Lec2 computer simulation Modeling and Simulation.pdf

Simulation Modeling and Analysis – Chapter 1 – Basic Simulation Modeling
• Ways to study a system
1.2 Systems, Models, and Simulation(cont’d.)
– Simulation is “method of
last resort?” Maybe …
– But with simulation there’s
no need (or less need) to
“look where the light is”

1.3 Discrete-Event Simulation(cont’d.)
• Example: Single-server queue
– Estimate expected average delay in queue (line, not
service)
– State variables
 Status of server (idle, busy) – needed to decide what to do
with an arrival
 Current length of the queue – to know where to store an
arrival that must wait in line
 Time of arrival of each customer now in queue – needed to
compute time in queue when service starts
– Events
 Arrival of a new customer
 Service completion (and departure) of a customer
 Maybe – end-simulation event (a “fake” event) – whether
this is an event depends on how simulation terminates (a
modeling decision)

1.3.1 Time-Advance Mechanisms
• Simulation clock: Variable that keeps the current value of
(simulated) time in the model
– Must decide on, be consistent about, time units
– Usually no relation between simulated time and (real) time needed
to run a model on a computer
• Two approaches for time advance
– Next-event time advance (usually used) … described in detail
below
– Fixed-increment time advance (seldom used)
 Generally introduces some amount of modeling error in terms of when
events should occur vs. do occur
 Forces a tradeoff between model accuracy and computational efficiency

1.3.1 Time-Advance Mechanisms(cont’d.)
• More on next-event time advance
– Initialize simulation clock to 0
– Determine times of occurrence of future events – event list
– Clock advances to next (most imminent) event, which is executed
 Event execution may involve updating event list
– Continue until stopping rule is satisfied (must be explicitly stated)
– Clock “jumps” from one event time to the next, and doesn’t “exist”
for times between successive events … periods of inactivity are
ignored

1.3.1 Time-Advance Mechanisms(cont’d.)
• Next-event time advance for the single-server queue
(assume the following notation for time variables, all the
time variables are random)
ti = time of arrival of ith customer (t0 = 0)
Ai = ti – ti-1 = interarrival time between (i-1)th and ith customers (usually
assumed to be a random variable from some probability distribution)
Si = service-time requirement of ith customer (another random variable)
Di = delay in queue of ith customer
Ci = ti + Di + Si = total time that ith customer completes service and departs
ej = time of occurrence of the jth event (of any type), j = 1, 2, 3, …

Representation of the system schematically

1.3.2 Components and Organization of a
Discrete-Event Simulation Model
• Each simulation model must be customized to target system
• But there are several common components
– System state – variables to describe state at a particular time
– Simulation clock – current value of simulated time
– Event list – times of future events (as needed)
– Statistical counters – to collect information about sys. performance
– Initialization routine – initialize model at time 0
– Timing routine – determine next event time, type; advance clock
– Event routines – updates system state when an event type occurs
– Library routines – utility routines to generate random variates, etc.
– Report generator – to summarize, report results at end
– Main program – ties routines together, executes them in right order

1.3.2 Components and Organization of a
Discrete-Event Simulation Model(cont’d.)
General Flow Control
in a Next Event Time
Advance Simulation

1.4 SIMULATION OF A SINGLE-SERVER QUEUEING SYSTEM
1.4.1 Problem Statement
• Consider the single-server queueing model
• Assume interarrival times are independent and
identically distributed (IID) random variables
(identically distributed means that the interarrival
times have the same probability distribution)
• Assume service times are IID, and are independent of
interarrival times
• Queue discipline is FIFO
• Start empty and idle at time 0
• First customer arrives after an interarrival time, not at
time 0
• Stopping rule: When nth customer has completed
delay in queue (i.e., enters service) … n will be
specified as input

1.4.1 Problem Statement(cont’d.)
• The objective of the model is to measure the performance of
the system by estimating the following quantities:
– Expected average delay in queue (excluding service time) of the n
customers completing their delays (will be denoted by d(n))
 Why “expected?”
– Expected average number of customers in queue (excluding any in
service) (will be denoted by q(n))
 A continuous-time average
 Area under Q(t) = queue length at time t, divided by T(n) = time simulation
ends.
– Expected utilization (proportion of time busy) of the server
 Another continuous-time average
 Area under B(t) = server-busy function (1 if busy, 0 if idle at time t), divided
by T(n) ,
• The first objective gives the performance from the customer
point of view while the second and third objectives are
typical requirements by the management of such a system.

Expected average no. of customers in the queue

Expected utilization of the server

Demonstration Example

1.4.2 Intuitive Explanation
• Given interarrival times (all times are in minutes):
0.4, 1.2, 0.5, 1.7, 0.2, 1.6, 0.2, 1.4, 1.9, …
• Given service times:
2.0, 0.7, 0.2, 1.1, 3.7, 0.6, …
• Arrival times = 0.4, 1.6, 2.1, 3.8, 4, 5.6, 5.8, 7.2
• Departure times = 2.4, 3.1, 3.3, 4.9, 8.6, 9.2
• n = 6 delays in queue desired
• “Hand” simulation:
– Display system, state variables, clock, event list, statistical
counters … all after execution of each event
– Use above lists of interarrival, service times to “drive” simulation
– Stop when number of delays hits n = 6, compute output
performance measures

1.4.2 Intuitive Explanation(cont’d)
Status
shown is
after all
changes
have
been
made in
each
case …
Interarrival times: 0.4, 1.2, 0.5, 1.7, 0.2, 1.6, 0.2, 1.4, 1.9, …
Service times: 2.0, 0.7, 0.2, 1.1, 3.7, 0.6, …

Service times: 2.0, 0.7, 0.2, 1.1, 3.7, 0.6, …

Service times: 2.0, 0.7, 0.2, 1.1, 3.7, 0.6, …
Final output performance measures:
Average delay in queue = 5.7/6 = 0.95 min./cust.
Time-average number in queue = 9.9/8.6 = 1.15 custs.
Server utilization = 7.7/8.6 = 0.90 (dimensionless)

Lec2 computer simulation Modeling and Simulation.pdf

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Lec2 computer simulation Modeling and Simulation.pdf