Waiting Lines Queues

1
Waiting LinesQueues
2
Queuing Theory

• Managers use queuing models to be more efficient
  in providing customer service.
• Models measure average waiting times and average
  length of waiting lines.

3
Historical Roots

• Agner Krarup Erlang, a Danish engineer who worked
  for the Copenhagen Telephone Exchange, published
  the first paper on queueing theory in 1909.
• David G. Kendall introduced an A/B/C queueing
  notation in 1953.

4
Three queuing disciplines used in Telephone
Networks

• First In First Out This principle states that
  customers are served one at a time and that the
  customer that has been waiting the longest is
  served first.5
• Last In First Out This principle also serves
  customers one at a time, however the customer
  with the shortest waiting time will be served
  first.5
• Processor Sharing Customers are served equally.
  Network capacity is shared between customers and
  they all effectively experience the same delay
• Source Wikipedia.org

5
FIFOFirst In First Out
6
LIFOLast in First Out
Elevators are a circumstance where this occurs.
7
SIROService In Random Order
8
Single-server Single-stage Queue
Service Facility
Customers In queue
Arrival Stream
Waiting for the Newest ????????
9
Multiple-server Single-stage Queue
10
Single-server Multiple-stage Queue
Pharmacy Conveyor System gtgtgtgtgt
11
Multiple-server Multiple-Stage Queue
Customers In queue
Service Facilities
12
Little's Theorem

• Little's theorem L ? / ?
• The average number of customers (N) can be
  determined from the following equation
• Here lambda (?) is the average customer arrival
  rate and mu (?) is the average service time for a
  customer.

13
Queuing system state probabilities(Basic Model)
?
P0

1 -
?
n
?
(
)
Probability distribution for the number of
customers in the system
Pn
P0

?
1, 2, 3,
n
14
Queuing Formulas(Basic Model)
?
Average of Customers in the system
L
? - ?
1
L
Average Customer Time spent in the system
W

?
? - ?
?2

Average of Customers waiting (length of line)
Lq
? (? - ?)
Lq
?

Average Customers waiting time
Wq
?
? (? - ?)
?
?
Server Utilization Factor
?
15
Phlebotomy Room Example

• A queuing system for blood draws.
• An average of 25 patients arrive for a blood draw
  each hour.
• One full-time (very experienced) phlebotomist can
  take one patient every two minutes, thus 30 draws
  per hour can be done.

16
Queuing Formulas(Basic Model)
? 25 Pts per hour
? 30 Pts per hour
?
25
Average of Patients in the system

5 customers
L
? - ?
30 - 25
1
1
1
Average Patient Time spent in the system
hour

W

? - ?

5

30 - 25
1
?2
(25)2
25
Average of Patients waiting
4


Lq


customers
6
30 (30 - 25)
? (? - ?)
6
Lq
?
25
1
Average Patients waiting time

hour
Wq


30 (30 - 25)
6
? (? - ?)
?
?
25
5
Server Utilization Factor
?


?
30
6
The phlebotomist is busy five-sixths of the
time.
17
The system state probabilities
25
?
P0

1 -

1 -
.1667

?
30
1
?
25
(
)
(
)
P1
P0


(.1667)

.1389
?
30
2
2
?
25
(
)
(
)
P2
P0


(.1667)

.1158
?
30
This formula provides the probability that n (0,
1, 2, 3, ) patients will be in the blood drawing
room. If you add the individual probabilities
for values of n cumulatively you would find 54 in
the number of patients where all probabilities of
n total 1.
18
Multiple server models

• Uses same notation as basic model but different
  formulas.
• Formulas are based on FIFO discipline.
• The customer at the head to waiting line proceeds
  to the first server.
• S Number of service channels

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Queuing system state probabilities(Multiple
Servers)
(? /?)s
1
(? /?)n
(
)
?
S-1
P0

1

S!
1 - ?/S?
n!
n0
(? /?)n
P0
If 0 lt n lt S
n!
Pn

(? /?)n
If n gt S
S!Sn-s
20
Phlebotomy Room Examplewith a second
PhlebotomistMultiple Servers

• A 2 service channel queuing system for blood
  draws.
• An average of 50 patients arrive for a blood draw
  each hour.
• Two full-time phlebotomists can take one patient
  each every two minutes, thus 60 draws per hour
  can be done.

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The Probability that there are no patients in the
system.
? 50 Pts per hour
? 60 Pts per hour
S 2 service channels

1

(? /?)0
(? /?)1
(
)
(? /?)2
P0

1

1 - ?/2?

0!
1!
2!

1

(50 /60)0
(50 /60)1
(50 /60)2
(
)

1

1 - 50/2(60)

0!
1!
2!


1
(.833)2
)
(

1
.833
1

1 - .416
2!
1 /1 .833 .594 1 / 2.427 .412
22
Queuing Formulas(Multiple Servers)
? 50 Pts per hour
? 60 Pts per hour
S 2 service channels
(?/?)2
(?/S?)
Average of Patients waiting

Lq
P0
S! (1 - ?/S?)2
Lq
Average Patients waiting time
Wq
?
1
Average Patient Time spent in the system
W
Wq
?
?
Average of Patients in the system
L
Lq
?
?
Server Utilization Factor
?
S?
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Queuing Formulas(Multiple Servers)
? 50 Pts per hour
? 60 Pts per hour
S 2 service channels
(50/2(60))
(50/60)2
Average of Patients waiting

Lq
(.412) .175
2! (1 - 50/2(60))2
Lq
.175
Average Patients waiting time
Wq
.0035 12.6 seconds

50
?
1
Average Patient Time spent in the system
W
Wq
.0035 1/60 .0035.016 .0195
?
.0195 hour 1.17 minutes
?
Average of Patients in the system
L
Lq
.175 50/60 .175 .833 1.008 pts
?
?
Server Utilization Factor
?
50/ 2(60) 0.416
S?
24
Two Fax machines example

• An organization is considering renting 2 fax
  machines.
• The 2006 model can send 100 faxes per minute.
• However, loading the originals and entering the
  receiving phone number slows the process. The
  vendor indicates the effective service rate is .5
  job per minute.
• The demand for fax service in the organization is
  projected at 3 jobs every 5 minutes (.6 job per
  minute)
• S 2 service channels
• ? .5 job per minute
• ? .6 job per minute

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The Probability that there are no patients in the
system.
? .5 job per minute
? .6 job per minute
S 2 service channels

1

(? /?)0
(? /?)1
(
)
(? /?)2
P0

1

1 - ?/2?

0!
1!
2!

1

(.6 /.5)0
(.6 /.5)1
(.6 /.5)2
(
)

1

1 - .6/2(.5)

0!
1!
2!


1
(1.2)2
)
(

1
1.2
1

1 - .6
2!
1 /1 1.2 1.8 1 / 4 . 25
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Queuing Formulas(Multiple Servers)
? .5 job per minute
? .6 job per minute
S 2 service channels
(?/?)2
(?/S?)
(.6/.5)2
.6/2(.5)
Average of Jobs waiting


Lq
P0
(. 25) .68 job
S! (1 - ?/S?)2
2! 1 - .6/2(.5)2
Lq
.68
Average job waiting time per job
Wq
1.13 minutes

.6
?
1
1
1
Average Job Time spent in the fax room
W
Wq
1.13
3.13 minutes
?
.5
.6
?
Average of jobs in the fax room
.68
1.88 jobs
L
Lq
.5
?
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Establishing a queuing system cost
Consider the average hourly cost of operating two
rented fax machines. Each job is personally
processed by the user. The average hourly
payroll cost is 10. Machine rental is a
straight .05 per copy, and an average job
involves 12 copies. The average number of jobs
per hour is .6 X 60 36 jobs Each employee
spends an average of W 3.13 minutes 3.13/60
.0522 hour Average cost of labor lost making
copies 10 X 36 X .0522 18.79 Hourly rental
cost .05 X 12 X 36 21.60 Total hourly
average cost of operating two machines 18.79
(labor cost) 21.60 (equipment rental) 40.39
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Two compared to one fax that is twice as fast.
One would would think that one server twice as
fast would produce identical results to two
servers. THIS IS NOT TRUE. If so we would not
need a different model for multi-channel
queues. If a 2008 model fax is twice as fast as
the 2006 model is there a difference? If ? 1
job per minute.
?2
(.6)2
.36


Lq
Average of jobs waiting


.9
Job (2008 model)
1(1 - .6)
? (? - ?)
.4
Lq
?
.9
Average Patients waiting time
1.5

Minutes (2008 model)
Wq


.6
? (? - ?)
?
1
1
Average job time spent in fax room

W
2.5
Minutes (2008 model)

? - ?


1 - .6
2.5 60
Results in a smaller hourly labor cost.
10 X 36 X 15.00
The new model might rent for a little more than
the older model, but would still be cheaper than
two 2006 models.
29
Single Server Model w/a finite queue
A waiting line of limited length is called a
finite queue. e.g., Hospital Emergency room with
a limited number of beds. If the number of
patients reaches a given point additional
patients are diverted. The patient (customer) who
does not enter the system does not return.
There are not limits on the number of patients
waiting for service
1 - ? /?
P0

Probabilities for of patients in the system
1 - (? /?)M 1
Pn

(? /?)n
P0
for 1 lt n lt M
? / ?
(M 1)(? / ?)M1
Average of patients in the system
L
1 - ?/?
1 - (? /?)M1
Average length of waiting line
Lq L (1 Po)
Lq
L
Average patient waiting times
Wq
W
?(1 PM)
?(1 PM)
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Summary

• What is queue? A waiting line.
• Queue disicplines
• FIFO
• LIFO
• SIRO
• Queuing models
• Single server Single stage
• Multiple server Single stage
• Single server Multiple stage
• Multiple server Multiple stage
• Single Server Model w/a finite queue