Transcript IE 324 SIMULATION

DISCRETE-EVENT SIMULATION CONCEPTS
and
EVENT SCHEDULING ALGORITHM
Discrete Event Simulation
Is concerned with modelling of a system as it
evolves over time by a representation in which the
system variables change at discrete points on time
Characteristics:
• the passage of time is a part of simulation (Dynamic)
• usually contains random elements (Stochastic)
Discrete-Event Simulation Concepts
• System: A collection of entities (e.g. people and
machines) that interact together over time to
accomplish one or more goals
• Model: An abstract representation of a system,
usually containing structural, logical, or
mathematical relationships that describe a system
in terms of its state, entities and their attributes,
sets, processes, events, activities and delays.
• System State: A collection of variables that
contain all the information necessary to describe a
system at any point in time
Discrete-Event Simulation Concepts
• Entity: Any object or component in a system which
requires explicit representation in the model (e.g. a server,
a customer, a machine)
• Attributes: Properties of a given entity (e.g. priority of a
waiting customer, the routing number of a job through a
job shop)
• List: A collection of (permanently or temporarily)
associated entities, ordered in some logical fashion (e.g.
customers currently waiting in a queue ordered according
to a queueing discipline) [sets, queues, chains]
Discrete-Event Simulation Concepts
• Event: An instantaneous occurrence that changes
the state of a system (e.g. arrival of a new
customer)
• Event Notice: A record of an event to occur at the
current or some future time, along with any
associated data to execute the event (at a
minimum: event type, event time)
• Event List: A list of event notices for future
events, ordered by time of occurrence, also know
as future event list (FEL)
Discrete-Event Simulation Concepts
• Activity: A duration of time of specified length (e.g.
service time), which is known when it begins.
– deterministic
– random
– function of state variables or entity attributes
(deterministic/random)
• Delay: A duration of time of unspecified length, which is
not directly known until it ends (an event triggers the end)
[conditional wait] (e.g. time a customer spends in a waiting
line)
• Clock: A variable representing simulated time (CLOCK)
Events and System State
A1
A2
S2
0
t1
t2
t3
t4
Time
S1
L(t)
C1
C2
2
1
0
t1
t2
t3
t4
t1
t2
t3
t4
Time
B(t)
1
0
Time
Example: A Doctor’s Office
Example: A Doctor’s Office
• System State
– Q(t), the number of patients waiting to be examined at time t
– B(t), 0 or 1 indicate the doctor being busy or not at time t
• Entities
– Neither the patients nor the doctor need to be explicitly
represented, except in terms of state variables unless certain patient
averages are desired
• Events
– Patient arrival event
– Examination completion event
Example: A Doctor’s Office
• Activities:
– Interarrival time
– Examination time
• Delay
– A patient’s wait until the doctor is free
• What more to specify?
–
–
–
–
Consequences of events ( a logical description)
Activity lengths (deterministic, probabilistic, a function)
Triggers for the beginning and the end of delays
Initial conditions (System state at time 0)
Time Advance
Advancing simulated time
Fixed time increments
Advances the clock by one time unit
& examines the events and
system state
Variable time increments
Advances the clock to the most
imminent event time & examine
the system state
EVENT ORIENTATION
(ACTIVITY SCANNING) Approach
Not an efficient approach for most systems
EVENT SCHEDULING
NEXT EVENT METHOD
Variable Time Increment
• Moves the system from one event time to the next
event time
• Produces a sequence of system snapshots (or
system images) which represent the system
evolution through time
• System evolution can be represented by these
snapshots since between events the system does
not change
Variable Time Increment
• Snapshots at each event time include:
– The current state of the system
– FEL for all activities currently in progress (their
completion times)
– The current membership to all lists
– The status of all entities
– Current values of all cumulative statistics and counters
• Thus only the last snapshot is needed to construct
the next snapshot (Markovian property)
Future Events List (FEL)
• In real world, the future events are not scheduled but
merely happen at a certain time (random arrivals, failures)
• Yet these event times can be modeled by a statistical
characterization (distribution) of the time to next
occurrence (arrival, failure)
• In simulation activity lengths are determined by obtaining
a random sample from the distribution associated with it.
• Completion Time = Starting Time + Activity Length
• The activity completion time is determined as the activity
begins and the activity completion event is added to FEL.
Event-Scheduling
• At any given time t, the FEL contains all previously
scheduled future events and their associated event times.
• The FEL is ordered by event time, which means that the
events are in chronological order.
t < t1 <= t2 <= t3 <= . . . <= tn
• The time t is the value of CLOCK, current value of
simulated time
• The event associated with time t1 is called the imminent
event, which means the next event to occur.
• The snapshot at CLOCK= t is updated
Event-Scheduling
•
•
•
•
Simulation time advances to CLOCK= t1
Imminent event notice about t1 is removed from the FEL
The event is executed
The execution creates a new snapshot of the system at t1
based on the previous snapshot and the nature of the event
• This snapshot may include new future event notices added
to FEL or not(depends on the nature of the executed event)
• And the cycle continues by repeating the same steps after
advancing CLOCK to the time of the new imminent event
until the end of the simulation
System Snapshot
CLOCK
System
State
t
(5,1,6)
...
Future Event List
(4, t1) – Type 4 event to occur at t1
(1, t2) – Type 1 event to occur at t2
(1, t3) – Type 1 event to occur at t3
.
.
.
(3, tn) – Type 3 event to occur at tn
...
System Snapshot
CLOCK
System
State
t1
(5,1,7)
...
Future Event List
(1, t2) – Type 1 event to occur at t2
(2, t*) – Type 2 event to occur at t*
(1, t3) – Type 1 event to occur at t3
.
.
.
(3, tn) – Type 3 event to occur at tn
...
Future Event Generation
• Exogenous Events (Events that are not caused by the
internal dynamics of the system. They happen outside the
system, yet effect the evolution of the system):
– The events that are in the initial system snapshot at time 0
(specified by initial conditions). These are placed to the FEL
before the simulation starts.
– Other external events such as arrivals to a queueing system. These
events are generated using a technique called Bootstrapping
• This technique is based on each event causing another event of the
same type
• The time between the events is an activity (a) (It may be a r.v.)
• When an event occurs, schedule the next event in the same stream of
events and place the associated notice in the FEL
• This next event in the stream is scheduled at t* = t + a
Bootstrapping
Future Event Generation
• Endogenous Events
– Service-completion event
• Each time a customer starts service a service-completion event
is scheduled (End of an activity)
• Note that service-start event is never scheduled, it is called a
conditional event. A service-start occurs within an arrival or a
service-completion event.
• The events that are placed on FEL are called primary events.
– Uptimes and downtimes for a system subject to
breakdown
• End-of-uptime and end-of-downtime events (primary events)
• Similar to bootstrapping
Stopping Simulation Runs
• At time 0, schedule a stop-simulation event at a
specified future time TE ( Simulation runs in the
interval [0, TE] )
• Stop the run at the time of occurrence of some
specified event E. (The completion of the 1000th
service at the server)
World Views
•
•
When using a simulation package or even when doing
hand simulation, a modeler adopts a world view or
orientation for developing a model.
The most prevalent world views are
1. the event-scheduling world view
2. the process-interaction world view
3. the activity-scanning world view.
•
Particular simulation packages (e.g. Arena) usually
support one of these world views.
Event-Scheduling World View
• Concentrates on events and their effects on system state
• It makes use of the Future Event List and variable time
increments
• This is the approach we have just discussed
• It is a very flexible approach using which any system can
be simulated
• It requires lots of mechanisms (e.g. FEL handling) to be
coded, which may slow down the development of the
simulation model for very simple systems.
Event-Scheduling World View
• We have not used this method on any example last week.
We are going to do it now on the doctor’s office system
• This is the world view that is usually used when the
simulation program is directly coded on a general-purpose
programming language such as Java.
• We are going to show its implementation in Java next
week
• It would be hard to use this world view with EXCEL since
the implementation of the standard mechanisms of EventScheduling is not easy in such environments (it could be
done with extensive use of EXCEL macros)
Process-Interaction World View
• Models the system in terms of processes
• Concentrates on entities, and their life cycles as
they flow through the system, demanding
resources and queueing to wait for resources.
• A process is a time-sequenced list of events,
activities and delays that define the life cycle of
one entity as it moves through a system.
• Some activities might require resources that are
limited, which may cause delays in the process
Process-Interaction World View
• This is the world view we adopted for the hand
simulation of the doctor’s office system last week
Process-Interaction World View
• This approach is easier for the modeler since
people usually perceive the systems in terms of the
processes the entities experience
• But this approach is usually harder to implement
directly, especially when the systems to be
simulated are complex
• Thus, many simulation packages (including
ARENA) use this view for the modeler to describe
the system (user-interface), but an eventscheduling approach is used operationally to
perform the simulation run, hidden from the user.
Activity-Scanning World View
• Concentrates on activities and conditions that
allow them to begin
• It uses fixed time increments
• At each clock advance, the conditions for each
activity are checked, and if the conditions hold,
then the corresponding activity starts.
• The (M,N) inventory system simulation we used
last week used activity-scanning approach. The
time increment used was a day.
Activity-Scanning World View
• Simple in concept and leads to modular models
that are easier to maintain, understand and modify.
• The repeated scanning (at each time increment)
results in slow runtime on computers
• The pure activity-scanning approach has been
modified by what is called the three-phase
approach, which combines the features of eventscheduling with activity scanning to allow for
variable time advance. (ARENA uses this as well)
Example: A Doctor’s Office
Example: A Doctor’s Office
• System State (Q(t),B(t))
– Q(t), the number of patients waiting to be examined at time t
– B(t), 0 or 1 indicate the doctor being busy or not at time t
• Entities
– (Pi,t), representing patient Pi who arrived at time t
• Events
– Patient arrival event (A)
– Patient departure event (D)
– Stopping event (E)
Example: A Doctor’s Office
• Event Notices
– (A,t,Pi) represents the arrival of patient Pi at future time t
– (D,t,Pi) represents the departure of patient Pi at future time t
– (E,19) represents the simulation-stop event at future time 19
• Activities
– Interarrival time, described in a following table
– Service time, described in a following table
• Delay
– Patient time spent in the waiting room
• Set
– “Waiting Room”, the set of all patients in the waiting room
– “Doctor’s Room”, the set of all patients in the doctor’s room
Arrival Event
Arrival event occurs
at CLOCK = t
NO
YES
B(t)=1
B(t)=1 ?
Generate service time s &
schedule new departure event at t+s
Generate interarrival time a;
schedule next arrival event at t+a
Collect statistics
Return control to time-advance routine
to continue simulation
Q(t)=Q(t)+1
Service Completion Event
Departure event occurs
at CLOCK = t
B(t)=0
NO
Q(t) 1 ?
YES
Q(t)=Q(t)-1
Generate service time s;
schedule new departure at t+s
Collect statistics
Return control to time-advance routine
to continue simulation
Input Data (Arrival)
Interarrival time
1
2
3
4
5
6
Probability
Cumulative probability
0.30
0.25
0.20
0.15
0.07
0.03
Probability
0.30
0.30
0.55
0.75
0.90
0.97
1.00
Cumulative probability
1.00
0.25
0.80
0.20
0.60
0.15
0.40
0.10
0.20
0.05
0.00
0.00
1 2 3 4 5 6
1 2 3 4 5 6
Input Data (Service)
Service time
1
2
3
4
Probability
Cumulative probability
0.25
0.25
0.25
0.25
Probability
0.25
0.50
0.75
1.00
Cumulative probability
0.25
1.00
0.20
0.80
0.15
0.60
0.10
0.40
0.05
0.20
0.00
0.00
1 2 3 4
1 2 3 4
Generation of interarrival times
Interarrival time
Label1
Cumulative probability
Random digit assignment
0.30
0.25
0.20
0.15
0.07
0.03
01-30
31-55
56-75
76-90
91-97
97-00
1
2
3
4
5
6
Customer No.
Random digit
Interarrival time
1
2
38
2
3
84
4
4
15
1
5
50
2
6
99
…
6
…
…
Generation of service times
Interarrival time
Cumulative probability
Random digit assignment
0.25
0.25
0.25
0.25
01-25
26-50
51-75
76-90
1
2
3
4
Customer No.
Random digit
Service time
1
2
28
24
2
1
3
73
3
4
46
2
5
04
1
6
85
…
4
…
…
Statistics
Discrete time statistics
Continuous time statistics
(Observational data)
Time dependent data
Continuous statistics
Time weighted statistics
Defined relative to the collection of
random variables with a discrete
time index (Observational data)
Defined on the random variables
{Q(t)},{B(t)}, each of which is
indexed on the continuous time
parameter^
Notation
TQ: time integral of Q(t) (from 0 to T)
TB: time integral of B(t) (from 0 to T)
TL: time integral of (B+Q)(t) (from 0 to T)
R: total response time for patients (time-in-the-system) who departed by T
W: total waiting time of patients (time-in-the-queue) who departed by T
N: total number of departures by T
Example: calculation of TL :
T B+Q
TLnew=TLold+(T-Told)(Bold +Qold)
TL
0
1
2
1
0+(2-0)*1=2
3
0
2+(3-2)*1=3
6
1
3+(6-3)*0=3
19
0
13+(19-15)*1=17
0
Simulation table
Clock B(t) Q(t)
0
2
3
6
7
9
11
12
15
19
1
1
0
1
1
1
1
0
1
0
0
0
0
0
1
1
0
0
0
0
DR
WR
(P1,0)
(P2,2)
(P3,6)
(P3,6)
(P4,7)
(P5,9)
(P6,15)
(P4,7)
(P5,9)
FEL
(A,2,P2),(D,2,P1),(E,19)
(D,3,P2),(A,6,P3),(E,19)
(A,6,P3),(E,19)
(A,7,P4),(D,9,P3),(E,19)
(A,9,P5),(D,9,P3),(E,19)
(D,11,P4),(A,15,P6),(E,19)
(D,12,P5),(A,15,P6),(E,19)
(A,15,P6),(E,19)
(D,19,P6),(E,19),(A,24,P7)
(A,24,P7)
R
W
N
TQ TB
0
2
3
3
3
6
10
13
13
17
0
0
0
0
0
2
4
4
4
4
0
1
2
2
2
3
4
5
5
6
0
0
0
0
0
2
4
4
4
4
0
2
3
3
4
6
8
9
9
13
17
4
6
4
13
Another Example of Hand
Simulation using Event-Scheduling
From Chapter 2 of the Arena Book
Machine
(Server)
Arriving
Blank Parts
7
6
5
Queue (FIFO)
4
Departing
Finished Parts
Part in Service
Model Specifics
• Initially (time 0) empty and idle
• Base time units: minutes
• Input data (assume given for now …), in minutes:
Part Number
1
2
3
4
5
6
7
8
9
10
11
.
.
Arrival Time
0.00
1.73
3.08
3.79
4.41
18.69
19.39
34.91
38.06
39.82
40.82
.
.
Interarrival Time
1.73
1.35
0.71
0.62
14.28
0.70
15.52
3.15
1.76
1.00
.
.
.
• Stop when 20 minutes of (simulated) time have passed
Service Time
2.90
1.76
3.39
4.52
4.46
4.36
2.07
3.36
2.37
5.38
.
.
.
44
Simulation by Hand:
Setup
System
Clock
Number of
completed waiting
times in queue
Total of
waiting times in queue
B(t)
Q(t)
Arrival times of
custs. in queue
Area under
Q(t)
Event calendar
Area under
B(t)
4
Q(t) graph
3
2
1
0
B(t) graph
0
5
10
15
20
0
5
10
15
20
2
1
0
Interarrival times
Time (Minutes)
1.73, 1.35, 0.71, 0.62, 14.28, 0.70, 15.52, 3.15, 1.76, 1.00, ...
Service times
2.90, 1.76, 3.39, 4.52, 4.46, 4.36, 2.07, 3.36, 2.37, 5.38, ...
Simulation by Hand:
t = 0.00, Initialize
System
Number of
completed waiting
times in queue
0
Clock
B(t)
Q(t)
0.00
0
0
Arrival times of
Event calendar
custs. in queue
[1, 0.00,
Arr]
<empty> [–, 20.00,
End]
Total of
waiting times in queue
Area under
Q(t)
Area under
B(t)
0.00
0.00
0.00
4
Q(t) graph
3
2
1
0
B(t) graph
0
5
10
15
20
0
5
10
15
20
2
1
0
Interarrival times
Time (Minutes)
1.73, 1.35, 0.71, 0.62, 14.28, 0.70, 15.52, 3.15, 1.76, 1.00, ...
Service times
2.90, 1.76, 3.39, 4.52, 4.46, 4.36, 2.07, 3.36, 2.37, 5.38, ...
Simulation by Hand:
t = 0.00, Arrival of Part 1
System
1
Number of
completed waiting
times in queue
1
Clock
B(t)
Q(t)
Total of
waiting times in queue
Arrival times of
Event calendar
custs. in queue
[2, 1.73,
Arr]
<empty> [1, 2.90,
Dep]
[–, 20.00,
End]
Area under
Area under
Q(t)
B(t)
0.00
1
0
0.00
0.00
0.00
4
Q(t) graph
3
2
1
0
B(t) graph
0
5
10
15
20
0
5
10
15
20
2
1
0
Interarrival times
Time (Minutes)
1.73, 1.35, 0.71, 0.62, 14.28, 0.70, 15.52, 3.15, 1.76, 1.00, ...
Service times
2.90, 1.76, 3.39, 4.52, 4.46, 4.36, 2.07, 3.36, 2.37, 5.38, ...
Simulation by Hand:
t = 1.73, Arrival of Part 2
System
2
1
Number of
completed waiting
times in queue
1
Clock
B(t)
Q(t)
Total of
waiting times in queue
Arrival times of
Event calendar
custs. in queue
[1, 2.90,
Dep]
(1.73) [3, 3.08,
Arr]
[–, 20.00,
End]
Area under
Area under
Q(t)
B(t)
1.73
1
1
0.00
0.00
1.73
4
Q(t) graph
3
2
1
0
B(t) graph
0
5
10
15
20
0
5
10
15
20
2
1
0
Interarrival times
Time (Minutes)
1.73, 1.35, 0.71, 0.62, 14.28, 0.70, 15.52, 3.15, 1.76, 1.00, ...
Service times
2.90, 1.76, 3.39, 4.52, 4.46, 4.36, 2.07, 3.36, 2.37, 5.38, ...
Simulation by Hand:
t = 2.90, Departure of Part 1
System
2
Number of
completed waiting
times in queue
2
Clock
B(t)
Q(t)
Total of
waiting times in queue
Arrival times of
Event calendar
custs. in queue
[3, 3.08,
Arr]
<empty> [2, 4.66,
Dep]
[–, 20.00,
End]
Area under
Area under
Q(t)
B(t)
2.90
1
0
1.17
1.17
2.90
4
Q(t) graph
3
2
1
0
B(t) graph
0
5
10
15
20
0
5
10
15
20
2
1
0
Interarrival times
Time (Minutes)
1.73, 1.35, 0.71, 0.62, 14.28, 0.70, 15.52, 3.15, 1.76, 1.00, ...
Service times
2.90, 1.76, 3.39, 4.52, 4.46, 4.36, 2.07, 3.36, 2.37, 5.38, ...
Simulation by Hand:
t = 3.08, Arrival of Part 3
System
3
2
Number of
completed waiting
times in queue
2
Clock
B(t)
Q(t)
Total of
waiting times in queue
Arrival times of
Event calendar
custs. in queue
[4, 3.79,
Arr]
(3.08) [2, 4.66,
Dep]
[–, 20.00,
End]
Area under
Area under
Q(t)
B(t)
3.08
1
1
1.17
1.17
3.08
4
Q(t) graph
3
2
1
0
B(t) graph
0
5
10
15
20
0
5
10
15
20
2
1
0
Interarrival times
Time (Minutes)
1.73, 1.35, 0.71, 0.62, 14.28, 0.70, 15.52, 3.15, 1.76, 1.00, ...
Service times
2.90, 1.76, 3.39, 4.52, 4.46, 4.36, 2.07, 3.36, 2.37, 5.38, ...
Simulation by Hand:
t = 3.79, Arrival of Part 4
System
4
3
2
Number of
completed waiting
times in queue
2
Clock
B(t)
Q(t)
Total of
waiting times in queue
Arrival times of
Event calendar
custs. in queue
[5, 4.41,
Arr]
(3.79, 3.08) [2, 4.66,
Dep]
[–, 20.00,
End]
Area under
Area under
Q(t)
B(t)
3.79
1
2
1.17
1.88
3.79
4
Q(t) graph
3
2
1
0
B(t) graph
0
5
10
15
20
0
5
10
15
20
2
1
0
Interarrival times
Time (Minutes)
1.73, 1.35, 0.71, 0.62, 14.28, 0.70, 15.52, 3.15, 1.76, 1.00, ...
Service times
2.90, 1.76, 3.39, 4.52, 4.46, 4.36, 2.07, 3.36, 2.37, 5.38, ...
Simulation by Hand:
t = 4.41, Arrival of Part 5
System
5
4
3
2
Number of
completed waiting
times in queue
2
Clock
B(t)
Q(t)
Total of
waiting times in queue
Arrival times of
Event calendar
custs. in queue
[2, 4.66,
Dep]
(4.41, 3.79, 3.08) [6, 18.69,
Arr]
[–, 20.00,
End]
Area under
Area under
Q(t)
B(t)
4.41
1
3
1.17
3.12
4.41
4
Q(t) graph
3
2
1
0
B(t) graph
0
5
10
15
20
0
5
10
15
20
2
1
0
Interarrival times
Time (Minutes)
1.73, 1.35, 0.71, 0.62, 14.28, 0.70, 15.52, 3.15, 1.76, 1.00, ...
Service times
2.90, 1.76, 3.39, 4.52, 4.46, 4.36, 2.07, 3.36, 2.37, 5.38, ...
Simulation by Hand:
t = 4.66, Departure of Part 2
System
5
4
3
Number of
completed waiting
times in queue
3
Clock
B(t)
Q(t)
Total of
waiting times in queue
Arrival times of
Event calendar
custs. in queue
[3, 8.05,
Dep]
(4.41, 3.79) [6, 18.69,
Arr]
[–, 20.00,
End]
Area under
Area under
Q(t)
B(t)
4.66
1
2
2.75
3.87
4.66
4
Q(t) graph
3
2
1
0
B(t) graph
0
5
10
15
20
0
5
10
15
20
2
1
0
Interarrival times
Time (Minutes)
1.73, 1.35, 0.71, 0.62, 14.28, 0.70, 15.52, 3.15, 1.76, 1.00, ...
Service times
2.90, 1.76, 3.39, 4.52, 4.46, 4.36, 2.07, 3.36, 2.37, 5.38, ...
Simulation by Hand:
t = 8.05, Departure of Part 3
System
5
4
Number of
completed waiting
times in queue
4
Clock
B(t)
Q(t)
Total of
waiting times in queue
Arrival times of
Event calendar
custs. in queue
[4, 12.57,
Dep]
(4.41) [6, 18.69,
Arr]
[–, 20.00,
End]
Area under
Area under
Q(t)
B(t)
8.05
1
1
7.01
10.65
8.05
4
Q(t) graph
3
2
1
0
B(t) graph
0
5
10
15
20
0
5
10
15
20
2
1
0
Interarrival times
Time (Minutes)
1.73, 1.35, 0.71, 0.62, 14.28, 0.70, 15.52, 3.15, 1.76, 1.00, ...
Service times
2.90, 1.76, 3.39, 4.52, 4.46, 4.36, 2.07, 3.36, 2.37, 5.38, ...
Simulation by Hand:
t = 12.57, Departure of Part 4
System
5
Number of
completed waiting
times in queue
5
Clock
B(t)
Q(t)
12.57
1
0
Arrival times of
custs. in queue
Total of
waiting times in queue
Area under
Q(t)
15.17
15.17
Event calendar
[5, 17.03,
Dep]
() [6, 18.69,
Arr]
[–, 20.00,
End]
Area under
B(t)
12.57
4
Q(t) graph
3
2
1
0
B(t) graph
0
5
10
15
20
0
5
10
15
20
2
1
0
Interarrival times
Time (Minutes)
1.73, 1.35, 0.71, 0.62, 14.28, 0.70, 15.52, 3.15, 1.76, 1.00, ...
Service times
2.90, 1.76, 3.39, 4.52, 4.46, 4.36, 2.07, 3.36, 2.37, 5.38, ...
Simulation by Hand:
t = 17.03, Departure of Part 5
System
Number of
completed waiting
times in queue
5
Clock
B(t)
Q(t)
17.03
0
0
Arrival times of
custs. in queue
()
Event calendar
[6, 18.69,
Arr]
[–, 20.00,
End]
Total of
waiting times in queue
Area under
Q(t)
Area under
B(t)
15.17
15.17
17.03
4
Q(t) graph
3
2
1
0
B(t) graph
0
5
10
15
20
0
5
10
15
20
2
1
0
Interarrival times
Time (Minutes)
1.73, 1.35, 0.71, 0.62, 14.28, 0.70, 15.52, 3.15, 1.76, 1.00, ...
Service times
2.90, 1.76, 3.39, 4.52, 4.46, 4.36, 2.07, 3.36, 2.37, 5.38, ...
Simulation by Hand:
t = 18.69, Arrival of Part 6
System
6
Number of
completed waiting
times in queue
6
Clock
B(t)
Q(t)
18.69
1
0
Arrival times of
custs. in queue
()
Total of
waiting times in queue
Area under
Q(t)
Event calendar
[7, 19.39,
Arr]
[–, 20.00,
End]
[6, 23.05,
Dep]
Area under
B(t)
15.17
15.17
17.03
4
Q(t) graph
3
2
1
0
B(t) graph
0
5
10
15
20
0
5
10
15
20
2
1
0
Interarrival times
Time (Minutes)
1.73, 1.35, 0.71, 0.62, 14.28, 0.70, 15.52, 3.15, 1.76, 1.00, ...
Service times
2.90, 1.76, 3.39, 4.52, 4.46, 4.36, 2.07, 3.36, 2.37, 5.38, ...
Simulation by Hand:
t = 19.39, Arrival of Part 7
System
7
6
Number of
completed waiting
times in queue
6
Clock
B(t)
Q(t)
Total of
waiting times in queue
Arrival times of
Event calendar
custs. in queue
[–, 20.00,
End]
(19.39) [6, 23.05,
Dep]
[8, 34.91,
Arr]
Area under
Area under
Q(t)
B(t)
19.39
1
1
15.17
15.17
17.73
4
Q(t) graph
3
2
1
0
B(t) graph
0
5
10
15
20
0
5
10
15
20
2
1
0
Interarrival times
Time (Minutes)
1.73, 1.35, 0.71, 0.62, 14.28, 0.70, 15.52, 3.15, 1.76, 1.00, ...
Service times
2.90, 1.76, 3.39, 4.52, 4.46, 4.36, 2.07, 3.36, 2.37, 5.38, ...
Simulation by Hand:
t = 20.00, The End
System
7
6
Number of
completed waiting
times in queue
6
Clock
B(t)
Q(t)
20.00
1
1
Arrival times of
Event calendar
custs. in queue
[6, 23.05,
Dep]
(19.39) [8, 34.91,
Arr]
Total of
waiting times in queue
Area under
Q(t)
Area under
B(t)
15.17
15.78
18.34
4
Q(t) graph
3
2
1
0
B(t) graph
0
5
10
15
20
0
5
10
15
20
2
1
0
Interarrival times
Time (Minutes)
1.73, 1.35, 0.71, 0.62, 14.28, 0.70, 15.52, 3.15, 1.76, 1.00, ...
Service times
2.90, 1.76, 3.39, 4.52, 4.46, 4.36, 2.07, 3.36, 2.37, 5.38, ...
Simulation by Hand:
Finishing Up
• Average waiting time in queue:
Total of times in queue 15.17

 2.53 minutes per part
No. of times in queue
6
• Time-average number in queue:
Area under Q(t ) curve 15.78

 0.79 part
Final clock value
20
• Utilization of drill press:
Area under B(t ) curve 18.34

 0.92 (dimensionless)
Final clock value
20
Complete Record of the Hand
Simulation