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QUEUING TEMPLATES 1995 by David W. Ashley
Revised May 21, 1997
This worksheet computes queuing results for the following
models:
M / M / sM / M / s with finite queue lengthM / M / s with finite
arrival populationM / G / 1
Click on the page tab to use the model of your choice. Enter the
required parameters in the boxes.
Parameters for all models are initially linked to those entered
for M/M/s.
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MMs
Page 2
M/M/s queuing computations lambda/muArrival rate 2 per hour
Assumes Poisson process for /sService rate 3 per hour arrivals and
services.Number of servers 1 (max of 40)
Time Unit hour P(0) =Utilization 66.67%P(0), probability that
the system is empty 0.3333 0 Lq, expected queue length 1.3333 1 L,
expected number in system 2.0000 2 Wq, expected time in queue
0.6667 hours 3 W, expected total time in system 1.0000 hours 4
Probability that a customer waits 0.6667 5
6 7 8 9
10 11 12 13 14 15 16 17 18 19 20 21 22 23
0
0.1
0.2
0.3
0.4
NUMBER IN SYSTEM
Prob
abilit
y
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MMs
Page 3
24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40
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MMs
Page 4
### s-1 0 THE ARRIVAL RATE SHOULD BE LESS THAN THE OVERALL
SERVICE RATE!###
2 s factorial = 1
### ### 1 P(n) 1 1
1 ### ### 1 1 0 ### 0 1 1 0 ### 0 1 1 0 ### 0 1 1 0 ### 0 1 1 0
### 0 1 1 0 ### 0 1 1 0 ### 0 1 1 0 ### 0 1 1 0 ### 0 1 1 0 ### 0 1
1 0 ### 0 1 1 0 ### 0 1 1 0 ### 0 1 1 0 ### 0 1 1 0 ### 0 1 1 0 ###
0 1 1 0 ### 0 1 1 0 ### 0 1 1 0 ### 0 1 1 0 ### 0 1 1
### 0 1 1 ### 0 1 1 ### 0 1 1
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MMs
Page 5
### 0 1 1 ### 0 1 1 ### 0 1 1 ### 0 1 1 ### 0 1 1 ### 0 1 1 ###
0 1 1 ### 0 1 1 ### 0 1 1 ### 0 1 1 ### 0 1 1 ### 0 1 1 ### 0 1 1
### 0 1 1 ### 0 1 1 ### 0 1 1 ### 0 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1
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MMs
Page 6
THE ARRIVAL RATE SHOULD BE LESS THAN THE OVERALL SERVICE
RATE!
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MMs
Page 7
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finite queue length
Page 8
M/M/s with Finite Queue Arrival rate 2 Service rate 3 Number of
servers 1 (max of 40) Maximum queue length 10 (max of 40
combined)Utilization 66.41%P(0), probability that the system is
empty 0.3359 Lq, expected queue length 1.2427 L, expected number in
system 1.9068 Wq, expected time in queue 0.6238 W, expected total
time in system 0.9571 Probability that a customer waits 0.6641
Probability that a customer balks 0.0039
0 0.1 0.2 0.3 0.4
NUMBER IN SYSTEM
Prob
abilit
y
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finite queue length
Page 9
0
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finite queue length
Page 10
lambda/mu ###/s ###
###
P(0) = ###
0 1 1 ###2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 0
10 0 11 0 12 0 13 0 14 0 15 0 16 0 17 0 18 0 19 0 20 0 21 0 22 0
23 0 24 0 25 0 26 0 27 0 28 0 29 0 30 0
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finite queue length
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31 0 32 0 33 0 34 0 35 0 36 0 37 0 38 0 39 0 40 0
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finite queue length
Page 12
11
s factorial = 1 comp of Lq 1
1 ### n P(n)1 1 0 0 ###1 1 0 1 ###
2 ### 1 1 ### 2 ###3 ### 1 1 ### 3 ###4 ### 1 1 ### 4 ###5 ### 1
1 ### 5 ###6 ### 1 1 ### 6 ###7 ### 1 1 ### 7 ###8 ### 1 1 ### 8
###9 ### 1 1 ### 9 ###
10 ### 1 1 ### 10 ###11 ### 1 1 ### 11 ###12 0 1 1 0 12 0 13 0 1
1 0 13 0 14 0 1 1 0 14 0 15 0 1 1 0 15 0 16 0 1 1 0 16 0 17 0 1 1 0
17 0 18 0 1 1 0 18 0 19 0 1 1 0 19 0 20 0 1 1 0 20 0 21 0 1 1 0 21
0 22 0 1 1 0 22 0 23 0 1 1 0 23 0 24 0 1 1 0 24 0 25 0 1 1 0 25 0
26 0 1 1 0 26 0 27 0 1 1 0 27 0 28 0 1 1 0 28 0 29 0 1 1 0 29 0 30
0 1 1 0 30 0 31 0 1 1 0 31 0
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finite queue length
Page 13
32 0 1 1 0 32 0 33 0 1 1 0 33 0 34 0 1 1 0 34 0 35 0 1 1 0 35 0
36 0 1 1 0 36 0 37 0 1 1 0 37 0 38 0 1 1 0 38 0 39 0 1 1 0 39 0 40
0 1 1 0 40 0 41 0 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1
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finite queue length
Page 14
computation of L prob wait### 0
### ### ###### 0 ###
### ### 0 0 0 ### ### 0 0 0 ### ### 0 0 0 ### ### 0 0 0 ### ###
0 0 0 ### ### 0 0 0 ### ### 0 0 0 ### ### 0 0 0 ### ### 0 0 0 ###
### 0 0 0 ### ### 0 0 0 ### ### 0 0 0 ### ### 0 0 0 ### ### 0 0 0
### ### 0 0 0 ### ### 0 0 0 ### ### 0 0 0 ### ### 0 0 0 ### ### 0 0
0 ### ### 0 0 0 ### ### 0 0 0 ### ### 0 0 0 ### ### 0 0 0 ### ### 0
0 0 ### ### 0 0 0 ### ### 0 0 0 ### ### 0 0 0 ### ### 0 0 0 ### ###
0 0 0 ### ### 0 0 0 ### ### 0 0 0
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finite queue length
Page 15
### ### 0 0 0 ### ### 0 0 0 ### ### 0 0 0 ### ### 0 0 0 ### ###
0 0 0 ### ### 0 0 0 ### ### 0 0 0 ### ### 0 0 0 ### ### 0 0 0
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finite population
Page 16
M/M/s with Finite Population overallArrival rate 0.2 (per
customer) 2 Service rate 3 (per server)Number of servers 1 (max of
40)Population size 10 (max of 100)
Utilization 58.97%P(0), probability that the system is empty
###Lq, expected queue length ###L, expected number in system ###Wq,
expected time in queue ###W, expected total time in system
###Probability that a customer waits ###
0 0.2 0.4 0.6
NUMBER IN SYSTEM
Prob
abilit
y
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finite population
Page 17
0
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finite population
Page 18
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finite population
Page 19
#REF!#REF!
lambda/mu ###/s ###
### ######
effective lambda P(0) = ### 1 n P(n)
-1 0 1 ###0 10 1 ### ###1 9 2 0.4 ###2 8 3 ### ###3 7 4 ### ###4
6 5 ### ###5 5 6 ### ###6 4 7 ### ###7 3 8 ### ###8 2 9 ### ###9 1
10 ### ###
10 0 11 0 0 11 -1 12 0 0 12 -2 13 0 0 13 -3 14 0 0 14 -4 15 0 0
15 -5 16 0 0 16 -6 17 0 0 17 -7 18 0 0 18 -8 19 0 0 19 -9 20 0 0 20
-10 21 0 0 21 -11 22 0 0 22 -12 23 0 0 23 -13 24 0 0 24 -14 25 0 0
25 -15 26 0 0 26 -16 27 0 0
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finite population
Page 20
27 -17 28 0 0 28 -18 29 0 0 29 -19 30 0 0 30 -20 31 0 0 31 -21
32 0 0 32 -22 33 0 0 33 -23 34 0 0 34 -24 35 0 0 35 -25 36 0 0 36
-26 37 0 0 37 -27 38 0 0 38 -28 39 0 0 39 -29 40 0 0 40 -30 41 0 0
41 -31 42 0 0 42 -32 43 0 0 43 -33 44 0 0 44 -34 45 0 0 45 -35 46 0
0 46 -36 47 0 0 47 -37 48 0 0 48 -38 49 0 0 49 -39 50 0 0 50 -40 51
0 0 51 -41 52 0 0 52 -42 53 0 0 53 -43 54 0 0 54 -44 55 0 0 55 -45
56 0 0 56 -46 57 0 0 57 -47 58 0 0 58 -48 59 0 0 59 -49 60 0 0 60
-50 61 0 0 61 -51 62 0 0 62 -52 63 0 0 63 -53 64 0 0 64 -54 65 0 0
65 -55 66 0 0 66 -56 67 0 0 67 -57 68 0 0 68 -58 69 0 0
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finite population
Page 21
69 -59 70 0 0 70 -60 71 0 0 71 -61 72 0 0 72 -62 73 0 0 73 -63
74 0 0 74 -64 75 0 0 75 -65 76 0 0 76 -66 77 0 0 77 -67 78 0 0 78
-68 79 0 0 79 -69 80 0 0 80 -70 81 0 0 81 -71 82 0 0 82 -72 83 0 0
83 -73 84 0 0 84 -74 85 0 0 85 -75 86 0 0 86 -76 87 0 0 87 -77 88 0
0 88 -78 89 0 0 89 -79 90 0 0 90 -80 91 0 0 91 -81 92 0 0 92 -82 93
0 0 93 -83 94 0 0 94 -84 95 0 0 95 -85 96 0 0 96 -86 97 0 0 97 -87
98 0 0 98 -88 99 0 0 99 -89 100 0 0
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finite population
Page 22
#REF!#REF!
s-1 0
Lq L### 0 ### ###
0 0 ### 0 0 0 0 0 1 ###
### 0 0 1 ###### 0 0 1 ###### 0 0 1 ###### 0 0 1 ###### 0 0 1
###### 0 0 1 ###### 0 0 1 ###### 0 0 1 ###### 0 0 1 ###
0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0
0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0
0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0
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finite population
Page 23
0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0
0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0
0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0
0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1
0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0
1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0
0 1 0 0 0 0 1 0
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finite population
Page 24
0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0
0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0
0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0
0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1
0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0
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finite population
Page 25
s factorial = 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
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finite population
Page 26
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1
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finite population
Page 27
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MG1
Page 28
M/G/1 queuing computationsaverage
Arrival rate 2 per hour service RATEAverage service TIME ###
hours 3 per hourStandard dev. of service time ### hoursTime unit
hour
Utilization 66.67%P(0), probability that the system is empty
0.3333 Lq, expected queue length 1.3333 L, expected number in
system 2.0000 Wq, expected time in queue 0.6667 hoursW, expected
total time in system 1.0000 hours
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MG1
Page 29
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MG1
Page 30
THE ARRIVAL RATE SHOULD BE LESS THAN THE OVERALL SERVICE
RATE!
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MG1
Page 31
THE ARRIVAL RATE SHOULD BE LESS THAN THE OVERALL SERVICE
RATE!
IntroMMsfinite queue lengthfinite populationMG1
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