Download - 4 Manual Simulation
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Kleber Barcia V.
Chapter 4
Manual Simulation
Manual simulation of discrete
events
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Important concepts ...
System
Model
System state
Entity
Attributes
Event
Activities
A list is a collection of permanent or temporary entities sorted under a certain logic. Example: Customer ordered under FIFO queue.
An event notice is the record of an event occurring or will occur in the future associated with the data necessary for the execution of the event.
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Important concepts ...
A event list is a list of event notices for future events sorted based on time of occurrence. Also known as Future Event List (FEL).
An activity is typically a service time or interarrival time.
The duration of an activity can be specified in three different ways:
1. Deterministic (5 minutes)
2. Statistical (as random numbers 2, 5, 7 with equal probability)
3. As a function depending on the variables of the system (the loading time of a vessel as a function of loading rate ton/hr)
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Example 1: Queue system with two servers
A computer technical support center is staffed by two people, Abel and
Baker, who take calls and try to answer questions and solve computer
problems. The time between calls ranges from 1 to 4 min. Service time
varies depending on the employee between 2 to 6 min. Abel is more
experienced and can provide service faster than Baker. Abel has
priority to serving customers.
The discrete event model has the following components:
System State
LQ (t) = Number of callers waiting to be served at time t
LA (t) = 0 or 1 indicating that Abel is idle or busy at time t
LB (t) = 0 or 1 indicating that Baker is idle or busy at time t
Entities
Neither clients nor servers are represented explicitly in the model
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Example 1: Queue system with two servers
Events
Events of arrival
The completion of service by Abel
The completion of service by Baker
Activities
Interarrival time defined in the simulation table
Abel service time defined in the simulation table
Baker service time defined in the simulation table
Delay
The customer waiting in queue until Abel and Baker are available to
address them
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The Event Scheduling/Time Advance Algorithm
It is the mechanism that ensures that all events occurring in the
chronological order in which advances the simulation time. It is
based on the list of future events (FEL)
Example 2: Single Server Queue
A shop has one dependent. Customers arrive randomly at
intervals of 1-8 min. with an equal chance of occurrence. The
service time varies from 1-6 min. with a given probability.
Find the total time the server is busy (B) and the number of
customers waiting in queue (MQ) in a simulation of 60 min.
The discrete event model has the following components:
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Example 2: Single Server Queue
System State
LQ (t) = number of customers waiting in queue at time t
LS (t) = number of customers being served at time t
Entities
Customers are not explicitly represented in the model
Events
Arrivals (A)
Outputs (D)
Stopping event (E) scheduled to occur at time 60
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Example 2: Single Server Queue
Future Event List
(A, t) Arrival Client will happen at some future time t
(D, t) out of the client that will occur at a future time t
(E, t) Stop the simulation will happen in time 60
Activities
Interarrival time
Service time
Delay
While the customer waits in queue
Interarrival time 8 6 1 8 3 8
Service time 4 1 4 3 2 4
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Example 2: Single Server Queue Simulation Table
System State Cumulative
Statistics
Clock LQ(t) LS(t) Future Event List Comment B MQ
0 0 1 (D,4) (A,8) (E,60) First A occurs 0 0
(a*=8) Schedule next A
(s*=4) Schedule first D
4 0 0 (A,8) (E,60) First D occurs: (D,4) 4 0
8 0 1 (D,9) (A,14) (E,60) Second A occurs:(A,8) 4 0
(a*=6) Schedule next A
(s*=1) Schedule next D
9 0 0 (A,14) (E,60) Second D occurs:(D,9) 5 0
14 0 1 (A,15) (D,18) (E,60) Third A occurs:(A,14) 5 0
(s*=4) Schedule next D
15 1 1 (D,18) (A,23) (E,60) Fourth A occurs:(A,15) 6 1
(Customer delayed)
18 0 1 (D,21) (A,23) (E,60) Third D occurs: (D,18) 9 1
(s*=3) Schedule next D
21 0 0 (A,23) (E,60) Fourth D occurs:(D,21) 12 1
a * = Interarrival time s * = service time Rev. 1
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Example 3: Extension of Example 2
Suppose that in the previous example is required to estimate the average time that the customer spends in the system and proportion of customers who spend 4 or more minutes in the system.
To perform this calculation the system is expanded with the following components:
Entities
(Ci, t) represents the client Ci at a time t. Customers are explicitly represented in the model.
Future Event List
(A, t, Ci) represents the arrival of client Ci at a future time t
(D, t, Cj) represents the output of Cj client at a future time t
Three new statistics should be calculated:
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Example 3: Extension of Example 2
S = the sum of the time customers spend in the system
F = The amount of customers who spend 4 or more minutes being attended
ND = The number of customers leaving the system
Considering the simulation table, the average time the customer spends in the system is:
S/ND = 15/4 = 3.75 minutes
The proportion of customers who spend 4 or more minutes in the system is:
F/ND = = 0.75
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Example 3: Extension of example 2 Simulation Table
System State List Future Event
Cumulative
Statistic
Clock LQ(t) LS(t) CHECKOUT LINE List S ND F
0 0 1 (C1,0) (D,4,C1) (A,8,C2) (E,60) 0 0 0
4 0 0 (A,8,C2) (E,60) 4 1 1
8 0 1 (C2,8) (D,9,C2) (A,14,C3) (E,60) 4 1 1
9 0 0 (A,14,C3) (E,60) 5 2 1
14 0 1 (C3,14) (A,15,C4) (D,18,C3) (E,60) 5 2 1
15 1 1 (C3,14)(C4,15) (D,18,C3) (A,23,C5) (E,60) 5 2 1
18 0 1 (C4,15) (D,21,C4) (A,23,C5) (E,60) 9 3 2
21 0 0 (A,23,C5) (E,60) 15 4 3
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Example 4: Dump Trucks (DT)
Six dump trucks are used to transport coal from the entrance of a
mine to a train transport. Each truck is loaded in two loading areas.
After loaded, the truck is moved to the weighing section. The system
works on the FIFO system. The travel time between the loading area
and weighing is negligible. After weighing, the trucks travel, unload
and return to the loading queue.
Estimate the loader and scale utilizations (percentage of time busy)
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Example 4: Dump Trucks (DT)
Loading
time Probability
Cumulative
Probability
Random
Digits
5 0.30 0.30 1-3
10 0.50 0.8 4-8
15 0.20 1.00 9-0
Weighing
Time Probability
Cumulative
Probability
Random
Digits
12 0.70 0.70 1-7
16 0.30 1.00 8-0
Travel
Time Probability
Cumulative
Probability
Random
Digits
40 0.40 0.40 1-4
60 0.30 0.70 5-7
80 0.20 0.90 8-9
100 0.10 1.00 0
Distribution of weighing time for trucks
Distribution of loading time for trucks
Distribution of travel time for trucks
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Example 4: Dump Trucks (DT)
The discrete event model has the following components:
System State
LQ (t) = Number of trucks waiting in line to load
L (t) = Number of trucks loading (0, 1, or 2)
WQ (t) = Number of trucks waiting in line to weigh
W (t) = Number of trucks weighing (0 or 1)
Entities
The six dump trucks (DT1, DT2, DT3, DT4, DT5 and DT6), are not represented explicitly in the model
Future Event List
(ALQ, t, DTi) Dti arrives at loader queue ALQ at time t
(EL, t, DTi) Dti ends loading EL at time t
(EW, t, DTi) DTi ends weighting EW at time t
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Example 4: Dump Trucks (DT)
List (checkout list)
Loader queue = All trucks waiting to be loading FIFO
Weigh queue = All trucks waiting to be weighed FIFO
Activities
Loading time
Weighing time
Travel time
Delays
Delay at loader queue
Delay at scale
assumptions:
For t = 0, three trucks are at loader queue, two trucks are loading, and one truck is weighing.
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0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150
DT1
DT2
DT3
DT4
DT5
DT6
Wait before loading
Loading
Wait before weighing
weighing
traveling
Loading
time 10 5 5 10 15 10 10
Weighing
time 12 12 12 16 12 16
Travel
time 60 100 40 40 80
5 12 24
25 64 72 36 76 52 92 144
Example 4: Dump Trucks (DT)
SYSTEM ACTIVITIES
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Simulation Table Dump Trucks
System State List Cumulative
Statistics Clock Loader Weigh Future Event
t LQ(t) L(t) WQ(t) W(T) Queue Queue List BL BS
0 3 2 0 1 DT4 (EL,5,DT3) 0 0
DT5 (EL,10,DT2)
DT6 (EW,12,DT1)
5 2 2 1 1 DT5 DT3 (EL,10,DT2) 10 5
DT6 (EL,5+5,DT4)
(EW,12,DT1)
10 1 2 2 1 DT6 DT3 (EL,10,DT4) 20 10
DT2 (EW,12,DT1)
(EL,10+10,DT5)
10 0 2 3 1 DT3 (EW,12,DT1) 20 10
DT2 (EL,20,DT5)
DT4 (EL,10+15,DT6)
where:
BL = Total busy time that spent the two loading areas
BS = total busy time that spent the weighing area
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Simulation Table Dump Trucks
System State List Cumulative
Statistics Clock Loader Weigh Future Event
t LQ(t) L(t) WQ(t) W(T) Queue Queue List BL BS
12 0 2 2 1 DT2 (EL,20,DT5) 24 12
DT4 (EW,12+12,DT3)
(EL,25,DT6)
(ALQ,12+60,DT1)
20 0 1 3 1 DT2 (EW,24,DT3) 40 20
DT4 (EL,25,DT6)
DT5 (ALQ,72,DT1)
24 0 1 2 1 DT4 (EL,25,DT6) 44 24
DT5 (EW,24+12,DT2)
(ALQ,72,DT1)
ALQ,24+100,DT3)
25 0 0 3 1 DT4 (EW,36,DT2) 45 25
DT5 (ALQ,72,DT1)
DT6 (ALQ,124,DT3)
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Simulation Table Dump Trucks
System State List Cumulative
Statistics Clock Loader Weigh Future Event
t LQ(t) L(t) WQ(t) W(T) Queue Queue List BL BS
36 0 0 2 1 DT5 (EW,36+16,DT4) 45 36
DT6 (ALQ,72,DT1)
(ALQ,36+40,DT2)
(ALQ,124,DT3)
52 0 0 1 1 DT6 (EW,52+12,DT5) 45 52
(ALQ,72,DT1)
(ALQ,76,DT2)
(ALQ,52+40,DT4)
(ALQ,124,DT3)
64 0 0 0 1 (ALQ,72,DT1) 45 64
(ALQ,76,DT2)
(EW,64+16,DT6)
(ALQ,92,DT4)
(ALQ,124,DT3)
(ALQ,64+80,DT5)
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Simulation Table Dump Trucks
System State List Cumulative
Statistics Clock Loader Weigh Future Event
t LQ(t) L(t) WQ(t) W(T) Queue Queue List BL BS
72 0 1 0 1 (ALQ,76,DT2) 45 72
(EW,80,DT6)
(EL,72+10,DT1)
(ALQ,92,DT4)
(ALQ,124,DT3)
(ALQ,144,DT5)
76 0 2 0 1 (EW,80,DT6) 49 76
(EL,82,DT1)
(EL,76+10,DT2)
(ALQ,92,DT4)
(ALQ,124,DT3)
(ALQ,144,DT5)
where:
BL = Total busy time that spent the two loading areas
BS = total busy time that spent the weighing area
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Example 4: Dump Trucks (DT)
Percentage of time busy of the loader area:
(49/2) / 76 = 0.32
Percentage of time busy of the scale area:
76/76 = 1.00
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