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