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Operations Management II

Waiting Lines

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Fundamental Trade-off in Waiting line

The Fundamental Trade-off in QueuingTheory/Waiting Line is that of Waiting Cost andService Cost.

If we add more servers (Cashiers, tellers,

equipment,) the waiting time andconsequently the waiting cost goes down but theservice cost goes up.

Conversely, if we reduce the service cost, the

waiting time and consequently the waiting costgoes up. (We dont want to lose our customers,do we?!That would cost us quite a considerablecost.)

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Elements of Waiting Line

CustomerArrivals

Servers

Waiting Line

Servicing System

Exit

Queue or

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Elements of Waiting Line

Arrivals ServiceWaitingline

Exit

Processingorder

System

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Major Issues in a Waiting Line

Queue Discipline

Length

Number of Lines&Line Structures

Service TimeDistribution

Queuing

System

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Waiting Time, Waiting Length

If l is the inter arrival rate, it means that l people( items in general e.g. cars, customers,

units,etc), join the line/queue every hour.This

implies that on average 1 personarrives

in, every 1/lhour. L , the length of the line is not the physical

length of the line/queue.It is the Number ofitems (people, objects,etc) in the line.

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Littles Law

1/l 1/l 1/l 1/l ServiceL m

(Average)WaitingTime= L(1/l)

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1) What is the average number of customers inline?

2)What is the average waiting time in the queue?

)-(=

2

ll

qL

-= l)(m

lW

Waiting Line and Waiting Time

m

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Waiting Time and SystemUtilization

By common sense, If we reduce thenumber of the servers in the system, weexpect the waiting time for the customers

to go up.

Reducing the number of the servers in thesystem higher occupation of the servers

lower service ratesIncrease inutilization.

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Waiting Time vs. Utilization

System Utilization

Av

eragenumbe

ron

tim

ewaitinginline

0 100%

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Avg. utilization rate/Traffic Intensity=

=

Avg # of customer served=

Arrival rate=

What is the average utilization of the server?

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Waiting Lines

Multiple channel

Multiple phaseChannel: A server in

a service system

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Customer Behaviors Models inWaiting Line

Patient

Customers enter the waiting line and remain untilserved

Reneging Waiting customers grow impatient and leave the

line

Jockeying

Customers may switch to another line

Balking

Upon arriving, decide the line is too long and

decide not to enter the line

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Length and Number of Lines

Waiting Lines

Length Number of Lines

InfiniteFinite

(Limited Capacity)

Single Multiple

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System Performance

1. Average number of customers waiting

2. Average time customers wait

3. System utilization

4. Implied cost

5. Probability that an arrival will have to

wait

Measured by:

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Priority Model (Emergency Room)

Arrivals ServiceWaitingline

Exit

Processingorder

System

11231

Arrivals are assigneda priority as they arrive

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Customer ServicePopulation Sources

Population Source

Finite Infinite

Example:

# of machinesneeding repair whena company only hasthree machines.

Example:

The number ofpeople who couldwait in a line forgasoline.

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Finite-Source Queuing

Not waiting orbeing served

WaitingBeingserved

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Service Pattern

ServicePattern

Constant Variable

Example:

Items coming downan automatedassembly line.

Example:

People spendingtime shopping.

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Line Structures

Single Channel

Multi-channel

SinglePhase

Multi-phase

One-personbarber shop

Car wash

Hospital

admissions

Bank tellers

windows