1. Modeling and Simulation
Simulation of queuing systems

2. Types of systems Discrete Continuous
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

3. Types of simulation models
Physical simulation models
Mathematical 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)
Simulation models
Physical Simulation models
Mathematical Simulation models
Discrete-event Simulation models
Continuous Simulation models
Static or dynamic Simulation models
Deterministic and stochastic
Simulation models

4. Single server queuing system simulation
Discrete-event simulation models
Single server queuing system simulation
What are the event of this system?
What are the system status ?

5. 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.

6. Simulation &Modeling

7. Simulation &Modeling

8. Simulation &Modeling

9. Simulation &Modeling

10. Examples
Simulation &Modeling

11. 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.

13. Simulation table : Bank teller example
customer
Inter arrival
arrival
Time Ser. Begins
Service time Duration
Time ser. ends
Time in queue
Time cust. Spends in sys.
Idle time service 1 2 3 4 6 9 7 11 5 12 15 19
Total 39 43 13 56 17
Average
6.5
2.1
0.67
2.8
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