian or Memoryless)

• Deterministic arrival (D)

• General arrival (G)

4. Service process

• Exponential service time (M)

(M stands for Markovian or Memoryless)

• Deterministic service time (D)

• General service time (G)

5. Number of Servers

• Single server (1)

• Multiple servers (k)

6. Limitation of Queueing Analysis

Queueing Model can only analyze steady-state properties of systems

7. Operating Characteristics

ρ : utilization (proportion of time server is busy)

P(0) : probability that the system is empty

λ : mean arrival rate

µ : mean service rate

Lq : mean number of customers in the queue

(L : mean number of customers in the system)

Wq : mean time a customer spends in the queue

(W : mean time a customer spends in the system)

P(Wq>0) : probability that a customer waits

8. Utilization

Utilization is a proportion of time server(s) are busy

λ = arrival rate

k μ = total service rate

Utilization ρ = λ/ (kμ)

9. L and W

There is a relationship (Little’s Law) among arrival rate, queue length and waiting time.

Lq = λWq       and       L =  λW

10. trade off

11. queueing notation