queuing theory - phl ched connect
TRANSCRIPT
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QUEUING THEORYLearning Objectives
University of the Philippines
By the end of this module, the students are expected to:
1. Identify the appropriate queuing model for a particular situation.
2. Evaluate the performance of a queuing system using different metrics.
3. Use a spreadsheet template to easily compute queuing-related performance measures.
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service facility
QUEUING THEORYThe Queuing System
University of the Philippines
customer server
queue
arriving
customer
customers
in service
system boundary
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QUEUING THEORYKendall-Lee Notation
University of the Philippines
A six-element classification system for queuing systems using the notation:
_____ / _____ / _____ / _____ / _____ / _____A S s D K N
Examples:
1. M / M / 2 / FCFS / 20 / 20
2. D / M / 1 / FCFS / โ / โ
3. G / G / 2 / LCFS / 100 / โ
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QUEUING THEORYKendall-Lee Notation
University of the Philippines
_____ / _____ / _____ / _____ / _____ / _____A S s D K N
A: Arrival (Input) Process. The arrival pattern of arriving customers. Let ฮป be
the arrival rate of customers.
M โ Markovian (exponential)
Ek โ Erlang
D โ constant (degenerate)
G โ general
S: Service (Output ) Process. The service time distribution. Let ยต be the
service rate of each server.
M โ Markovian
Ek โ Erlang
D โ constant (degenerate)
G โ general
s: Servers. The number of servers in a service facility.
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QUEUING THEORYKendall-Lee Notation
University of the Philippines
_____ / _____ / _____ / _____ / _____ / _____A S s D K N
D: Queuing Discipline. Determines which customer will be served next.
FSFC โ first come, first served
LCFS โ last come, first served
SIRO โ service in random order
GD โ general discipline
K: System Capacity. The maximum number of customers that are allowed to
enter the system.
N: Input Source / Calling Population. The number of customers that may
enter the system.
The last three may be omitted in the notation if FCFS/โ/โ.
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QUEUING THEORYReview of Relevant Probability Distributions
University of the Philippines
Degenerate Normal Erlang Exponential
Increasing Randomness (Variability)
Mathematicallytractable
Limited Mathematical Analysis(may require simulation)
Mathematicallytractable
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QUEUING THEORYReview of Relevant Probability Distributions
University of the Philippines
Degenerate Normal Erlang Exponential
10.0
10.0
10.0
10.0
10.0
10.0
10.0
10.0
9.7
9.7
9.6
9.6
9.3
11.3
9.3
10.1
17.6
17.9
15.0
7.0
6.4
7.2
39.3
4.0
7.6
8.6
40.0
36.1
8.3
34.2
3.4
9.0
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QUEUING THEORYReview of Relevant Probability Distributions
University of the Philippines
Exponential
Degenerate
Normal
Erlang
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QUEUING THEORYKendall-Lee Notation
University of the Philippines
Identify the Kendall-Lee notation of these queuing systems.
1. A canteen operates an espresso stand. Customers arrive to the standwith an exponential inter-arrival time. The espresso vending machinerequires exactly 45 seconds for each customer to operate.
2. A company has provided specialized machines to 12 different clients.Part of the agreement between the company and the clients is that thecompany will send a technician whenever the machine breaks. Thetechnician visits the site and will spend an average of 3 days per site tofix the machine. Assume normally distributed fix time. The company has2 technicians. Mean time to failure (MTTF) of each machine isexponential with mean of 60 days.
M/D/1/FCFS/โ/โ or M/D/1
M/G/2/FCFS/12/12
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QUEUING THEORYKendall-Lee Notation
University of the Philippines
Identify the Kendall-Lee notation of these queueing systems.
3. An eat-all-you-can dining place is usually full (can seat only 45 groups ofpeople). That is why it usually has a number of groups waiting in theholding area. A group of customers will balk (go to other dining place) ifthey see that there are 10 groups already in the holding area. Eatingtime per group is roughly 1.5 hrs. Assume Poisson arrival and randomeating time.
4. Students return books to the library at a rate of 6 books per hour. Theborrowerโs section librarian returns the books, one at a time, to theirrespective bookshelves. This takes an erlang distributed time per book.Since a newly-returned book is placed on top of the pile, it is the firstone to be given attention by the librarian.
M/M/45/FCFS/55/โ
M/Ek/1/LCFS/โ/โ
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QUEUING THEORYPerformance Measure of Queuing Systems
University of the Philippines
โช Ls. Average number of customers being served.
โช Lq. Average length of customers in the line (queue).
โช L = Ls + Lq. Average number of customers in the system
โช Ws. Average service time of a customer.
โช Wq. Average waiting time of customer in the line (queue).
โช W = Ws + Wq. Average waiting time of customer in the system.
โช Probability of Balking. Balking happens when a customer cannot (orrefuses to) enter the line because it is either full or too long.
โช Server Utilization. The proportion of time that a given server is busy as itattends to customers.
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QUEUING THEORYPerformance Measure of Queuing Systems
University of the Philippines
Steady State(reported values)
Transient Period(not reported)
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QUEUING THEORYPerformance Measure of Queuing Systems
University of the Philippines
Let Pn be the steady-state probability that the queuing system has n
customers.
๐ฟ =
๐
๐๐๐
๐ฟ๐ =
๐>๐
เตซ๐ โ ๐ )๐๐
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QUEUING THEORYLittleโs Queuing Formulas
University of the Philippines
Littleโs Law states that the average number of customers in a queuingsystem is the product of the average entry rate of customers and theaverage time a customer spends in the system.
The following relationships are likewise true.
๐ฟ = ๐๐๐ฃ๐๐
๐ฟ๐ = ๐๐๐ฃ๐๐๐
๐ฟ๐ = ๐๐๐ฃ๐๐๐
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QUEUING THEORYLittleโs Queuing Formulas
University of the Philippines
L WL = ฮปaveW
Wq
W = Wq + 1/ยต
Lq
Lq = ฮปaveWq
L = Lq + ฮป/ยต
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QUEUING THEORYLittleโs Queuing Formulas
University of the Philippines
In the Bills Payment section of a utility company, customers arrive to settletheir bills at a rate of 60 per hour. These customers join a single queueleading to three cashiers. Each cashier, if working continuously, can servecan serve a customer 24 customers in an hour. It is estimated that acustomer waits in line for an average of 1.5 mins before service begins.Determine the average number of customers in queuing system.
๐ = 60 ๐๐ข๐ ๐ก๐๐๐๐๐ /โ๐
๐๐ = 1.5 ๐๐๐๐
ฮผ = 24 ๐๐ข๐ ๐ก๐๐๐๐๐ /โ๐
๐๐ =1
๐=
1
24= 0.042 โ๐ = 2.5 ๐๐๐๐
๐ = ๐๐ + ๐๐ =1.5
60+ 0.042 = 0.067 โ๐ = 4 ๐๐๐๐
๐ฟ = ๐๐๐ฃ๐๐ = 60 0.067 = ๐ ๐๐๐๐๐๐๐๐๐
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QUEUING THEORYBirth-and-Death Model of Markovian Queues
University of the Philippines
0 1 2 3 4 5 . . .
ฮป0 ฮป1 ฮป2 ฮป3 ฮป4 ฮป5
ยต1 ยต2 ยต3 ยต4ยต5 ยต6
Let the state N(t) of the queuing system be the number of customers in the
system at time t.
Birth is the transition from state n to state n+1. If the system is in state n,
the remaining time until the next birth is exponential with rate ฮปn, n โฅ 0.
Death is the transition from state n to state n-1. If the system is in state n,
the remaining time until the next death is exponential with rate ยตn, n โฅ 1.
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QUEUING THEORYBirth-and-Death Model of Markovian Queues
University of the Philippines
0 1 2 3 4 5 . . .
ฮป0 ฮป1 ฮป2 ฮป3 ฮป4 ฮป5
ยต1 ยต2 ยต3 ยต4ยต5 ยต6
๐1 =๐0๐1๐0
๐2 =๐0๐1๐1๐2
๐0 =๐1๐2๐1
๐3 =๐0๐1๐2๐1๐2๐3
๐0 =๐2๐3๐2
๐4 =๐0๐1๐2๐3๐1๐2๐3๐4
๐0 =๐3๐4๐3
๐5 =๐0๐1๐2๐3๐4๐1๐2๐3๐4๐5
๐0 =๐4๐5๐4
๐0 = 1 +๐0๐1
+๐0๐1๐1๐2
+๐0๐1๐2๐1๐2๐3
+๐0๐1๐2๐3๐1๐2๐3๐4
+๐0๐1๐2๐3๐4๐1๐2๐3๐4๐5
+ โฆ
โ1
1 +๐0๐1
+๐0๐1๐1๐2
+๐0๐1๐2๐1๐2๐3
+๐0๐1๐2๐3๐1๐2๐3๐4
+๐0๐1๐2๐3๐4๐1๐2๐3๐4๐5
+ โฆ
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QUEUING THEORYBirth-and-Death Model of Markovian Queues
University of the Philippines
0 1 2 3 4 5
5/hr 5/hr 5/hr 5/hr 5/hr
4/hr 8/hr 8/hr 8/hr 8/hr
๐1 =๐0๐1๐0 =
5
40.249 = 0.311
๐2 =๐1๐2๐1 =
5
80.311 = 0.195
๐3 =๐2๐3๐2 =
5
80.195 = 0.122
๐4 =๐3๐4๐3 =
5
80.122 = 0.076
๐5 =๐4๐5๐4 =
5
80.076 = 0.048
๐0 = 1 +๐0๐1
+๐0๐1๐1๐2
+๐0๐1๐2๐1๐2๐3
+๐0๐1๐2๐3๐1๐2๐3๐4
+๐0๐1๐2๐3๐4๐1๐2๐3๐4๐5
+
โ1
1 +5
4+5(5)
4(8)+5(5)(5)
4(8)(8)+5(5)(5)(5)
4(8)(8)(8)+5(5)(5)(5)
4(8)(8)(8)
Consider the rate diagram of a Markovian queuing system. Find the steady-
state probabilities Pnโs.
= 0.249
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QUEUING MODELSM/M/1
University of the Philippines
๐ =๐
๐< 1 ๐0 = 1 โ ๐
๐ฟ =๐
1 โ ๐=
๐
๐ โ ๐๐ฟ๐ =
๐2
1 โ ๐=
๐2
)๐(๐ โ ๐
๐๐ = ๐๐(1 โ ๐)
๐ =๐
)๐(1 โ ๐=
1
๐ โ ๐๐๐ =
๐2
)๐(1 โ ๐=
๐
)๐(๐ โ ๐
0 1 2 3 . . .
ฮป ฮป ฮป ฮป
ฮผ ฮผ ฮผ ฮผ
๐ > ๐ก = ๐๐โ๐ 1โ๐ ๐กWq
Note: is the waiting time of customer in queue. Wq is the average waiting time of
customer in queue or E( ).
WqWq
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QUEUING MODELSM/M/1
University of the Philippines
A regional airport has a single runway. Airplanes requiring the use of therunway arrive at a Poisson rate of 12 per hour. Each plane uses the runwayfor an exponential time with mean of 4 mins.
a. Find the average waiting time of airplanes before they can use therunway.
b. What is the utilization of the runway?
c. Compute the average number of airplanes currently using or waiting touse the runway.
d. Find the probability that an airplane does not need to queue to use therunway.
e. What is the probability that an airplane needs to wait for more than 6mins before it can use the runway?
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QUEUING MODELSM/M/1
University of the Philippines
0 1 2 3 4 5 . . .
12/hr 12/hr 12/hr 12/hr 12/hr 12/hr
15/hr 15/hr 15/hr 15/hr 15/hr 15/hr
M/M/1 with ฮป = 12/hr and ยต = 15/hr
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QUEUING MODELSM/M/1
University of the Philippines
a. Find the average waiting time of airplanes before they can use therunway.
b. What is the utilization of the runway?
c. Compute the average number of airplanes currently using or waiting touse the runway.
๐๐ =๐
)๐(๐ โ ๐=
12
)15(15 โ 12= ๐. ๐๐๐๐ ๐๐ = ๐๐๐๐๐๐
๐ =๐
๐=12
15= ๐. ๐๐
๐ฟ =๐
๐ โ ๐=
12
15 โ 12= ๐ ๐๐๐๐๐๐
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QUEUING MODELSM/M/1
University of the Philippines
d. Find the probability that an airplane does not need to queue to use therunway.
e. What is the probability that an airplane needs to wait for more than 6mins before it can use the runway?
๐0 = 1 โ ๐ = 1 โ 0.80 = ๐. ๐๐
๐ > ๐ก = ๐๐โ๐ 1โ๐ ๐ก = 0.80๐โ15 1โ0.80 (660) = ๐. ๐๐๐Wq
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QUEUING MODELSM/M/1
University of the Philippines
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QUEUING MODELSM/M/1
University of the Philippines
A small grocery store has a single cashier lane where grocers line up to payfor their purchases. Inter-arrival time of customers is exponentiallydistributed with mean of 3 mins. Due to the variety in type and number ofitems purchased, transaction time is approximately exponentially distributedwith mean 2.5 mins.
a. Compute the average number of customers in the cashier lane.
b. Find the probability that there are at most 4 customers in the lane.
c. What is the proportion of time that the cashier is idle?
d. Find the average time a customer spends in the cashier, includingqueuing time and transaction time.
e. Buying a POS machine will help reduce average transaction time to only2 mins (20% reduction) with the aid of a barcode reader โ eliminatingthe need to type item codes manually. If implemented, find the percentreduction in the average time a customer spends in the cashier.
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QUEUING MODELSM/M/1
University of the Philippines
0 1 2 3 4 5 . . .
20/hr 20/hr 20/hr 20/hr 20/hr 20/hr
24/hr 24/hr 24/hr 24/hr 24/hr 24/hr
M/M/1 with ฮป = 20/hr and ยต = 24/hr
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QUEUING MODELSM/M/1
University of the Philippines
a. Compute the average number of customers in the cashier lane.
b. Find the probability that there are at most 4 customers in the lane.
c. What is the proportion of time that the cashier is idle?
๐ฟ =๐
๐ โ ๐=
20
20 โ 24= ๐ ๐๐๐๐๐๐๐๐๐
๐=0
4
๐๐ =
๐=0
4
)๐๐(1 โ ๐ =
๐=0
420
24
๐
1 โ20
24= ๐. ๐๐๐
๐0 = 1 โ ๐ = 1 โ20
24= ๐. ๐๐๐
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QUEUING MODELSM/M/1
University of the Philippines
d. Find the average time a customer spends in the cashier, includingqueuing time and transaction time.
e. Buying a POS machine will help reduce average transaction time to only2 mins (20% reduction) with the aid of a barcode reader โ eliminatingthe need to type item codes manually. If implemented, find the percentreduction in the average time a customer spends in the cashier.
๐ =1
๐ โ ๐=
1
24 โ 20= 0.25 โ๐ = ๐๐๐๐๐๐
๐๐๐๐ค =1
๐๐๐๐ค โ ๐=
1
30 โ 20= 0.10 โ๐ = ๐๐๐๐๐
% ๐ ๐๐๐ข๐๐ก๐๐๐ =15 ๐๐๐๐ โ 6 ๐๐๐๐
15 ๐๐๐๐ = ๐๐%
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QUEUING MODELSM/M/1
University of the Philippines
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QUEUING MODELSM/M/s
University of the Philippines
0 1 2 s-1. . .
ฮป ฮป ฮป ฮป
ฮผ 2ฮผ 3ฮผ sฮผ
. . .s
ฮป
s+1
sฮผ(s-1)ฮผ
ฮป
sฮผ
ฮป
๐ =๐
๐ ๐< 1 ๐ ๐ โฅ ๐ =
ฮค๐ ๐ ๐ ๐0)๐ ! (1 โ ๐
๐0 = ๐=0
๐ โ1 ฮค๐ ๐ ๐
๐!+
ฮค๐ ๐ ๐
)๐ ! (1 โ ๐
โ1
๐๐ =
ฮค๐ ๐ ๐๐0๐!
, ๐ โค ๐ .ฮค๐ ๐ ๐๐0๐ ! ๐ ๐โ๐
, ๐ โฅ ๐
๐ฟ๐ =๐ ๐ โฅ ๐ ๐
1 โ ๐
๐๐ =๐ ๐ โฅ ๐
๐ ๐ โ ๐
๐ > ๐ก = ๐ ๐ โฅ ๐ ๐โ๐ ๐ 1โ๐ ๐กWq
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QUEUING MODELSM/M/s
University of the Philippines
A customer support contact center employs 15 employees (or agents). Callsfrom customers who wish to made an inquiry enter the companyโs IT systemat a Poisson rate of 94 calls per hour. The system places the call (or ticket)in a single queue and directs it to an agent once available. Customer callslast for an exponential time with mean of 8 mins
a. Find the probability that all agents are busy in a given time.
b. Find the average time a call is put on hold before it is attended by anagent.
c. On the average, how may calls are in the IT system (both being attendedto and put on hold).
d. Since waiting time is not acceptable, find the number of additionalagents required to ensure that average time a call is put on hold isreduced to at most 30 seconds.
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QUEUING MODELSM/M/s
University of the Philippines
0 1 2 โฆ 14 15 16 โฆ
94/hr 94/hr 94/hr 94/hr 94/hr 94/hr 94/hr
7.5/hr 2(7.5)/hr 3(7.5)/hr 14(7.5)/hr 15(7.5)/hr 15(7.5)/hr 15(7.5)/hr
M/M/15 with ฮป = 94/hr and ยต = 7.5/hr
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QUEUING MODELSM/M/s
University of the Philippines
a. Find the probability that all agents are busy in a given time.
๐ =๐
๐ ๐=
94
15(7.5)= 0.836
๐0 = ๐=0
๐ โ1 ฮค๐ ๐ ๐
๐!+
ฮค๐ ๐ ๐
)๐ ! (1 โ ๐
โ1
= ๐=0
14 ฮค94 7.5 ๐
๐!+
ฮค94 7.5 15
)15! (1 โ 0.836
โ1
= 2.96 ๐ฅ 10โ6
๐ ๐ โฅ ๐ =ฮค๐ ๐ ๐ ๐0
)๐ ! (1 โ ๐=
94/7.5 15 2.96 ๐ฅ 10โ6
)15! (1 โ 0.836= ๐. ๐๐๐
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QUEUING MODELSM/M/s
University of the Philippines
b. Find the average time a call is put on hold before it is attended by anagent.
c. On the average, how may calls are in the IT system (both being attendedto and put on hold).
๐๐ =๐ ๐ โฅ ๐
๐ ๐ โ ๐=
0.407
15(7.5) โ 94= ๐. ๐๐๐ ๐๐ = ๐. ๐๐๐๐๐๐๐
๐ฟ๐ =๐ ๐ โฅ ๐ ๐
1 โ ๐=0.407 โ 0.836
1 โ 0.836= 2.069 ๐๐๐๐๐
๐ฟ = ๐ฟ๐ +๐
๐= 2.069 +
94
7.5= ๐๐. ๐๐๐ ๐๐๐๐๐
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QUEUING MODELSM/M/s
University of the Philippines
d. Since waiting time is not acceptable, find the number of additionalagents required to ensure that average time a call is put on hold isreduced to at most 30 seconds (0.5 min).
s Wq
15 1.321 mins
16 0.619 min
17 0.306 min
18 0.155 min
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QUEUING MODELSM/M/s
University of the Philippines
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QUEUING MODELSM/M/s
University of the Philippines
A dental clinic has three dentists. Patients arrive at a Poisson rate of 2.7patients / hr. Due to the varying nature of services required by patients, thetime a dentist needs to serve a patient is found to be exponentiallydistributed with mean of 40 mins per patient. Compute the fourperformance measures (L, Lq, W, Wq) of the queuing system for the twoassumptions below:
a. Assume that patients are indifferent
among the three dentists and are
willing to be attended by whoever is
immediately available. There is a
single queue of patients.
b. Assume each patient has a preferred
dentist. Thus, there are three
separate queues, each with identical
arrival rate of 0.9 patient per hour.
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QUEUING MODELSM/M/s
University of the Philippines
a. Assume that patients are indifferent among the three dentists and arewilling to be attended by whoever is immediately available. There is asingle queue of patients.
0 1 2 3 4 5 . . .
2.7/hr 2.7/hr 2.7/hr 2.7/hr 2.7/hr 2.7/hr
1.5/hr 3.0/hr 4.5/hr 4.5/hr 4.5/hr 4.5/hr
M/M/3 with ฮป = 2.7/hr and ยต = 1.5/hr
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QUEUING MODELSM/M/s
University of the Philippines
a. Assume that patients are indifferent among the three dentists and arewilling to be attended by whoever is immediately available. There is asingle queue of patients.
๐ =๐
๐ ๐=
2.7
3(1.5)= 0.60
๐0 = ๐=0
๐ โ1 ฮค๐ ๐ ๐
๐!+
ฮค๐ ๐ ๐
)๐ ! (1 โ ๐
โ1
= ๐=0
2 ฮค2.7 1.5 ๐
๐!+
ฮค2.7 1.5 ๐
)3! (1 โ 0.60
โ1
= 0.146
๐ ๐ โฅ ๐ =ฮค๐ ๐ ๐ ๐0
)๐ ! (1 โ ๐=
2.7/1.5 30.146
)3! (1 โ 0.60= 0.355
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QUEUING MODELSM/M/s
University of the Philippines
a. Assume that patients are indifferent among the three dentists and arewilling to be attended by whoever is immediately available. There is asingle queue of patients.
๐ฟ๐ =๐ ๐ โฅ ๐ ๐
1 โ ๐=0.355 โ 0.60
1 โ 0.60= ๐. ๐๐๐ ๐๐๐๐๐๐๐๐
๐ฟ = ๐ฟ๐ +๐
๐= 0.532 +
2.7
1.5= ๐. ๐๐๐ ๐๐๐๐๐๐๐๐
๐๐ =๐ ๐ โฅ ๐
๐ ๐ โ ๐=
0.355
3(1.5) โ 2.7= ๐. ๐๐๐ ๐๐ = ๐๐. ๐๐๐๐๐๐๐
๐ = ๐๐ +1
๐= 0.197 +
1
1.5= ๐. ๐๐๐ ๐๐ = ๐๐. ๐๐๐๐๐๐๐
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QUEUING MODELSM/M/s
University of the Philippines
b. Assume each patient has a preferred dentist. Thus, there are threeseparate queues, each with identical arrival rate of 0.9 patient per hour.
M/M/1 with ฮป = 0.9/hr, ยต = 1.5/hr
๐ฟ๐ =๐2
๐(๐ โ ๐)=
0.92
1.5(1.5 โ 0.9)= ๐. ๐๐ ๐๐๐๐๐๐๐
๐ฟ =๐
๐ โ ๐=
0.9
1.5 โ 0.9= ๐. ๐ ๐๐๐๐๐๐๐๐
๐ =1
๐ โ ๐=
1
1.5 โ 0.9= ๐. ๐๐ ๐๐๐ = ๐๐๐๐๐๐๐
๐๐ =๐
๐(๐ โ ๐)=
0.9
1.5(1.5 โ 0.9)= ๐ ๐๐ = ๐๐๐๐๐๐
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QUEUING MODELSM/M/s
University of the Philippines
(a) 1 M/M/3 (b) 3 M/M/1
Whole Individual Whole
Lq (patients) 0.532 0.90 2.70
L (patients) 2.332 1.50 4.50
Wq (hours) 0.197 1.00 1.00
W (hours) 0.864 1.67 1.67
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QUEUING MODELSM/M/s
University of the Philippines
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QUEUING MODELSM/M/s
University of the Philippines
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QUEUING MODELSG/G/โ (Self-Service Model)
University of the Philippines
๐๐ = ๐ฟ๐ = 0 ๐ =1
๐๐ฟ =
๐
๐
For G/G/โ/GD/โ/โ
Only for M/G/โ/GD/โ/โ
๐๐ =๐ )โ( ฮค๐ ๐ ( )ฮค๐ ๐ ๐
๐!
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QUEUING MODELSG/G/โ (Self-Service Model)
University of the Philippines
A gym is available to its members 24/7. Members arrive at a Poisson rate of4 per hour. On the average, each member stays in the gym for a normallydistributed time with mean of 2.5 hrs and standard deviation of 0.5 hr.
a. How many members are expected to be found in the gym at any givenpoint in time?
b. What is the probability that there are at most 14 members in the gym?
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QUEUING MODELSG/G/โ (Self-Service Model)
University of the Philippines
a. How many members are expected to be found in the gym at any givenpoint in time?
b. What is the probability that there are at most 14 members in the gym?
๐ฟ =๐
๐=
4
0.4= ๐๐๐๐๐๐๐๐๐
๐=0
14
๐๐ =
๐=0
14๐โ(๐/๐) ๐/๐ ๐
๐!=
๐=0
14๐โ(4/0.4) 4/0.4 ๐
๐!= ๐. ๐๐๐
M/G/โ with ฮป = 4/hr and ยต = 0.4/hr
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QUEUING MODELSG/G/โ (Self-Service Model)
University of the Philippines
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QUEUING MODELSM/M/1/GD/K/โ
University of the Philippines
0 1 2 . . .
ฮป ฮป ฮป
ฮผ ฮผ ฮผ
K-1
ฮป
ฮผ
K
ฮผ
ฮป
๐ =๐
๐
๐๐ =
1 โ ๐ ๐๐
1 โ ๐๐พ+1, ๐ โ 1
1
๐พ + 1, ๐ = 1
For n > K, Pn = 0.
For n โค K:
๐ฟ๐ = ๐ฟ โ (1 โ ๐0)
๐๐๐ฃ๐ = าง๐ = ๐(1 โ ๐๐พ
๐ฟ =
๐
1 โ ๐โ
๐พ + 1 ๐๐พ+1
1 โ ๐๐พ+1, ๐ โ 1
.๐พ
2, ๐ = 1
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QUEUING MODELSM/M/s/GD/K/โ
University of the Philippines
0 1 s. . .
ฮป ฮป ฮป
ฮผ 2ฮผ sฮผ
K-1
ฮป
K
sฮผsฮผ
ฮป
. . .
sฮผ
ฮป
๐ =๐
๐ ๐, ๐ โค ๐พ ๐0 =
๐=0
๐ ฮค๐ ๐ ๐
๐!+
ฮค๐ ๐ ๐
๐ !
๐=๐ +1
๐พ
๐๐โ๐ โ1
๐๐ =
ฮค๐ ๐ ๐๐0๐!
, ๐ โค ๐ .ฮค๐ ๐ ๐๐0๐ ! ๐ ๐โ๐
, ๐ โฅ ๐
For n > K, Pn = 0.
For n โค K:
๐ฟ๐ =๐ ฮค๐ ๐ ๐ ๐0 )1 โ ๐๐พโ๐ โ ๐พ โ ๐ ๐๐พโ๐ (1 โ ๐
๐ ! 1 โ ๐ 2
๐ฟ = ๐ฟ๐ + ๐=0
๐ โ1
๐๐๐ + ๐ 1 โ ๐=0
๐ โ1
๐๐
เตฏ๐๐๐ฃ๐ = าง๐ = ๐(1 โ ๐๐พ
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QUEUING MODELSM/M/s/GD/K/โ
University of the Philippines
A drive-thru of a fast food chain has a single window that services itscustomers. Attending a customer takes an exponential time with mean of 4mins. Including the space for the car/customer being served, the drive-thruhas a total of 6 available spaces. If an arriving car finds that all spaces aretaken, it will opt to get meals elsewhere. Cars arrive at the drive-thru at aPoisson rate of 15 customers per hour (but not all can enter as pointed out).
a. Find the average number of customers in the queuing system.
b. On the average, how long does a car stay in the drive-thru?
c. What is the probability that all the available spaces are occupied?
d. If each served customer generates a revenue of PhP 200, find theexpected total revenue in 4 hours of operations?
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QUEUING MODELSM/M/s/GD/K/โ
University of the Philippines
0 1 2 3 4 5 6
15/hr 15/hr 15/hr 15/hr 15/hr 15/hr
15/hr 15/hr 15/hr 15/hr 15/hr 15/hr
M/M/1/FCFS/6/โ with ฮป = 15/hr and ยต = 15/hr
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QUEUING MODELSM/M/s/GD/K/โ
University of the Philippines
a. Find the average number of customers in the queueing system.
b. On the average, how long does a car stay in the drive-thru?
๐ =๐
๐=15
15= ๐
๐ฟ =๐พ
2=6
2= ๐ ๐๐๐๐
๐๐พ =1
๐พ + 1=
1
6 + 1= ๐. ๐๐๐
)๐๐๐ฃ๐ = ๐(1 โ ๐๐พ = 15 1 โ 0.143 = ๐๐. ๐๐๐ ๐๐๐๐
๐ =๐ฟ
๐๐๐ฃ๐=
3
12.857= ๐. ๐๐๐ ๐๐ = ๐๐๐๐๐๐
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QUEUING MODELSM/M/s/GD/K/โ
University of the Philippines
c. What is the probability that all the available spaces are occupied?
d. If each served customer generates a revenue of PhP 200, find theexpected total revenue in 4 hours of operations?
๐๐พ =1
๐พ + 1=
1
6 + 1= ๐. ๐๐๐
)๐๐๐ฃ๐ = ๐(1 โ ๐๐พ = 15 1 โ 0.143 = ๐๐. ๐๐๐ ๐๐๐๐
๐๐๐ก๐๐ ๐ ๐๐ฃ๐๐๐ข๐ = 200(4)๐๐๐ฃ๐ = 200(4) 12.857 = ๐๐ก๐ ๐๐, ๐๐๐. ๐๐
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QUEUING MODELSM/M/s/GD/K/โ
University of the Philippines
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QUEUING MODELSM/M/s/GD/K/โ
University of the Philippines
An amusement arcade has 5 videoke booths where customers can belttheir hearts out. Customers arrive at a Poisson rate of 4 groups per hour.Each group occupies one booth for an exponential time with mean of 1 hour.If a group finds no vacant videoke booth, the group will then leave.
a. Find the average number of occupied videoke booths.
b. What is the average time a group spends in queue before it can use thebooth?
c. Find the proportion of time that a given booth is occupied.
d. On the average, how many groups balk in an hour?
e. Recompute average time a group spent in queue assuming this timethat arriving groups will only balk if there are two groups already waiting,in addition to the five groups in the booths.
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QUEUING MODELSM/M/s/GD/K/โ
University of the Philippines
0 1 2 3 4 5
4/hr 4/hr 4/hr 4/hr 4/hr
1/hr 2/hr 3/hr 4/hr 5/hr
M/M/5/FCFS/5/โ with ฮป = 4/hr and ยต = 1/hr
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QUEUING MODELSM/M/s/GD/K/โ
University of the Philippines
a. Find the average number of occupied videoke booths.
๐ =๐
๐ ๐=
4
5(1)= 0.80
๐0 = ๐=0
๐ ฮค๐ ๐ ๐
๐!+
ฮค๐ ๐ ๐
๐ !
๐=๐ +1
๐พ
๐๐โ๐ โ1
๐0 = ๐=0
5 ฮค4 1 ๐
๐!+
ฮค4 1 5
5!
๐=6
5
0.80๐โ5โ1
= 0.0233
๐ฟ๐ =๐ ฮค๐ ๐ ๐ ๐0 )1 โ ๐๐พโ๐ โ ๐พ โ ๐ ๐๐พโ๐ (1 โ ๐
๐ ! 1 โ ๐ 2
๐ฟ๐ =0.80 ฮค4 1 5(0.0233) )1 โ 0.85โ5 โ 5 โ 5 0.85โ5(1 โ 0.80
5! 1 โ 0.80 2 = 0
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QUEUING MODELSM/M/s/GD/K/โ
University of the Philippines
a. Find the average number of occupied videoke booths.
๐1 =(๐/๐)๐๐0
๐!=(4/1)10.0233
1!= 0.0933
๐2 =(๐/๐)๐๐0
๐!=(4/1)20.0233
2!= 0.1866
๐3 =(๐/๐)๐๐0
๐!=(4/1)30.0233
3!= 0.2488
๐4 =(๐/๐)๐๐0
๐!=(4/1)40.0233
4!= 0.2488
๐5 =(๐/๐)๐๐0
๐!=(4/1)50.0233
5!= 0.1991
๐ฟ = ๐ฟ๐ + ๐=0
๐ โ1
๐๐๐ + ๐ 1 โ ๐=0
๐ โ1
๐๐ = 0 +๐=0
4
๐๐๐ + 5 1 โ ๐=0
4
๐๐
๐ฟ = ๐. ๐๐๐๐ ๐๐๐๐๐๐
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QUEUING MODELSM/M/s/GD/K/โ
University of the Philippines
b. What is the average time a group spends in queue before it can use thebooth?
c. Find the proportion of time that a given booth is occupied.
d. On the average, how many groups balk in an hour?
)๐๐๐ฃ๐ = ๐(1 โ ๐๐พ = 4 1 โ 0.1991 = 3.204 ๐๐๐๐ข๐๐ /โ๐
๐๐ =๐ฟ๐๐๐๐ฃ๐
=0
3.204= ๐ ๐๐ = ๐๐๐๐
% ๐๐ก๐๐ =๐๐๐ฃ๐๐ ยต
=3.204
5(1)= ๐๐. ๐๐%
๐๐๐๐๐ = ๐๐๐พ = 4 0.1991 = ๐. ๐๐๐ ๐๐๐๐๐/๐๐
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QUEUING MODELSM/M/s/GD/K/โ
University of the Philippines
๐1 =4
1๐0 =
4
1(0.0233) = 0.0933
๐2 =4
2๐1 =
4
2(0.0933) = 0.1866
๐3 =4
3๐2 =
4
3(0.1866) = 0.2488
๐4 =4
4๐3 =
4
4(0.2488) = 0.2488
๐5 =4
5๐4 =
4
5(0.2488) = 0.1991
๐0 = 1 +4
1+4(4)
1(2)+4(4)(4)
1(2)(3)+4(4)(4)(4)
1(2)(3)(4)+4(4)(4)(4)(4)
1(2)(3)(4)(5)
โ1
= 0.0233
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QUEUING MODELSM/M/s/GD/K/โ
University of the Philippines
a. Find the average number of occupied videoke booths.
b. What is the average time a group spends in queue before it can use thebooth?
๐ฟ = ๐=0
K
๐๐๐ = 0 0.0233 + 1 0.0933 + โฆ+ 5 0.1991 = ๐. ๐๐๐๐ ๐๐๐๐๐๐
๐๐ =๐ฟ๐๐๐๐ฃ๐
=0
3.204= ๐ ๐๐ = ๐๐๐๐
๐ฟ๐ = ๐>s
(๐ โ ๐ )๐๐ = 0 ๐๐๐๐ข๐
๐๐๐ฃ๐ =๐=0
K
๐๐๐๐ = 4 0.0233 + 4 0.0933 + โฆ+ 0 0.1991 = 3.204 ๐๐๐๐ข๐๐ /โ๐
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QUEUING MODELSM/M/s/GD/K/โ
University of the Philippines
c. Find the proportion of time that a given booth is occupied.
d. On the average, how many groups balk in an hour?
%๐๐ก๐๐ =0
50.0233 +
1
50.0933 +
2
50.1866 + โฏ+
5
50.1991 = ๐๐. ๐๐%
๐๐๐๐๐ = 0 0.0233 + 0 0.0933 + โฆ+ 4 0.1991 = ๐. ๐๐๐ ๐๐๐๐๐/๐๐
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QUEUING MODELSM/M/s/GD/K/โ
University of the Philippines
0 1 2 3 4 5 6 7
4/hr 4/hr 4/hr 4/hr 4/hr 4/hr 4/hr
1/hr 2/hr 3/hr 4/hr 5/hr 5/hr 5/hr
M/M/5/FCFS/7/โ with ฮป = 4/hr and ยต = 1/hr
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QUEUING MODELSM/M/s/GD/K/โ
University of the Philippines
e. Recompute average time spent in queue assuming this time thatarriving groups will only balk if there are two groups already waiting, inaddition to the five groups in the booths.
๐0 = 1 +4
1+4(4)
1(2)+4(4)(4)
1(2)(3)+ โฏ+
4(4)(4)(4)(4)(4)
1(2)(3)(4)(5)(5)+4(4)(4)(4)(4)(4)(4)
1(2)(3)(4)(5)(5)(5)
โ1
๐0 = 0.0181
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QUEUING MODELSM/M/s/GD/K/โ
University of the Philippines
e. Recompute average time spent in queue assuming this time thatarriving groups will only balk if there are two groups already waiting, inaddition to the five groups in the booths.
๐1 =4
1๐0 =
4
1(0.181) = 0.0725
๐2 =4
2๐1 =
4
2(0.0725) = 0.1450
๐3 =4
3๐2 =
4
3(0.1450) = 0.1934
๐4 =4
4๐3 =
4
4(0.1934) = 0.1934
๐5 =4
5๐4 =
4
5(0.1934) = 0.1547
๐6 =4
5๐5 =
4
5(0.1547) = 0.1238
๐7 =4
5๐6 =
4
5(0.1238) = 0.0990
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QUEUING MODELSM/M/s/GD/K/โ
University of the Philippines
e. Recompute average time spent in queue assuming this time thatarriving groups will only balk if there are two groups already waiting, inaddition to the five groups in the booths.
๐ฟ๐ = ๐>s
(๐ โ ๐ )๐๐ = 1๐6 + 2๐7 = 1 0.1238 + 2 0.0990 = 0.3218 ๐๐๐๐ข๐
๐๐๐ฃ๐ =๐=0
K
๐๐๐๐ = 4 ๐0 + ๐1 + ๐2 + ๐3 + ๐4 + ๐5 + ๐6 + 0๐7 = 4 1 โ ๐7
๐๐๐ฃ๐ = 4 1 โ 0.990 = 3.604 ๐๐๐๐ข๐๐ /โ๐
๐๐ =๐ฟ๐๐๐๐ฃ๐
=0.3218
3.604= ๐. ๐๐๐๐ ๐๐ = ๐. ๐๐๐๐๐๐๐
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QUEUING MODELSM/M/s/GD/K/โ
University of the Philippines
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QUEUING MODELSM/M/1/GD/N/N
University of the Philippines
For n > N, Pn = 0.
For n โค N:
0 1 2 . . .
Nฮป (N-1)ฮป
ฮผ ฮผ ฮผ
N-1
ฮป
ฮผ
N
ฮผ
2ฮป(N-2)ฮป
๐0 = ๐=0
๐ ๐!
๐ โ ๐ !
๐
๐
๐ โ1
๐๐ =๐!
๐ โ ๐ !
๐
๐
๐
๐0
๐ฟ = ๐ โ๐
๐1 โ ๐0
๐ฟ๐ = ๐ โ๐ + ๐
๐1 โ ๐0
เตฏ๐๐๐ฃ๐ = าง๐ = ๐(๐ โ ๐ฟ
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QUEUING MODELSM/M/s/GD/N/N
University of the Philippines
For n > N, Pn = 0.
For n โค N:
0 1 s. . .
Nฮป (N-1)ฮป 2ฮป
ฮผ 2ฮผ sฮผ
N-1
ฮป
N
sฮผsฮผ
(N-s+1)ฮป
. . .
sฮผ
(N-s)ฮป
๐0 = ๐=0
๐ โ1
๐ถ๐๐
๐
๐
๐
+ ๐=๐
๐ ๐ถ๐๐๐!
๐ ! ๐ ๐โ๐ ๐
๐
๐ โ1
๐๐ =
๐ถ๐๐
๐
๐
๐
๐0, ๐ โค ๐
.
๐ถ๐๐๐!
๐ ! ๐ ๐โ๐ ๐
๐
๐
๐0, ๐ โฅ ๐
๐ฟ๐ = ๐=๐
๐
เตซ๐ โ ๐ )๐๐
๐ฟ = ๐ฟ๐ + ๐=0
๐ โ1
๐๐๐ + ๐ 1 โ ๐=0
๐ โ1
๐๐
เตฏ๐๐๐ฃ๐ = าง๐ = ๐(๐ โ ๐ฟ
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QUEUING MODELSM/M/s/GD/N/N
University of the Philippines
A printing business has five printing machines used for high-volume orders.All throughout the day, these machines are running, except for a few timeswhen they are required to be attended by a printing employee to setmachine parameters. In some instances, the machines may wait for anemployee to be available since the employees may be attending othermachines for setup. The business employs two printing employees. Eachemployee can setup a machine at an exponential time with mean of 20mins. After setup, each machine runs for an exponential time with mean 2hrs before requiring another setup.
a. What is the proportion of time that the both employees are idle as allmachines are working?
b. On the average, how long does a machine queue before being attendedby an employee for setup?
c. A cost of PhP 1,000 per hour is incurred if a machine is not running dueto lost machine productivity, find the total expected cost of waiting in anhour considering all the machines.
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QUEUING MODELSM/M/s/GD/N/N
University of the Philippines
0 1 2 3 4 5
2.5/hr 2.0/hr 1.5/hr 1.0/hr 0.5/hr
3/hr 6/hr 6/hr 6/hr 6/hr
M/M/2/FCFS/5/5 with ฮป = 0.5/hr and ยต = 3/hr
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QUEUING MODELSM/M/s/GD/N/N
University of the Philippines
a. What is the proportion of time that the both employees are idle as allmachines are working?
b. On the average, how long does a machine queue before being attendedby an employee for setup?
๐0 = ๐=0
๐ โ1
๐ถ๐๐
๐
๐
๐
+ ๐=๐
๐ ๐ถ๐๐๐!
๐ ! ๐ ๐โ๐ ๐
๐
๐ โ1
๐0 = ๐=0
1
๐ถ๐5
0.5
3
๐
+ ๐=2
5 ๐ถ๐5๐!
2! 2๐โ20.5
3
๐ โ1
= ๐. ๐๐๐
๐1 = ๐ถ๐๐
๐
๐
๐
๐0 = ๐ถ15
0.5
3
1
0.456 = 0.3800
๐2 = ๐ถ๐๐
๐
๐
๐
๐0 = ๐ถ25
0.5
3
2
0.456 = 0.1267
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QUEUING MODELSM/M/s/GD/N/N
University of the Philippines
b. On the average, how long does a machine queue before being attendedby an employee for setup?
๐3 =๐ถ๐๐๐!
๐ ! ๐ ๐โ๐ ๐
๐
๐
๐0 =๐ถ353!
2! 23โ20.5
3
๐
0.456 = 0.0317
๐4 =๐ถ๐๐๐!
๐ ! ๐ ๐โ๐ ๐
๐
๐
๐0 =๐ถ454!
2! 24โ20.5
3
๐
0.456 = 0.0053
๐5 =๐ถ๐๐๐!
๐ ! ๐ ๐โ๐ ๐
๐
๐
๐0 =๐ถ555!
2! 25โ20.5
3
๐
0.456 = 0.0004
๐ฟ๐ = ๐=๐
๐
เตซ๐ โ ๐ )๐๐ = 0๐2 + 1๐3 + 2๐4 + 3๐5 = 0.0435 ๐๐๐โ๐๐๐๐
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QUEUING MODELSM/M/s/GD/N/N
University of the Philippines
b. On the average, how long does a machine queue before being attendedby an employee for setup?
๐ฟ = ๐ฟ๐ + ๐=0
๐ โ1
๐๐๐ + ๐ 1 โ ๐=0
๐ โ1
๐๐ = 0.0435 + ๐=0
1
๐๐๐ + 2 1 โ ๐=0
1
๐๐
๐ฟ = 0.0435 + 0 0.3800 + 1 0.1267 + 2 1 โ 0.3800 โ 0.1267 = 0.7516 ๐๐๐โ๐๐๐๐
)๐๐๐ฃ๐ = ๐(N โ ๐ฟ = 0.5 5 โ 0.7516 = 2.1242 ๐๐๐โ๐๐๐๐ /โ๐
๐๐ =๐ฟ๐๐๐๐ฃ๐
=0.0435
2.1242= ๐. ๐๐๐๐ ๐๐ = ๐. ๐๐๐๐๐๐๐๐
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QUEUING MODELSM/M/s/GD/N/N
University of the Philippines
c. A cost of PhP 1,000 per hour is incurred if a machine is not running dueto lost machine productivity, find the total expected cost of waiting in anhour considering all the machines.
๐ป๐๐ข๐๐๐ฆ ๐๐๐๐ก๐๐๐ ๐ถ๐๐ ๐ก = 1000๐๐๐๐ฃ๐ = 1000๐ฟ = 1000 0.7516 = ๐ท๐๐ท ๐๐๐. ๐๐
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QUEUING MODELSM/M/s/GD/N/N
University of the Philippines
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QUEUING MODELSM/M/s/GD/N/N
University of the Philippines
A hospital wing has a supply station where nurses retrieve items importantto their patients. The station is manned by a single supply officer. The winghas 6 nurses. On the average, each nurse will use be back to the station1.25 hours after his or her last visit. Assume exponential time. The supplyofficer takes an exponential time with mean of 6 minutes to service therequirement of a nurse.
a. Find the probability that there are exactly 2 nurses, including the onebeing served, in the supply station.
b. Find the proportion of time the supply officer is busy.
c. How long, on the average, does a nurse stay in the supply station beforeher transaction is completed.
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QUEUING MODELSM/M/s/GD/N/N
University of the Philippines
0 1 2 3 4 5 6
4.8/hr 4.0/hr 3.2/hr 2.4/hr 1.6/hr 0.8/hr
10/hr 10/hr 10/hr 10/hr 10/hr 10/hr
M/M/1/FCFS/6/6 with ฮป = 0.8/hr and ยต = 10/hr
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QUEUING MODELSM/M/s/GD/N/N
University of the Philippines
๐1 =4.8
10๐0 =
4.8
10(0.5712) = 0.2742
๐2 =4.0
10๐1 =
4.0
10(0.2742) = 0.1097
๐3 =3.2
10๐2 =
3.2
10(0.1097) = 0.0351
๐4 =2.4
10๐3 =
2.4
10(0.0351) = 0.0084
๐5 =1.6
10๐4 =
1.6
10(0.0084) = 0.0013
๐0 = 1 +4.8
10+4.8(4.0)
10(10)+ โฏ+
4.8(4.0)(3.2)(2.4)(1.6)(0.8)
10(10)(10)(10)(10)(10)
โ1
= 0.5712
๐6 =0.8
10๐5 =
0.8
10(0.0013) = 0.0001
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QUEUING MODELSM/M/s/GD/N/N
University of the Philippines
a. Find the probability that there are exactly 2 nurses, including the onebeing served, in the supply station.
b. Find the proportion of time the supply officer is busy.
๐2 = ๐. ๐๐๐๐
% ๐๐ก๐๐ =0
1๐0 +
1
1๐1 + ๐2 + ๐3 + ๐4 + ๐5 + ๐6 = 1 โ ๐0
%๐๐ก๐๐ = 1 โ 0.5712 = ๐๐. ๐๐%
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QUEUING MODELSM/M/s/GD/N/N
University of the Philippines
c. How long, on the average, does a nurse stay in the supply station beforeher transaction is completed?
๐ฟ = ๐=0
N
๐๐๐ = 0 0.5712 + 1 0.2742 + โฆ+ 6 0.0001 = 0.6399 ๐๐ข๐๐ ๐
๐๐๐ฃ๐ =๐=0
K
๐๐๐๐ = 4.8๐0 + 4.0๐1 + 3.2๐2 + 2.4๐3 + 1.6๐4 + 0.8๐5 + 0๐6
๐๐๐ฃ๐ = 4.8 0.05712 + 4.0 0.2742 +โฏ+ 0 0.0001 = 4.2881 ๐๐ข๐๐ ๐๐ /โ๐
๐ =๐ฟ
๐๐๐ฃ๐=0.6399
4.2881= ๐. ๐๐๐๐ ๐๐ = ๐. ๐๐๐๐๐๐๐๐
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QUEUING MODELSM/M/s/GD/N/N
University of the Philippines
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QUEUING MODELSM/G/1
University of the Philippines
Pollaczek-Khinchin Equation, where variance of service time is given as ฯ2.
๐ฟ๐ =(๐๐)2+ ๐/๐ 2
2 1โ๐
๐
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QUEUING MODELSM/G/1
University of the Philippines
Customers arrive randomly to a bank at a rate of 1 customer every 3minutes. To service customers, the bank is considering two alternatives: (a)a human teller with random service time of mean 2 minutes, and (b) ATMwith a normal service time with mean 2 minutes and standard deviation of0.3 mins. Random times are modelled using the exponential distribution.Which one should be preferred in order to minimize customer queuing time?
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QUEUING MODELSM/G/1
University of the Philippines
a. human teller with random service time of mean 2 minutes
M/M/1 with ฮป = 20/hr and ยต = 30/hr
b. ATM with a normal service time with mean 2 minutes and standarddeviation of 0.3 mins.
M/G/1 with ฮป = 20/hr, ยต = 30/hr, and ฯ = 0.3/60 hr = 0.005 hr
๐๐ =๐
)๐(๐ โ ๐=
20
)30(30 โ 20= ๐. ๐๐๐ ๐๐ = ๐๐๐๐๐
๐ฟ๐ =(๐๐)2+ ๐/๐ 2
2(1 โ ๐/๐ )=(20 โ 0.005)2+ 20/30 2
2(1 โ 20/30 )= 0.682 ๐๐ข๐ ๐ก๐๐๐๐
๐๐ =๐ฟ๐๐๐๐ฃ๐
=0.682
20= ๐. ๐๐๐ ๐๐ = ๐. ๐๐๐๐๐๐๐
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QUEUING MODELSM/G/1
University of the Philippines
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QUEUING MODELSM/G/1
University of the Philippines
Compare the performance measures โ L, Lq, W, Wq โ of the followingqueueing systems: (A) M/M/1, (B) M/D/1, and (C) M/G/1 (where theservice time is uniformly distributed from 2 to 4 minutes). Assume similaraverage service rate of 20/hr, and similar arrival rate of 10/hr. Include a 4thcase (D) where the server performs two tasks on a single customer. Theservice time for each task is exponentially distributed with mean of 1.5minutes for this last case.
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QUEUING MODELSM/G/1
University of the Philippines
Computations of standard deviation ฯ:
a. exponential with rate ฮปexpo = 20/hr
b. Constant (deterministic)
c. continuous uniform with minimum a = 2/60 hr and maximum b =4/60 hr.
๐ =1
๐๐๐ฅ๐๐2 =
1
202= ๐. ๐๐๐๐ ๐๐
๐ = ๐ ๐๐
๐ =๐ โ ๐ 2
12=
(4/60) โ (2/60) 2
12= ๐. ๐๐๐๐ ๐๐
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QUEUING MODELSM/G/1
University of the Philippines
Computations of standard deviation ฯ:
d. exponential with ฮปexpo = 40/hr + exponential with ฮปexpo = 40/hr =erlang with shape parameter k = 2 and ฮปexpo = 40/hr
๐ =๐
๐๐๐ฅ๐๐2 =
2
402= ๐. ๐๐๐๐ ๐๐
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QUEUING MODELSM/G/1
University of the Philippines
Sample computations for case (D)
๐ฟ๐ =(๐๐)2+ ๐/๐ 2
2(1 โ ๐/๐ )=(10 โ 0.0354)2+ 10/20 2
2(1 โ 10/20 )= ๐. ๐๐๐ ๐๐๐๐๐๐๐๐
๐๐ =๐ฟ๐๐๐๐ฃ๐
=0.375
10= ๐. ๐๐๐๐ ๐๐
๐ = ๐๐ +1
๐= 0.0375 +
1
20= ๐. ๐๐๐๐ ๐๐
๐ฟ = ๐๐๐ฃ๐๐ = 10 0.0875 = ๐. ๐๐๐ ๐๐
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QUEUING MODELSM/G/1
University of the Philippines
Case A B C D
Kendall-Lee Notation M/M/1 M/D/1 M/G/1 M/E2/1
Service Time Distribution exponential constant uniform erlang
Arrival Rate, ฮป 10/hr 10/hr 10/hr 10/hr
Service Rate, ยต 20/hr 20/hr 20/hr 20/hr
Service Time Std Dev, ฯ 0.0500 hr 0.0000 hr 0.0167 hr 0.0354 hr
Lq (customers) 0.500 0.250 0.278 0.375
L (customers) 1.000 0.750 0.778 0.875
Wq (hrs) 0.050 0.025 0.028 0.038
W (hrs) 0.100 0.075 0.078 0.088
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QUEUING MODELSM/G/1
University of the Philippines
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QUEUING MODELSOther Models: Cooperating Servers
University of the Philippines
A car wash shop has 3 employees. Each employee can wash a car in anexponential time with mean of 45 mins. However, when there are only twocars in the shop, one of the employees will help one of the other two,reducing the average time from 45 mins to 30 mins. Meanwhile, thecustomer being served by one employee will still have a service timeaverage of 45 mins. Note than when a new vehicle arrives, one of thecooperating employee will leave its partner and will now assist the newlyarrived customer. In that case, service time average goes back to 45 minsper customer.
If there is a single customer in the shop, only two employees will wash thevehicle, with service time average of 30 mins. The third employee will beidle during this time.
Cars arrive to this system at a Poisson rate of 5 customers per hour.However, customers balk when two cars are already waiting in the shop(excluding those being washed).
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QUEUING MODELSOther Models: Cooperating Servers
University of the Philippines
a. Find the expected revenue per hour of the shop if each vehicle washedgenerates PhP 200.
b. Find the average number of vehicles present in the shop.
c. On the average, how long does a vehicle stay in the car wash?
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QUEUING MODELSOther Models: Cooperating Servers
University of the Philippines
0 1 2 3 4 5
5/hr 5/hr 5/hr 5/hr 5/hr
2/hr (1.33+2)/hr 3(1.33)/hr 3(1.33)/hr 3(1.33)/hr
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QUEUING MODELSOther Models: Cooperating Servers
University of the Philippines
๐1 =5
2๐0 =
5
2(0.0398) = 0.0994
๐2 =5
3.33๐1 =
5
3.33(0.0994) = 0.1493
๐3 =5
4๐2 =
5
4(0.1493) = 0.1866
๐4 =5
4๐3 =
5
4(0.1866) = 0.2333
๐5 =5
4๐4 =
5
4(0.2333) = 0.2916
๐0 = 1 +5
2+
5(5)
2(3.33)+
5(5)(5)
2(3.33)(4)+ โฏ+
5(5)(5)(5)(5)
2(3.33)(4)(4)(4)
โ1
= 0.0398
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QUEUING MODELSOther Models: Cooperating Servers
University of the Philippines
a. Find the expected revenue per hour of the shop if each vehicle washedgenerates PhP 200.
b. Find the average number of vehicles present in the shop.
๐๐๐ฃ๐ =๐=0
5
๐๐๐๐ = 5 ๐0 + ๐1 + ๐2 + ๐3 + ๐4 + 0๐5 = 5 1 โ ๐5
๐๐๐ฃ๐ = 5 1 โ 0.2916 = 3.542 ๐ฃ๐โ๐๐๐๐๐ /โ๐
๐ป๐๐ข๐๐๐ฆ ๐ ๐๐ฃ๐๐๐ข๐ = 200๐๐๐ฃ๐ = 200 3.542 = ๐ท๐๐ท ๐๐๐. ๐๐
๐ฟ = ๐=0
5
๐๐๐ = 0 0.0398 + 1 0.0994 + โฆ+ 5 0.2916 = ๐. ๐๐๐ ๐๐๐๐๐๐๐๐
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QUEUING MODELSOther Models: Cooperating Servers
University of the Philippines
c. On the average, how long does a vehicle stay in the car wash?
๐ =๐ฟ
๐๐๐ฃ๐=3.349
3.542= ๐. ๐๐๐ ๐๐ = ๐๐. ๐๐๐๐๐๐๐
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QUEUING MODELSOther Models: Balking Customers
University of the Philippines
A newly-opened tea shop is very popular to mall-goers. The Poisson arrivalsof customers to the said shop has rate of ฮป = 26 per hour. The shop has asingle employee. The mean service time is 2.5 mins. Assume that theservice time is exponentially distributed.
Though popular, some clients may not go inside when there is a relativelylong queue. They will instead go to other tea shops in the mall. Specifically,an arriving customer will balk with probability 0.30 if there are 3 or 4customers inside (including the one being served). This probability ofbalking goes up to 0.75 if there are 5 or 6 customers inside. No customerenters the queue when there are 7 customers.
a. What is the probability that the employee is idle?
b. Find the average waiting time before a customer is entertained by theemployee.
c. How many customers are lost per hour due to balking?
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QUEUING MODELSOther Models: Balking Customers
University of the Philippines
0 1 2 3 4 5 6 7
26/hr 26/hr 26/hr 18.2/hr 18.2/hr 6.5/hr 6.5/hr
24/hr 24/hr 24/hr 24/hr 24/hr 24/hr 24/hr
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QUEUING MODELSOther Models: Balking Customers
University of the Philippines
๐0 = 0.1544
๐1 = 0.1673
๐2 = 0.1812
๐3 = 0.1963
๐4 = 0.1489
๐5 = 0.1129
๐6 = 0.0306
๐7 = 0.0083
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QUEUING MODELSOther Models: Balking Customers
University of the Philippines
a. What is the probability that the employee is idle?
b. Find the average waiting time before a customer is entertained by theemployee.
๐0 = ๐. ๐๐๐๐
๐ฟ๐ = ๐=๐
7
เตซ๐ โ ๐ )๐๐ = 0๐1 + 1๐2 + 2๐3 +โฏ+ 6๐7
๐ฟ๐ = 0 0.1673 + 1 0.1812 + 2 0.1963 + โฏ+ 6 0.0083 = 1.675 ๐๐ข๐ ๐ก๐๐๐๐๐
๐๐๐ฃ๐ =๐=0
7
๐๐๐๐ = 26๐0 + 26๐1 + 26๐2 + 18.2๐3 + 18.2๐4 + 6.5๐5 + 6.5๐6 + 0๐7
๐๐๐ฃ๐ = 26 0.1544 + 26 0.1673 + โฏ+ 0 0.0083 = 20.294 ๐๐ข๐ ๐ก๐๐๐๐๐ /โ๐
๐๐ =๐ฟ๐๐๐๐ฃ๐
=1.675
20.294= ๐. ๐๐๐ ๐๐ = ๐. ๐๐๐๐๐๐๐
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QUEUING MODELSOther Models: Balking Customers
University of the Philippines
c. How many customers are lost per hour due to balking?
๐๐๐๐๐ = ๐ โ ๐๐๐ฃ๐ = 26 โ 20.294 = ๐. ๐๐๐ ๐๐๐๐๐๐๐๐๐ ๐๐๐ ๐๐๐๐
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QUEUING MODELSReferences
University of the Philippines
โช Hillier F. and G. Lieberman. Introduction to Operations Research, 9th ed.New York: McGraw-Hill Higher Education, 2010.
โช Taha, Hamdy. Operations Research: An Introduction, 8th ed. New Jersey:Pearson Education Ltd, 2007.
โช Wayne, Winston. Operations Research: Applications and Algorithms, 4th
ed. Belmont, CA: Thomson Brooks / Cole, 2004.