1

Queuing Systems (2)

Queueing Models (Henry C. Co) 2

Queuing Analysis

Cost ofservicecapacity

Cost ofcustomerswaiting

Co

st

Service capacity

Totalcost

Customerwaiting cost

Capacitycost= +

Queueing Models (Henry C. Co) 3

The Basic Model

Queueing Models (Henry C. Co) 4

A Basic Queue

Server

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

Queueing Models (Henry C. Co) 5

A Basic Queue

CustomerArrivals Server

Queueing Models (Henry C. Co) 6

A Basic Queue

Server

Queueing Models (Henry C. Co) 7

A Basic Queue

CustomerDepartures

Server

Queueing Models (Henry C. Co) 8

A Basic Queue

Queue(waiting line)Customer

Arrivals

CustomerDepartures

Server

Queueing Models (Henry C. Co) 9

A Basic Queue

Queue(waiting line)Customer

Arrivals

CustomerDepartures

Server

Line too long?Customer balks

(never enters queue)

Queueing Models (Henry C. Co) 10

A Basic Queue

Queue(waiting line)Customer

Arrivals

CustomerDepartures

Line too long?Customer reneges(abandons queue)

Server

Line too long?Customer balks

(never enters queue)

Queueing Models (Henry C. Co) 11

Queuing Analysis

Single Channel (or Single Server) Queue

Queueing Models (Henry C. Co) 12

Queuing Analysis

Service

Rate (

Queueing Models (Henry C. Co) 13

Queuing Analysis

ArrivalRate (

Service

Rate (

Queueing Models (Henry C. Co) 14

Queuing Analysis

ArrivalRate (

Average Waiting

Time in Queue (Wq )Service

Rate (

Queueing Models (Henry C. Co) 15

Queuing Analysis

Arrival

Rate (Average Number of

People in Queue (Lq )

Average Waiting

Time in Queue (Wq )Service

Rate (

Queueing Models (Henry C. Co) 16

Queuing Analysis

Arrival

Rate (Average Number of

People in Queue (Lq )

Average Time in System (W )

Average Number in System (L )

Average Waiting

Time in Queue (Wq )Service

Rate (

Queueing Models (Henry C. Co) 17

Characteristics of a Queue

Queueing Models (Henry C. Co) 18

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

Queueing Models (Henry C. Co) 19

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).

Queueing Models (Henry C. Co) 20

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).

Queueing Models (Henry C. Co) 21

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.

Queueing Models (Henry C. Co) 22

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

capacity? Number of lines; configuration of the

lines; jockeying.

Queueing Models (Henry C. Co) 23

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).

Queueing Models (Henry C. Co) 24

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

Queueing Models (Henry C. Co) 25

Exit Return to source population

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

Appendectomy-only-once case.

Queueing Models (Henry C. Co) 26

Steady State

Queueing Models (Henry C. Co) 27

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.

Queueing Models (Henry C. Co) 28

Waiting Time vs Utilization

System Utilization

Ave

rag

e n

um

ber

on

tim

e w

aiti

ng

in

lin

e

0 100%

Queueing Models (Henry C. Co) 29

M/M/1 Queues

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

Queueing Models (Henry C. Co) 30

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

Queueing Models (Henry C. Co) 31

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

Queueing Models (Henry C. Co) 32

00.05

0.10.15

0.20.25

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

Poisson Distribution

Queueing Models (Henry C. Co) 33

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.

Queueing Models (Henry C. Co) 34

Operating Characteristics

Utilization (fraction of time server is busy)

Queueing Models (Henry C. Co) 35

Operating Characteristics

Utilization (fraction of time server is busy)

Expected (Average) waiting times

W 1

W Wq

Queueing Models (Henry C. Co) 36

Operating Characteristics

Utilization (fraction of time server is busy)

Average waiting times

Average numbers

W 1

W Wq

L LLq

Queueing Models (Henry C. Co) 37

Fundamental Relationship

Little’s Law: L=W or Lq= Wq

Queueing Models (Henry C. Co) 38

Example

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

Queueing Models (Henry C. Co) 39

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.

Queueing Models (Henry C. Co) 40

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.

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

Queueing Models (Henry C. Co) 41

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.

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

= 6/10 = 0.6 or 60%

Queueing Models (Henry C. Co) 42

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.

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

= 6/10 = 0.6 or 60%

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

Queueing Models (Henry C. Co) 43

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.

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

= 6/10 = 0.6 or 60%

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

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

Queueing Models (Henry C. Co) 44

Example (cont.)

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

Queueing Models (Henry C. Co) 45

Example (cont.)

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

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

Queueing Models (Henry C. Co) 46

Example (cont.)

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

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

= 9/10 = 0.9 or 90%

Queueing Models (Henry C. Co) 47

Example (cont.)

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

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

= 9/10 = 0.9 or 90%

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

Queueing Models (Henry C. Co) 48

Example (cont.)

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

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

= 9/10 = 0.9 or 90%

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

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

Queueing Models (Henry C. Co) 49

Resource Utilization

service rate =

Queueing Models (Henry C. Co) 50

Resource Utilization

service rate ==

Lq =

Queueing Models (Henry C. Co) 51

Resource Utilization

Arrival Rate = 10.0= 1.0)

= 0.0= 0.0)

service rate ==

Lq =

Lq

Queueing Models (Henry C. Co) 52

Resource Utilization

Arrival Rate = 10.0= 1.0)

= 0.0= 0.0)

service rate ==

Lq =

Lq

Queueing Models (Henry C. Co) 53

Resource Utilization

Arrival Rate = 10.0= 1.0)

= 0.0= 0.0)

service rate ==

Lq =

Lq

Queueing Models (Henry C. Co) 54

Flexibility/Utilization Trade-off

Utilization = 1.0= 0.0

Queueing Models (Henry C. Co) 55

Flexibility/Utilization Trade-off

Utilization = 1.0= 0.0

L Lq

WWq

Queueing Models (Henry C. Co) 56

Flexibility/Utilization Trade-off

Utilization = 1.0= 0.0

L Lq

WWq

Queueing Models (Henry C. Co) 57

Flexibility/Utilization Trade-off

Utilization = 1.0= 0.0

L Lq

WWq

High utilizationLow flexibilityPoor service

Low utilizationHigh flexibilityGood service

Queueing Models (Henry C. Co) 58

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

Queueing Models (Henry C. Co) 59

Cost Trade-offs

Utilization = 1.0= 0.0

Cost

Queueing Models (Henry C. Co) 60

Cost Trade-offs

Utilization = 1.0= 0.0

Cost

Cost ofWaiting

Queueing Models (Henry C. Co) 61

Cost Trade-offs

Utilization = 1.0= 0.0

Cost

Cost ofWaiting

Cost ofService

Queueing Models (Henry C. Co) 62

Cost Trade-offs

Utilization = 1.0= 0.0

Cost

CombinedCombinedCostsCosts

Cost ofWaiting

Cost ofService

Queueing Models (Henry C. Co) 63

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 !

Queueing Models (Henry C. Co) 64

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.

Questions

Queueing Models (Henry C. Co) 65