1. Simulating Single server queuing models

2. Consider the following sequence of activities that each customer undergoes: 1.Customer arrives 2.Customer waits for service if the server is busy. 3.Customer receives service. 4.Customer departs the system.

3. Example Arrival rate ( =15) customers per hour Service time = 3 minutes (service rate (  = 20) customer per hour Arrival and Service times are exponentially distributed Note: the generation of Exponential Random Variable is: –Generate uniform [0,1] RN: RAND() –Return X = -1/ * ln(RAND())

4. Analytical Solutions Analytical solutions for W, L, Wq, Lq exist (see Lecture 05) However, analytical solution exist at infinity which cannot be reached. Therefore, Simulation is a most.

5. Flowchart of an arrival event IdleBusy An Arrival Status of Server Customer joins queue Customer enters service More

6. Flowchart of a Departure event NOYes A Departure Queue Empty ? Set system status to idle Remove customer from Queue and begin service More

7. An example of a hand simulation Consider the following IAT’s and ST’s: A1=0.4, A2=1.2, A3=0.5, A4=1.7, A5=0.2, A6=1.6, A7=0.2, A8=1.4, A9=1.9, … S1=2.0, S2=0.7, S3=0.2, S4=1.1, S5=3.7, S6=0.6 Want: Average delay in queue Utilization

8. Initialization Time = 0 system Server 0 0 0 0 0 00 0 0.4 999 System state Server status # in que Times of Arrival Time Of Last event Clock Eventlist Number delayed Total delay Area Under Q(t) Area Under B(t) A D Statistical Counters

9. Arrival Time = 0.4 system 01 0.4 1000 1.6 2.4 System state Server status # in que Times of Arrival Time Of Last event Clock Eventlist Number delayed Total delay Area Under Q(t) Area Under B(t) A D 0.4 A1=0.4, A2=1.2, A3=0.5, A4=1.7, A5=0.2, A6=1.6, A7=0.2, A8=1.4 S1=2.0, S2=0.7, S3=0.2, S4=1.1, S5=3.7, S6=0.6 Statistical Counters

10. Arrival Time = 1.6 system 11 1.6 1 00 1.2 1.6 2.1 2.4 System state Server status # in que Times of Arrival Time Of Last event Clock Eventlist Number delayed Total delay Area Under Q(t) Area Under B(t) A D 0.4 1.6 A1=0.4, A2=1.2, A3=0.5, A4=1.7, A5=0.2, A6=1.6, A7=0.2, A8=1.4 S1=2.0, S2=0.7, S3=0.2, S4=1.1, S5=3.7, S6=0.6 Statistical Counters

11. Arrival Time = 2.1 21 2.1 1.6 2.1 100.5 1.7 2.1 3.8 2.4 System state Server status # in que Times of Arrival Time Of Last event Clock Eventlist Number delayed Total delay Area Under Q(t) Area Under B(t) A D 0.4 1.6 System 2.1 A1=0.4, A2=1.2, A3=0.5, A4=1.7, A5=0.2, A6=1.6, A7=0.2, A8=1.4 S1=2.0, S2=0.7, S3=0.2, S4=1.1, S5=3.7, S6=0.6 Statistical Counters

12. Departure Time = 2.4 11 2.4 2.1 20.81.1 2.0 2.4 3.8 3.1 System state Server status # in que Times of Arrival Time Of Last event Clock Eventlist Number delayed Total delay Area Under Q(t) Area Under B(t) A D 1.6 2.1 System A1=0.4, A2=1.2, A3=0.5, A4=1.7, A5=0.2, A6=1.6, A7=0.2, A8=1.4 S1=2.0, S2=0.7, S3=0.2, S4=1.1, S5=3.7, S6=0.6 Statistical Counters

13. Departure Time = 3.1 01 3.1 31.8 2.7 3.1 3.8 3.3 System state Server status # in que Times of Arrival Time Of Last event Clock Eventlist Number delayed Total delay Area Under Q(t) Area Under B(t) A D 2.1 System A1=0.4, A2=1.2, A3=0.5, A4=1.7, A5=0.2, A6=1.6, A7=0.2, A8=1.4 S1=2.0, S2=0.7, S3=0.2, S4=1.1, S5=3.7, S6=0.6 Statistical Counters

14. Departure Time = 3.1 00 3.3 31.8 2.9 3.3 3.8 999. System state Server status # in que Times of Arrival Time Of Last event Clock Eventlist Number delayed Total delay Area Under Q(t) Area Under B(t) A D System A1=0.4, A2=1.2, A3=0.5, A4=1.7, A5=0.2, A6=1.6, A7=0.2, A8=1.4 S1=2.0, S2=0.7, S3=0.2, S4=1.1, S5=3.7, S6=0.6 Statistical Counters

15. Departure Time = 3.1 01 3.8 41.8 2.9 3.8 4.0 4.9 System state Server status # in que Times of Arrival Time Of Last event Clock Eventlist Number delayed Total delay Area Under Q(t) Area Under B(t) A D System 3.8 A1=0.4, A2=1.2, A3=0.5, A4=1.7, A5=0.2, A6=1.6, A7=0.2, A8=1.4 S1=2.0, S2=0.7, S3=0.2, S4=1.1, S5=3.7, S6=0.6 Statistical Counters

16. Departure Time = 3.1 11 4.0 41.8 3.1 4.0 5.6 4.9 System state Server status # in que Times of Arrival Time Of Last event Clock Eventlist Number delayed Total delay Area Under Q(t) Area Under B(t) A D System 3.8 4.0 A1=0.4, A2=1.2, A3=0.5, A4=1.7, A5=0.2, A6=1.6, A7=0.2, A8=1.4 S1=2.0, S2=0.7, S3=0.2, S4=1.1, S5=3.7, S6=0.6 Statistical Counters

17. Departure Time = 3.1 01 4.9 52.7 4.0 4.9 5.6 8.6 System state Server status # in que Times of Arrival Time Of Last event Clock Eventlist Number delayed Total delay Area Under Q(t) Area Under B(t) A D System 4.0 A1=0.4, A2=1.2, A3=0.5, A4=1.7, A5=0.2, A6=1.6, A7=0.2, A8=1.4 S1=2.0, S2=0.7, S3=0.2, S4=1.1, S5=3.7, S6=0.6 Statistical Counters