© 2007 Pearson Education Waiting Lines Supplement C

© 2007 Pearson Education

Waiting Lines

Supplement C


Waiting Lines

• Waiting line: One or more “customers” waiting for service.

• Customer population: An input that generates potential customers.

• Service facility: A person (or crew), a machine (or group of machines), or both, necessary to perform the service for the customer.

• Priority rule: A rule that selects the next customer to be served by the service facility.

• Service system: The number of lines and the arrangement of the facilities.


Waiting Line ModelsBasic Elements

Service systemService system

Customer Customer populationpopulation

Waiting line

Priority rule

Service facilities

Served Served customerscustomers

© 2007 Pearson Education© 2007 Pearson Education

Waiting Line Arrangements

Single lineSingle line

Service facilitiesService facilities

Multiple linesMultiple lines

Service facilitiesService facilities


Service Facility Arrangements

Channel: One or more facilities required to perform a given service.

Phase: A single step in providing a service.

Priority rule: The policy that determines which customer to serve next.



Single channel, single phaseSingle channel, single phase

Service facility


Single channel, multiple phaseSingle channel, multiple phase


Service facility 1

Service facility 2


Multiple channel, single phaseMultiple channel, single phase


Service facility 1

Service facility 2


Multiple channel, multiple phaseMultiple channel, multiple phase


Service facility 3

Service facility 4

Service facility 1

Service facility 2


Service facility 3

Service facility 4

Service facility 1

Service facility 2

Routing for : 1–2–4Routing for : 1–2–4Routing for : 2–4–3Routing for : 2–4–3Routing for : 3–2–1–4Routing for : 3–2–1–4


Mixed ArrangementMixed Arrangement


Priority Rule

The priority rule determines which customer to serve next.

Most service systems use the first-come, first-serve (FCFS) rule. Other priority rules include:

Earliest promised due date (EDD)

Customer with the shortest expected processing time (SPT)

Preemptive discipline: A rule that allows a customer of higher priority to interrupt the service or another customer.


Probability DistributionsArrival Times

Example C.1Example C.1 Arrival rate = Arrival rate = 22/hour/hour

Customer ArrivalsCustomer Arrivals are usually random and can be are usually random and can be described by a Poisson distribution.described by a Poisson distribution.

Probability that Probability that nn customers customers will arrive…will arrive…

PPnn = e = e--TT((TT))nn

nn!!

PP44 = e = e-2(1)-2(1)[[22(1)](1)]44

44!! PP44 = e = e-2-2 = 0.090 = 0.09016162424

Interarrival times: The time between customer arrivals.

Probability that Probability that 44 customers customers will arrive…will arrive…

Mean = T Variance = T


The exponential distribution describes the probability that the service time will be no more than T time periods.

Probability DistributionsService time

If the customer service rate is three per hour, what is the probability If the customer service rate is three per hour, what is the probability that a customer requires less than 10 minutes of service?that a customer requires less than 10 minutes of service?

PP((t ≤Tt ≤T)) = = 1 – 1 – ee--TTμ = average number of customers completing service per periodt = service time of the customerT = target service time

PP((t ≤ t ≤ 0.167 hr)0.167 hr) = = 1 – 1 – ee-3(0.167) -3(0.167) = 1 – 0.61 = 0.39= 1 – 0.61 = 0.39

Example C.2 Example C.2

Mean = 1/ Variance = (1/ )2


Operating Characteristics

Line Length: Number of customers in line. Number of Customers in System: Includes

customers in line and being serviced. Waiting Time in Line: Waiting for service to begin. Total Time in System: Elapsed time between

entering the line and exiting the system. Service Facility Utilization: Reflects the percentage

of time servers are busy.


Single-ServerModel

The simplest waiting line model involves a single server and a single line of customers.

Assumptions: The customer population is infinite and patient. The customers arrive according to a Poisson distribution,

with a mean arrival rate of The service distribution is exponential with a mean

service rate of The mean service rate exceeds the mean arrival rate.

Customers are served on a first-come, first-served basis.

The length of the waiting line is unlimited.


= Average utilization of the system =

L = Average number of customers in the service system = –

Lq = Average number of customers in the waiting line = L

W = Average time spent in the system, including service =1

–

Wq = Average waiting time in line = W

n = Probability that n customers are in the system = (1 – )n

Single-ServerModel


Single-Channel, Single-Channel, Single-Phase SystemSingle-Phase System

Arrival rate (= 30/hour, Service rate ( = 35/hour

Average time in line = Average time in line = WWqq = = 0.8570.857(0.20) = 0.17 hour, or 10.28 minutes(0.20) = 0.17 hour, or 10.28 minutes

Average time in system = Average time in system = WW = = 0.20 hour, or 12 minutes = = 0.20 hour, or 12 minutes11

3535 – – 3030

Average number in line = Average number in line = LLqq = = 0.8570.857(6) = 5.14 customers(6) = 5.14 customers

Average number in system = Average number in system = LL = = 6 customers = = 6 customers3030

3535 – – 3030

Utilization = Utilization = = = = = = = 0.8570.857, or 85.7%, or 85.7%

30303535

Example C.3


Single-Channel, Single-Channel, Single-Phase SystemSingle-Phase System

Arrival rate (= 30/hour Service rate ( = 35/hour/hour


Application C.1

8.025

20

42025

20

L

2.348.0 LLq

2.02025

11

W

16.02.08.0 WWq


Example C.4


Example C.4


Example C.4


Application C.2

17.01

W

117.0

88.25

17.0

2017.01

20


Multiple-ChannelMultiple-Channel,, Single-Phase SystemSingle-Phase System

With the multiple-server model, customers form a single line and choose one of s servers when one is available.The service system has only one phase.

There are s identical servers.

The service distribution for each is exponential.

Mean service time is 1/The service rate (s exceeds the arrival rate ().


American Parcel Service is concerned about the amount of time the American Parcel Service is concerned about the amount of time the company’s trucks are idle, waiting to be unloaded. company’s trucks are idle, waiting to be unloaded.

The terminal operates with four unloading bays. Each bay requires The terminal operates with four unloading bays. Each bay requires a crew of two employees, and each crew costs $30/hr.a crew of two employees, and each crew costs $30/hr.

The estimated cost of an idle truck is $50/hr. Trucks arrive at an The estimated cost of an idle truck is $50/hr. Trucks arrive at an average rate of three per hour, according to a Poisson distribution.average rate of three per hour, according to a Poisson distribution.

Unloading a truck averages one hour with exponential service Unloading a truck averages one hour with exponential service times.times.

4 Unloading bays4 Unloading bays Crew costs $30/hourCrew costs $30/hour2 Employees/crew2 Employees/crew Idle truck costs $50/hourIdle truck costs $50/hourArrival rate = 3/hourArrival rate = 3/hour Service time = 1 hourService time = 1 hour

Multiple-ServerMultiple-Server Model ModelExample C.5Example C.5


Multiple-Server Model4 Unloading bays Crew costs $30/hour2 Employees/crew Idle truck costs $50/hourArrival rate = 3/hour Service time = 1 hour

Utilization = =3

1(4)= 0.75

0 = [∑ + ( )]-1(3/1)n

n!(3/1)4

4!1

1 – 0.75= 0.0377

Average trucks in line = Lq =0()ss!(1 – )2

0.0377(3/1)4(0.75)4!(1 – 0.75)2= = 1.53 trucks

Average time in line = Wq =Lq

1.53

3= 0.51 hours=

Average time in system = W = Wq + 1

= 0.51 + 11

= 1.51 hours

Average trucks in system = L = W = 3(1.51) = 4.53 trucks


Multiple-Server ModelMultiple-Server Model

Labor costs: $30(s) = $30(4) = $120.00Idle truck cost: $50(L) = $50(4.53) = 226.50

Total hourly cost = $346.50

4 Unloading bays Crew costs $30/hour2 Employees/crew Idle truck costs $50/hourArrival rate = 3/hour Service time = 1 hour


Application C.3


Application C.3

8.05.122

20

s

11.0

8.01

1

2

5.12

20

5.12

201

1

1

11

120

s

Ps

408.18.01!2

8.05.12

2011.0

1! 2

2

20

s

PL

s

q

0704.020

408.1

q

q

LW hrs. (or 4.224 minutes)


Little’s Law

Little’s Law relates the number of customers in a waiting-line system to the waiting time of customers.

L = WL is the average number of customers in the

system.

is the customer arrival rate.

W is the average time spent in system, including service.


In the finite-source model, the single-server model assumptions are changed so that the customer population is finite, with N potential customers.

If N is greater than 30 customers, then the single-server model with an infinite customer population is adequate.

Finite-Source ModelFinite-Source Model


Finite-Source Model

= Average utilization of the server = 1 – 0

Lq = Average number of customers in line = N – (1 – 0) +

L = Average number of customers in the system = N – (1 – 0)

Wq = Average waiting time in line = Lq [(N – L)]–1

W = Average time in the system = L[(N – L)]–1

0 = probability of zero customers [∑ ( )n

]–1N!

(N – n)!

N

n=0


Number of robots = 10 Loss/machine hour = $30Service time = 10 hrs Maintenance cost = $10/hrTime between failures = 200 hrs

Example C.6


Finite-Source ModelFinite-Source ModelExample C.6Example C.6 Solution Solution

= 1 – 0.538 = 0.462= 1 – 0.538 = 0.462

LLqq = 0.30 robots = 0.30 robots LL = = 0.76 robot0.76 robot

WW = 16.43 hours = 16.43 hours

Number of robots = 10 Loss/machine hour = $30Service time = 10 hrs Maintenance cost = $10/hrTime between failures = 200 hrs

00 = 0.538 = 0.538 WWqq = 6.43 hours = 6.43 hours

Labor cost: ($10/hr)(8 hrs/day)(0.462 utilization) = $ 36.96Idle robot cost: (0.76 robot)($30/robot hr)(8 hrs/day) = 182.40

Total daily cost = $219.36


Decision Areas for Management

Using waiting-line analysis, management can improve the service system in one or more of the following areas.

Arrival RatesNumber of Service FacilitiesNumber of PhasesNumber of Servers Per FacilityServer EfficiencyPriority RuleLine Arrangement


Application C.4

44.0

08.00

!808.0

!7

!808.0

!8

!8

1

!

!

810

1

00

N

n

n

nN

NP

56.044.011 0 P

144.0102.0

25.081 0 PNL


Application C.5

667.030

20

111.0667.0333.01

222.0667.0333.01

333.0667.0333.01

222

111

000

P

P

P


Application C.5

333.12030

20667.0

LLq

min6min601.02030

11

hrhrW

Documents

© 2007 Pearson Education Waiting Lines Supplement C