Modeling and Simulation
Simulation of queuing systems
• Discrete
Types of systems
– State variables change
instantaneously at separated
points in time
– Bank model: State changes
occur only when a customer
arrives or departs
– Continuous
• State variables change continuously
as a function of time
• Head of water behind the dam :
State variables Head amount change
continuously
• Many systems are partly discrete, partly continuous
Types of simulation models
• Physical simulation
models
• Mathematical
simulation models
Simulation models
Physical Simulation
models
Mathematical
Simulation models
Discrete-event
Simulation models
Static or dynamic
Simulation models
Continuous
Simulation models
Deterministic and
stochastic
Simulation models
– Static vs. dynamic
– Deterministic vs.
stochastic
– Continuous vs.
discrete
(Most operational models
are dynamic, stochastic,
and discrete – will be
called discrete-event
simulation models)
Discrete-event simulation models
Single server queuing system simulation
What are the event of this system?
What are the system status ?
Single server queuing system simulation
• In the single-channel queue, the calling population is infinite.
• Arrivals and services are defined by the distribution of the time
between arrivals and the distribution of service times, respectively.
• For any simple single-or multi-channel queue, the overall effective
arrival rate must be less than the total service rate, or the waiting
line will grow without bound. When queues grow without bound,
they are termed (explosive) or unstable.
• Prior to introducing several simulations of queuing systems, it is
necessary to understand the concepts of system state, events, and
simulation clock.
– The state of the system is the number of units in the system and the
status of the server, busy or idle.
– An event is a set of circumstances that cause an instantaneous change
in the state of the system. In a single-channel queuing system there
are only two possible events that can affect the state of the system.
They are the entry of a unit into the system (the arrival event) or the
completion of service on a unit (the departure event).
– The queuing system includes the server, the unit being serviced (if one
is being serviced), and units in the queue (if any are waiting).
– The simulation clock is used to track simulated time.
Simulation &Modeling
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Examples
Simulation &Modeling
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Simulate Bank system: with 6 expected customers
In a single-channel queuing system interarrival times and service times are
generated from the distributions of these random variables. For simplicity, assume
that the times between arrivals were generated by rolling a die five times and
recording the up face. These five interarrival times are used to compute the arrival
times of six customers at the queuing system.
Simulation table : Bank teller example
customer Inter
arrival
arrival
Time
Ser.
Begins
Service
time
Duration
Time
ser.
ends
Time
in
queue
Time cust. Idle
Spends in time
sys.
service
1
0
0
0
2
2
0
2
0
2
2
2
2
1
3
0
1
0
3
4
6
6
3
9
0
3
3
4
1
7
9
2
11
2
4
0
5
2
9
11
1
12
2
3
0
6
6
15
15
4
19
0
4
3
Total
15
39
Average
3
6.5
43
13
2.1
56
4
17
6
0.67
2.8
1
The average time between arrival = (sum time between arrival) /total number of arrivals (customers)-1
Compare between the expected and calculated average time between arrival E(A)= the
mean of the discrete uniform distribution for rolling a die whose endpoints are a = 1 and b =
6. is equal 3.5 !!! ( (1+6)/2 =3.5 )
The average time Customer spends on the system , will be compare with the total of:
Average service time+ time in queue