Modeling & Simulation
System Models and Simulation
 Framework
for
Modeling
and
Simulation
The framework defines the entities and their Relationships that are central to the
M&S enterprise
Experimental Frame
Simulator
Source
System
Modeling
Relation
Model
Simulation
Relation
Figure 1 Basic Entities in M&S Framework and their Relationships
1- The source system : is the real or virtual environment
that we interested in modeling (it is viewed as a source of
observable data).
2- Experimental Frame : is a specification of the conditions
under which the system is observed or experimented with.
3- Simulation Model : It is a set of instructions, rules,
equations, or constrains for generating I/O behavior of the
system.
4- Simulator: any computation system that Capable of
executing the model to generate its behavior (Processor,
Human Mind, or an algorithm).
Definitions:
 System’s state: is
the collection of variables (state variables)
necessary to describe a system at a particular time, relative to the
objectives of a study.
 SystemsTypes:
We categorize systems to be of two types, discrete and continuous.
 A discrete system : is one for which the state variables change
instantaneously at separated points in time.
 A continuous system : is one for which the state variables change
continuously with respect to time.
 Static vs. Dynamic Simulation Models: A static
simulation model is a representation of a system at a
particular time, a dynamic simulation model represents a
system as it evolves over time, such as a conveyor system in a
factory.
 Continuous vs. Discrete Simulation Models: a
discrete model is not always used to model a discrete system
and vice versa. To use a discrete or a continuous model for a
particular system depends on the specific objectives of the
study.
 Deterministic vs. Stochastic Simulation Models: If a
simulation model does not contain any probabilistic (i.e.,
random) components, it is called deterministic;
Stochastic simulation models: Having at least some random
input components produce output that is itself random.
Discrete-Event Simulation
 Discrete-event simulation concerns the modeling of a
system as it evolves over time by a representation in
which the state variables change instantaneously at
separate points in time. These points in time are the ones
at which an event occurs.
 Event is defined as an instantaneous occurrence that
may change the state of the system.
Example 1 : Consider a service facility
with a single server
 State variables: that may be used to simulate this
system are:
- Status of the server (idle or busy)
- Number of customers waiting in a queue
- Time of arrival for each customer.
 Events for this system:
- The arrival of a customer
- The completion of service for a customer,
Simulation of A Single-Server Queuing System
 Problem Statement
Consider a single-server queuing system :
D2
 ti: times of arrival of the ith customer (t0=0)
 Ai= ti -ti-1




the interarrival time between (i-1)st and ith arrivals of
customers.
Si : service time for the ith customer
Di : Delay in queue for the ith customer
C i = t i + Di + S i
ei : time of occurrence of ith event.
 the interarrival times A1, A2, … are independent, identically
distributed (IID) random variables.
 the service times S1, S2, … of the successive customers are IID
random variables that are independent of the interarrival times.
 We wish to simulate this system until a fixed number (n) of
customers have completed their delays in queue.
Measures of performance
To measure the performance of this system, we will look at
estimates of three quantities:
1-The expected average delay in queue of the n
customers completing their delays during the simulation;
we denote this quantity by d (n).
 From a single run of the simulation resulting in customer
delays D1, D2, ….., Dn, an obvious estimator of d(n) is
n

d ( n)


i 1
Di
n
•Which is just the
average of the n Di’s
2- The expected average number of
customers in the queue, denoted by q(n )
 Let Q (t) : denote the number of customers in queue at
time t, for any real number t ≥ 0
 T (n) : be the total time required to observe our n
delays in queue of length i.
Ť(n) = T0 + T1 + T2 + ….

q ( n) 


ip
i 0
i
(1.1)

 pi: is the observed proportion of the time during the
simulation that there were i customers in the queue.

 pi= Ti / T(n), so that we can rewrite Eq. (1.1) above as :


q ( n) 

i  0
iTi
T ( n)
(1.2)
3- The expected utilization of the server
 Which is the expected proportion of time during the
simulation that the server is busy (i.e., not idle), denote it by
u (n).
 our estimate of u(n) is û(n) = the observed proportion of
time during the simulation that the server is busy.
(Figure 3 )
(3.3  0.4)  (8.6  3.8) 7.7
u ( n) 
  0.90
8.6
8.6

(1.5)
Components and Organization of a
Discrete-Event Simulation Model
In particular, the following components will be found in most discreteevent simulation models using the next-event time-advance approach:
 System state:The collection of state variables necessary to describe the




system at a particular time.
Simulation clock: A variable giving the current value of simulated time.
Event list: A list containing the next time when each type of event will
occur.
Statistical counters:Variables used for storing statistical information
about system performance.
Initialization routine: A subprogram to initialize the simulation
model at time zero.
 Timing routine: A subprogram that determines the next event from



the event list and then advances the simulation clock to the time when
that event is to occur.
Event routine: A subprogram that updates the system state when a
particular type of event occurs.
Report generator: A subprogram that computes estimates (from the
statistical counters) of the desired measures of performance and
produces a report when the simulation ends.
Main program: A subprogram that invokes the timing routine to
determine the next event and then transfers state control to the
corresponding event routine to update the system state appropriately.
The main program may also check for termination and invoke the
report generator when simulation is over
Flowchart for arrival routine, queuing system
Flowchart for departure routine, queuing system