Queueing Theory

Dr. Ron Lembke

Operations Management

Queues

In England, they don’t ‘wait in line,’ they ‘wait on queue.’

So the study of lines is called queueing theory.

Cost-Effectiveness

How much money do we lose from people waiting in line for the copy machine? Would that justify a new machine?

How much money do we lose from bailing out (balking)?

We are the problem Customers arrive randomly. Time between arrivals is called the “interarrival

time” Interarrival times often have the “memoryless

property”: On average, interarrival time is 60 sec. the last person came in 30 sec. ago, expected time

until next person: 60 sec. 5 minutes since last person: still 60 sec.

Variability in flow means excess capacity is needed

Memoryless Property

Interarrival time = time between arrivals Memoryless property means it doesn’t matter how long

you’ve been waiting. If average wait is 5 min, and you’ve been there 10 min,

expected time until bus comes = 5 min Exponential Distribution Probability time is t =

tetf )(

Poisson Distribution

Assumes interarrival times are exponential

Tells the probability of a given number of arrivals during some time period T.

Ce n'est pas les petits poissons.Les poissons Les poissons How I love les poissons Love to chop And to serve little fish First I cut off their heads Then I pull out the bones Ah mais oui Ca c'est toujours delish Les poissons Les poissons Hee hee hee Hah hah hah With the cleaver I hack them in two I pull out what's inside And I serve it up fried God, I love little fishes Don't you?

Simeon Denis Poisson "Researches on the probability

of criminal and civil verdicts" 1837 

looked at the form of the binomial distribution when the number of trials was large. 

He derived the cumulative Poisson distribution as the limiting case of the binomial when the chance of success tend to zero.

Binomial Distribution

The binomial distribution tells us the probability of having x successes in n trials, where p is the probability of success in any given

attempt.

xnx ppx

npnxb

1),,(

Binomial Distribution The probability of getting 8 tails in 10 coin

flips is:

b(8,10,0.5)10

8

(0.5)8 1 0.5 10 8

10*92*1

*0.0039062*0.254.4%

Poisson Distribution

x

k

k

x

k

eCUMPOISSON

x

ePOISSON

0 !

!

POISSON(x,mean,cumulative)

X   is the number of events. Mean   is the expected numeric value. Cumulative   is a logical value that determines

the form of the probability distribution returned. If cumulative is TRUE, POISSON returns the cumulative Poisson probability that the number of random events occurring will be between zero and x inclusive; if FALSE, it returns the Poisson probability mass function that the number of events occurring will be exactly x.

Larger average, more normal

Queueing Theory Equations

Memoryless Assumptions: Exponential arrival rate = = 10

• Avg. interarrival time = 1/ • = 1/10 hr or 60* 1/10 = 6 min

Exponential service rate = = 12• Avg service time = 1/ = 1/12

Utilization = = /• 10/12 = 5/6 = 0.833

Avg. # in System

Lq = avg # in line =

Ls = avg # in system =

Prob. n in system = Because We can also write it as

2

qL

qs LL

Pn 1

n

nnP 1

Example

Customers arrive at your service desk at a rate of 20 per hour, you can help 35 per hr. What % of the time are you busy? How many people are in the line on average? How many people are there, in total on avg? What are the odds you have 3 or more

people there?

Queueing Example

λ=20, μ=35 so ρ=20/35 = 0.571 Lq = avg # in line =

Ls = avg # in system = Lq + ρ = 0.762 + 0.571 = 1.332

762.0525

400

203535

2022

qL

Prob. Given # in System

Prob. n people in system, ρ = 0.571

Prob 0-3 people = 0.429 + 0.245 + 0.140 + 0.080 = 0.894

Prob 4 or more = 1-0.894 = 0.106

n

nP

1 nnP 1

n (1-ρ)*ρ n Value

0 0.429 * 0.5710 =0.429 * 1 0.429

1 0.429 * 0.5711 = 0.429*0.571 0.245

2 0.429 * 0.5712 = 0.429*0.326 0.140

3 0.429 * 0.5713 = 0.429*0.186 0.080

4 0.429 * 0.5714 = 0.429*0.106 0.045

Probability of n in systemn Pn Pr(<=n) Pr(>n)0 0.429 0.429 0.571 1 0.245 0.674 0.326 2 0.140 0.814 0.186 3 0.080 0.894 0.106 4 0.046 0.939 0.061 5 0.026 0.965 0.035 6 0.015 0.980 0.020 7 0.008 0.989 0.011 8 0.005 0.994 0.006 9 0.003 0.996 0.004

10 0.002 0.998 0.002 11 0.001 0.999 0.001 12 0.001 0.999 0.001

0 1 2 3 4 5 6 7 8 9 10 11 12 -

0.20

0.40

0.60

0.80

1.00

1.20

Pn Pr(<=n) Pr(>n)

Average Time

Wq = avg wait in line

Ws = avg time in systemq

q

LW

s

s

LW

How Long is the Wait?

Time waiting for service =• Lq = 0.762, λ=20

• Wq = 0.762 / 20 = 0.0381 hours

• Wq = 0.0381 * 60 = 2.29 min

Total time in system =• Ls = 1.332, λ=20

• Ws = 1.332 / 20 = 0.0666 hours

• Ws = 0.0666 * 60 = 3.996 = 4 min

• μ=35, service time = 1/35 hrs = 1.714 min• Ws = 2.29 + 1.71 = 4.0 min

q

q

LW

s

s

LW

What did we learn? Memoryless property means exponential

distribution, Poisson arrivals Results hold for simple systems: one line,

one server Average length of time in line, and system Average number of people in line and in

system Probability of n people in the system