Previously

• Optimization

• Probability Review

• Inventory Models

• Markov Decision Processes

Agenda

• Queues

Performance Measures

• T time in system• Tq waiting time (time in

queue)• N #customers in system• Nq #customers in queue

system

arrivals departures

queue servers

• W = E[T]• Wq= E[Tq]

• L = E[N]• Lq= E[Nq] fraction of time servers are

busy (utilization)

Plain-Vanilla Queue

• 1 queue,

• 1 class of customers

• identical servers

• time between customer arrivals is independent

• mean rate of arrivals, rate of serviceconstant

What Are We Ignoring?

• Rush-hour effects

• Priority classes

• Balking, Reneging, Jockeying

• Batching

• Multi-step processes

• Queue capacity

Parameters

mean arrival rate of customers (per unit time)

• c servers• µ mean service rate (per unit time)

Some Relations

= /(cµ) utilization• c = /µ average # of busy servers

• W = Wq + 1/µ

• L = Lq + c

• Little’s Law: Lq = Wq and L = W

So What is Lq?

• Depends on details.

0

10

20

30

40

50

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Lq

M/M/1 queue(exponential arrival times, exponential processing times, 1 server)

Qualitatively

• Dependence on 1 means Lq

• Increased variability (arrival / service times)– Lq increases

• Pooling queues– Lq decreases

Queue Notation

M / M / 1

M = ‘Markov’ exponential distributionD = ‘Deterministic’ constantG = ‘General’ other

distribution of the time between arrivals

distribution of the processing time

number of servers: 1, 2, …

M/M/1D/M/1M/G/3…

Back to Lq…

• Lq=E[Nq]

• M/G/1 2 = variance of the service time

– M/D/1– M/M/1

€

E[Nq ] =ρ 2 + λ2σ 2

2(1− ρ)

M/M/1

• N+1 has distribution Geometric(1-)– Var[Nq] = (1+-2) 2/(1-)2

– StDev[Nq] > E[Nq] /

• Pooling queues decreases Lq

Ex. (p501) 3 clinics with 1 nurse each (M/M/1)=4/hr, µ=1/13min

=87%, Lq=5.6, Wq=1.4hrConsolidated (M/M/3)=12/hr, µ=60/13hr, =87%

Lq=4.9, Wq=0.4