Service Operations
and
Waiting Lines
Dr. Everette S. Gardner, Jr.
Case study: Single-server model
Reference
Vogel, M. A., “Queuing Theory Applied to Machine Manning,”
Interfaces, Aug. 79.
Company
Becton - Dickinson, mfg. of hypodermic needles and syringes
Bottom line
Cash savings = $575K / yr.
Also increased production by 80%.
Problem
High-speed machines jammed frequently. Attendants cleared jams. How
many machines should each attendant monitor?
Model
Basic single-server:
Server—Attendant
Customer—Jammed machine
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Case study (cont.)
Solution procedure
Each machine jammed at rate of λ = 60/hr.
With M machines, arrival rate to each attendant is λ = 60M
Service rate is μ = 450/hr.
Utilization ratio = 60M/450
Experimenting with different values of M produced an arrival rate that
minimized costs (wages + lost production)
M = 5 was optimal, compared to M = 1 before queuing study
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Case study: Multiple-server model
Reference
Deutch, H. and Mabert, V. A., “Queuing Theory Applied to Teller Staffing,”
Interfaces, Oct., 1980.
Company
Bankers Trust Co. of New York
Bottom line
Annual cash savings of $1,000,000 in reduced wages. Cost to develop
model of $110,000.
Problem
Determine number of tellers to be on duty per hour of day to meet goals for
waiting time. Staffing decisions needed at 100 branch banks.
Model
Straightforward application of multi-channel model in text.
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Case study (cont.)
Analysis
Development of arrival and service distributions by hour and day
of week at each bank.
Arrival and service shown to be Poisson / Exponential.
Experimentation with number of servers in model showed that full-time
tellers were idle much of the day.
Result
Elimination of 100 full-time tellers. Increased use of part-time tellers.
Today, the multi-channel model is a standard tool for staffing
decisions in banking.
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Queuing model structures
Single-server model
Source
pop.
Pop.
can be
finite or
infinite
Service
facility
Arrival
rate
must be
Poisson
Queue
capacity can
be finite
or infinite
Waiting Lines
Service time
usually exp.,
but can be
anything
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Queuing model structures (cont.)
Multiple-server model
Source
pop.
Pop.
must be
infinite
Arrival
rate
must be
Poisson
Queue
capacity
must be
infinite
Note: There is only one queue
regardless of nbr. of servers
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Service
facility
#1
Service
facility
#2
Service time
for each
server must
have same
mean and
be exp.
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Applying the single-server model
1. Analyze service times.
- plot actual vs. exponential distribution
- if exponential good fit, use it
- otherwise compute σ of times
2. Analyze arrival rates.
- plot actual vs. Poisson Distribution
- if Poisson good fit, use it
- if not, stop—only alternative is simulation
3. Determine queue capacity.
- infinite or finite?
- if uncertain, compare results from alternative models
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Applying the single-server model
(cont.)
4. Determine size of source population.
- infinite or finite?
- if uncertain, compare results from alternative models
5. Choose model from SINGLEQ worksheet.
Waiting Lines
SINGLEQ.xls
9
Applying the multiple-server model
1. Analyze service times.
- Must be exponential
2. Analyze arrival rates.
- Must be Poisson
3. Queue capacity must be infinite.
4. Source population must be infinite.
5. Apply MULTIQ worksheet.
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MULTIQ.xls
10
Single-server equations
Arrival rate
= λ
Service rate
= μ
Mean number in queue
= λ2/(μ(μ-λ))
Mean number in system
= λ /(μ-λ)
Mean time in queue
= λ /(μ(μ-λ))
Mean time in system
= 1/(μ-λ)
Utilization ratio
(Prob. server is busy)
= λ /μ
Waiting Lines
SINGLEQ.xls
11
Utilization ratio vs. queue length
λ
5
10
15
19
19.5
19.6
19.7
19.8
19.9
19.95
19.99
μ
20
20
20
20
20
20
20
20
20
20
20
λ/μ
.25
.50
.75
.95
.975
.98
.985
.99
.995
.997
.999
20
20
1.000
Queue length
0.08 people
0.50
2.25
18.05
38.03
48.02
64.68
98.01
198.01
398.00
1,998.00

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SINGLEQ.xls
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Single-server queuing identities
A. Number units in system = arrival rate * mean time in system
B. Number units in queue = arrival rate * mean time in queue
C. Mean time in system
= mean time in queue + mean service time
Note: Mean service time = 1/ mean service rate
If we can determine only one of the following, all other values can be
found by substitution:
Number units in system or queue
Mean time in system or queue
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State diagram: single-server model
A
# in system
0
A
2
1
S
A
S
3
S
●
# in system also called state.
●
To get from one state to another, an arrival (a) must occur or
a service completion (s) must occur.
●
In long-run, for each state:
Rate in = Rate out
Mean # A = Mean # S
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Balance equations for each state
State
0
Rate in
SP1
=
Rate out
AP0
Probability in
state 1
The only way
into state 0
is service
completion from 1
Probability in
state 0
The only way
out of state 0
is to have
an arrival
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Balance equations for each state
(cont.)
State
1
Rate in
AP0 + SP2
=
=
Can arrive
state 1 by
arrival from 0
or service
completion from 2
Rate out
AP1 + SP1
Two ways
out of state 1,
arrival or
service completion
2
AP1 + SP3
=
AP2 + SP2
3
AP2 + SP4
=
AP3 + SP3
etc.
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Solution of balance equations
Expected number in system = ΣnPn
Solve equations simultaneously to get each probability.
Given number in system, all other values are found by substitution in
queuing identities.
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