4/11: queuing models collect homework, roll call queuing theory, situations single-channel waiting...

4/11: Queuing Models• Collect homework, roll call• Queuing Theory, Situations• Single-Channel Waiting Line System

– Distribution of arrivals– Distribution of service times– Queue discipline: FCFS– Steady-state operation– Operating characteristics

• Multiple-Channel Waiting Line System

Queuing Theory, Situations• Waiting line for a

roller coaster

• Waiting line at a restaurant

• Need to find an acceptable balance between few workers and few lost customers.

Image courtesy of ohiomathworks.org

Structure• Single-channel waiting line

ServerCustomers in line

Distribution of Arrivals• When customers arrive

• Assume random & independent arrivals.

• A Poisson probability distribution:

ServerCustomers in line

!)(

x

exP

x

x : # of arrivals in period : mean # of arrivals per periode : 2.71828

Distribution of Arrivals: Example• Assume that it has been calculated that 30

customers arrive per hour. Calculate the likelihood that no customers, 1 customer, and 2 customers will arrive in the next two minutes.

ServerCustomers in line

!)(

x

exP

x

Distribution of Arrivals: Example• = 30 cust./60 min. = 0.50 cust./ 1 min.

• No customers in next minute:

ServerCustomers in line

6065.0!0

)50.0()0( 50.0

50.00

ee

P

Distribution of Arrivals: Example• = 30 cust./60 min. = 0.50 cust./ 1 min.

• One customer in next minute:

ServerCustomers in line

3032.)50(.!1

)50(.)1( 50.0

50.01

ee

P

Distribution of Arrivals: Example• = 30 cust./60 min. = 0.50 cust./ 1 min.

• Two customers in next minute:

ServerCustomers in line

0759.2

)25(.

!2

)50(.)2(

50.050.02

ee

P

Distribution of Arrivals: Example• = 30 cust./ hour ?• Two customers in next TWO minutes:

ServerCustomers in line

?)2( P

= 1 customer / 2 min.

1840.2

)1(

!2

)1()2(

112

ee

P

Distribution of Service Times• Service times follow an exponential

probability distribution.

• = average # of units that can be served per time periode = 2.71828

tettimeserviceP 1)(

Prob. of Service Time: Example• Our server can take care of, on average, 45

customers per hour.

• What is the probability of an order taking less than one minute?

tetimeservP

1.)min1.(

.min/.cu75.0.min60

.cust45

Prob. of Service Time: Example• Our server : 45 customers per hour.

• What is the probability of an order taking less than one minute?

5276.04724.011

.)min1.()1(75.0

e

timeservP

Prob. of Service Time: Example• Our server : 45 customers per hour.

• What is the probability of an order taking less than two minutes?

7769.02231.011

.)min2.()2(75.0

e

timeservP

Queue Discipline• We must define how the units are arranged

for service.

• We will assume that our lines are FCFS, or First-Come, First-Served.

• Balking, reneging, jockeying

Steady-State Operation• There is a transient period when a line

starts, when things cannot be predicted.

• The system must be stable, that is, able to keep up with the customers.

• Our models apply only to the normal operation, or the steady-state operation.

Operating Characteristics• Probability of no units in system

• Average number of units in waiting line

• Average number of units in system

• Average time unit is in waiting line

• Average time a unit is in the system

• Probability that an arriving unit will have to wait for service

• Probability of n units in system

Probability of no units in system• = mean arrival rate (mean # arriv./time)

• = mean service rate (mean # serv./time)

• The system includes the customer being serviced.

10P

Average # of units in waiting line• = mean arrival rate (mean # arriv./time)

• = mean service rate (mean # serv./time)

• The waiting line does NOT include the customer being serviced.

)(

2

qL

Average # of units in system• = mean arrival rate (mean # arriv./time)

• = mean service rate (mean # serv./time)

qLL

Avg. time a unit is in waiting line• = mean arrival rate (mean # arriv./time)

• = mean service rate (mean # serv./time)

q

q

LW

Average time a unit is in system• = mean arrival rate (mean # arriv./time)

• = mean service rate (mean # serv./time)

1

qWW

Probability that an arriving unit will have to wait for service

• = mean arrival rate (mean # arriv./time)

• = mean service rate (mean # serv./time)

wP

Probability of n units in system• = mean arrival rate (mean # arriv./time)

• = mean service rate (mean # serv./time)

0PPn

n

Group Exercise• A bank has a drive-up teller window.

Arrivals are Poisson-distributed, with a mean rate of 24 customers / hour. Service times are exponential-probability distributed, with a mean rate of 36 customers / hour.

• Calculate the operating characteristics of the system.

Multiple-Channel Waiting Lines

ServerCustomers in line

Server

Server

Multiple-Channel Waiting Lines• Assumptions:

• Arrivals follow Poisson distribution.

• Service times follow exponential prob. dist.

• The mean service rate is the same for each channel (server).

• Arrivals wait in single line, then move to first open channel.

Multiple-Channel Waiting Lines• = mean arrival rate

• = mean service rate for each channel

• k = number of channels

Operating Characteristics• Probability of no units in system

• Average number of units in waiting line

• Average number of units in system

• Average time unit is in waiting line

• Average time a unit is in the system

• Probability that an arriving unit will have to wait for service

• Probability of n units in system

Probability of no units in system• = mean arrival rate (mean # arriv./time)

• = mean service rate (mean # serv./time)

• k = # of channels

• The system includes the customer being serviced.

1

0

0

!)/(

!)/(

1k

n

kn

kk

kn

P

Average # of units in waiting line• = mean arrival rate (mean # arriv./time)

• = mean service rate (mean # serv./time)

• k = # of channels

• The waiting line does NOT include the customer being serviced.

02)()!1(

)/(P

kkL

k

q

Average # of units in system• = mean arrival rate (mean # arriv./time)

• = mean service rate (mean # serv./time)

qLL

Avg. time a unit is in waiting line• = mean arrival rate (mean # arriv./time)

• = mean service rate (mean # serv./time)

q

q

LW

Average time a unit is in system• = mean arrival rate (mean # arriv./time)

• = mean service rate (mean # serv./time)

1

qWW

Probability that an arriving unit will have to wait for service

• = mean arrival rate (mean # arriv./time)

• = mean service rate (mean # serv./time)

• k = # of channels

0!

1P

k

k

kP

k

w

Probability of n units in system• = mean arrival rate, = mean service

rate, k = # of channels

knforPkk

P

knforPn

P

kn

n

n

n

n

0)(

0

!

)/(

!

)/(

Homework due 4/18• Ch. 12 #5 (answer all questions)

• Ch. 12 #6 (answer all questions)

• Ch. 12 #7 (answer all questions)

• Ch. 12 #12 (answer all questions)

• Ch. 12 #13 (answer all questions)

• Turn in ON PAPER (you can do it by hand or using Excel).