1. Queuing Theory

2. Model Customers arrive randomly in accordance with some arrival time distribution. One server services customers in order of arrival. The service time is random following some service time distribution.

3. Model Ntcustomersinsystematt()#  avgarrivalrate N (t) t a t. lim   

4. Model Ntcustomersinsystematt()#  avgarrivalrate N (t) t a t. lim     s avgservicerate .

5. Model Measures of Performance

6. Model Measures of Performance L=avg. # customers in system L q = avg. # customers in queue W = avg. waiting time in the system W q = avg. waiting time in the queue

7. Model Little’s Formula LW  LW qq  WW q   1

8. Model Steady State Ntcustomersinsystematt()#  Plongrunprobabilitythatthere arencustomersinsystem n  PNtn t lim{()}  

9. M/M/1 Queue M/M/1 Queue assumes exponential interarrival times and exponential service times Ae i A i   Se i S i    

10. M/M/1 Queue M/M/1 Queue assumes exponential interarrival times and exponential service times Ae i A i   Se i S i     Exponential Review Expectations Memoryless Property Inverse Functions

11. M/M/1 Queue Relation to Poisson ifXtarrivals int()#(,]  0PXtPfirstarrivalt Pxt e t {()}{} {}     0

12. M/M/1 Queue Relation to Poisson PXtPfirstarrivalt Pxt e t {()}{} {}     0 PXtn te n nt {()} () !   miracle 37

13. M/M/1 Queue Inverse Function

14. M/M/1 Queue Inverse Function 2.032 1.951 1.349.795.539.347

15. M/M/1 Queue 2.032 1.951 1.349.795.539.347 0.305 0.074 0.035 0.520 1.535 0.159

16. M/M/1 Queue 2.032 1.951 1.349.795.539.347 0.305 0.074 0.035 0.520 1.535 0.159

17. M/M/1 Queue.347

18. M/M/1 Queue.347

19. M/M/1 Queue.539.347 0.305

20. M/M/1 Queue.5390.652

21. M/M/1 Queue.539 0.074 0.652

22. M/M/1 Queue 0.726

23. M/M/1 Queue.795 0.035 0.726

24. M/M/1 Queue 0..830

25. M/M/1 Queue 1.349 0.520 0.830

26. M/M/1 Queue 2.032 1.951 1.349.795.539.347 0.305 0.074 0.035 0.520 1.535 0.159

27. M/M/1 Event Calendar

28. M/M/1 Performance Measures