09 Queueing Theory

Queueing Models (Henry C. Co) 1

Queueing Models

Henry C. CoTechnology and Operations Management, California Polytechnic and State University


Queueing Models Theory

Danish engineer A. K. Erlang (early 1900’s) Studied telephone switchboards in Copenhagen

for the Danish Telephone Company. Objective

Balance cost of waiting (and/or balking) with cost of adding resources (capacity).

Priority selection process What priority rule or procedure should be used to

select the next customer to be served or job to be worked on?


Queuing Analysis

Cost ofservicecapacity

Cost ofcustomerswaiting

Co

st

Service capacity

Totalcost

Customerwaiting cost

Capacitycost= +


The Basic Model


A Basic Queue

Server

Stephen R. LawrenceLeeds School of BusinessUniversity of ColoradoBoulder, CO 80309-0419


A Basic Queue

CustomerArrivals Server


A Basic Queue

Server


A Basic Queue

CustomerDepartures

Server


A Basic Queue

Queue(waiting line)Customer

Arrivals

CustomerDepartures

Server


A Basic Queue


Arrivals

CustomerDepartures

Server

Line too long?Customer balks

(never enters queue)


A Basic Queue


Arrivals

CustomerDepartures

Line too long?Customer reneges(abandons queue)

Server

Line too long?Customer balks

(never enters queue)


Queuing Analysis

Single Channel (or Single Server) Queue


Queuing Analysis

Service

Rate ( m )


Queuing Analysis

ArrivalRate ( l )

Service

Rate ( m )


Queuing Analysis

ArrivalRate ( l )

Average Waiting

Time in Queue (Wq )Service

Rate ( m )


Queuing Analysis

Arrival

Rate ( l )Average Number of

People in Queue (Lq )

Average Waiting


Rate ( m )


Queuing Analysis

Arrival

Rate ( l )Average Number of

People in Queue (Lq )

Average Time in System (W )

Average Number in System (L )

Average Waiting


Rate ( m )


Characteristics of a Queue


Source population Arrival characteristics Physical features of lines Selection from the waiting line Service facility Exit

Elements of Queuing System

Arrivals ServiceWaitingline

Exit

Processingorder

System


Source Population May be finite or infinite. For practical intent and purposes,

when the population is large in comparison to the service system, we assume the source population to be infinite (e.g., in a small barber shop, 200 potential customers per day may be treated as an infinite population).


Arrival Pattern of arrivals

Controllable arrival pattern Movie theatres offering Monday specials. Department stores running sales. Airlines offering off-season rates. Overseas telecom rates from 1:00 a.m. To 7:00

a.m. Uncontrollable arrival pattern

Emergency operations. Fire department.

Size of arrivals: single or batch arrival? Probability distribution pattern of arrivals.

Periodic: constant time-between-arrivals (TBA). Purely random TBA (e.g., exponential distribution).


Degree of patients A patient arrival is one who waits as long

as necessary until the service facility is ready to serve him/her (even if the customer grumble and behave discourteously or Impatiently).

Impatient arrivals. Balking: views the situation (length of queue)

and then decides to leave. Reneging: views the situation, joins the

queue, after some time, departs without being served.


Physical Features of Waiting Line Length of line: infinite or finite waiting

capacity? Number of lines; configuration of the

lines; jockeying.


Selection from the Waiting Line Queue discipline: priority rule(s) for

determining the order of service to customers in a waiting line FIFO. By reservations/appointment only/first. Emergencies first. Highest profit customer first. Largest orders first. Best customer first. Longest waiting time in line first. Soonest promised date first. Shortest processing time first.

Line structuring: express checkouts (supermarkets); “commercial transactions only” (banks).


Service Facility Structure

Single-channel single-phase. Single-channel multi-phase. Multi-channel single-phase. Multi-channel multi-phase. Mixed.

Service rate Constant Random (probability distribution).

Queuing SystemsMultiple channel

Multiple phase


Exit Return to source population

Recurring-common-cold case. Low probability of re-service

Appendectomy-only-once case.


Steady State


A stable system: The queue will never increase to infinity. An empty state is reached for sure after some time period.

Condition for Stability: >. This condition MUST be met to make all formulas valid.

The steady state: Probability {n customers in the system} does not depend on the time.


Waiting Time vs Utilization

System Utilization

Ave

rag

e n

um

ber

on

tim

e w

aiti

ng

in

lin

e

0 100%


M/M/1 Queues

1st M (for “Markovian) – Arrival Distribution is Exponential2nd M – Service Distribution is Exponential1 – Single Channel


Population Time horizon: an infinite horizon. Source Population: infinite.


Arrival Process The inter-arrival time is an

exponentially-distributed random variable with average arrival rate = .

If the inter-arrival time is an exponentially-distributed random variable, then the number of arrivals during the fixed period of time is a Poisson distribution.

No balking or reneging


00.05

0.10.15

0.20.25

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

Poisson Distribution


Service Process The service time is also assumed to be

exponentially distributed with mean service rate .

Only 1 server First-come-first-served (FCFS) queue

priority Mean length of service = 1/ No limit on the queue size.


Operating Characteristics

Utilization (fraction of time server is busy)




Expected (Average) waiting times

W 1

W Wq




Average waiting times

Average numbers

W 1

W Wq

L LLq


Fundamental Relationship

Little’s Law: L=W or Lq= Wq


Example

Stephen R. LawrenceLeed School of BusinessUniversity of ColoradoBoulder, CO 80309-0419


Example

Boulder Reservoir has one launching ramp for small boats.On summer weekends, boats arrive for launching at a mean rate of 6 boats per hour. It takes an average of s=6 minutes to launch a boat. Boats are launched FCFS.


Example


l = 6/hr m = 1/s =1/6 = 0.167/min or 10/hr


Example


l = 6/hr m = 1/s =1/6 = 0.167/min or 10/hr

= / r l m = 6/10 = 0.6 or 60%


Example


l = 6/hr m = 1/s =1/6 = 0.167/min or 10/hr

= / r l m = 6/10 = 0.6 or 60%

L = /( - ) l m l = 6/(10-6) = 1.5 boatsLq = Lr = 1.5(0.6) = 0.9 boats


Example


l = 6/hr m = 1/s =1/6 = 0.167/min or 10/hr

= / r l m = 6/10 = 0.6 or 60%

L = /( - ) l m l = 6/(10-6) = 1.5 boatsLq = Lr = 1.5(0.6) = 0.9 boats

W = 1/( - ) m l = 1/(10-6) = 0.25 hrs or 15 minsWq = Wr = 0.25(0.6) = 0.15 hrs or 9 mins


Example (cont.)

During the busy Fourth of July weekend, boats are expectedto arrive at an average rate of 9 per hour.


Example (cont.)


l = 9/hr m = 1/s =1/6 = 0.167/min or 10/hr


Example (cont.)


l = 9/hr m = 1/s =1/6 = 0.167/min or 10/hr

= / r l m = 9/10 = 0.9 or 90%


Example (cont.)


l = 9/hr m = 1/s =1/6 = 0.167/min or 10/hr

= / r l m = 9/10 = 0.9 or 90%

L = /( - ) l m l = 9/(10-9) = 9.0 boatsLq = Lr = 9(0.6) = 5.4 boats


Example (cont.)


l = 9/hr m = 1/s =1/6 = 0.167/min or 10/hr

= / r l m = 9/10 = 0.9 or 90%

L = /( - ) l m l = 9/(10-9) = 9 boatsLq = Lr = 9(0.6) = 5.4 boats

W = 1/( - ) m l = 1/(10-9) = 1.0 hrs or 60 minsWq = Wr = 1(0.9) = 0.9 hrs or 54 mins


Resource Utilization

service rate m = 10



service rate m = 10 r = / l m

Lq = / ( - )rl m l



Arrival Rate l l = 10.0( r = 1.0)

l = 0.0( r = 0.0)


Lq = / ( - )rl m l

Lq




l = 0.0( r = 0.0)


Lq = / ( - )rl m l

Lq




l = 0.0( r = 0.0)


Lq = / ( - )rl m l

Lq


Flexibility/Utilization Trade-off

Utilization r r = 1.0 r = 0.0




L Lq

WWq




L Lq

WWq




L Lq

WWq

High utilizationLow flexibilityPoor service

Low utilizationHigh flexibilityGood service


Queues and Flexibility Low utilization levels ( < 0.6 ) provide

better service levels greater flexibility lower waiting costs (e.g., lost business)

High utilization levels ( > 0.9 ) provide better equipment and employee utilization fewer idle periods lower production/service costs

Must trade off benefits of high utilization levels with benefits of flexibility and service


Cost Trade-offs


Cost


Cost Trade-offs


Cost

Cost ofWaiting


Cost Trade-offs


Cost

Cost ofWaiting

Cost ofService


Cost Trade-offs


Cost

CombinedCosts

Cost ofWaiting

Cost ofService


Queues and Simulation Only simple queues can be

mathematically analyzed “Real world” queues are often very

complex multiple servers, multiple queues balking, reneging, queue jumping machine breakdowns networks of queues, ...

Need to analyze, complex or not Computer simulation !


Adding an extra server Reduces the expected queue length and

waiting time greatly. Reduces the server’s utilization level

significantly. In some cases, a manager wants the

expected customer waiting time is below certain critical level. Otherwise, he may lose customers.

Documents

09 Queueing Theory