1. QUEUING THEORY 1

2.  - means the number of arrivals per second   - service rate of a device  T - mean service time for each arrival   = ( ) Utilization, percentage of time a device is in use  q – mean number of customers in the system (either waiting or in service)  Q – number of customers in the system waiting or being served  N – number of servers  Response time – time from when a process is entered, until it completes  Throughput – Work / Time  Bottlenecks – resource that is in limiting the use of the system because it can’t get it’s work done.  Saturated System – the bottleneck device has reached 100 utilization. 2 NOTATION/DEFINITIONS USED IN QUEUING THEORY

3. A/B/c/K/m/z  A – interarrival distribution  B – service time distribution  c – number of servers  K – the capacity of the queue  m – number of customers in the system  z – queuing discipline 3 KENDALL NOTATION

4.  A/B/c is used when -There is no limit on the length of the queue -The source is infinite -The queue discipline is FCFS  A and B may be any of the following: -GI for general independent interarrival time -G for general service time -E for Erlang-k interarrival or service time distribution -M for exponential interarrival or service time distribution -D for deterministic interarrival or service time distribution -H or hyperexponential (with k stages) interarrival or service time distribution. 4 EXAMPLES OF KENDALL NOTATION

5. 5 M/D/3/4/12/FCFS

6. 6 LITTLE’S LAW (A.K.A. LITTLE’S RESULT)

7. 7 M/M/1 QUEUE EXAMPLE

8. 8 M/D/3

9. 9 DISK DRIVE EXAMPLE

10. Questions: What is the expected number of visits to the disk.  E = (0.5) ( 1 + E)  E = ½ + E/2  ½ E = ½  E = 1 Question: What is the expected number of visits to the cpu?  E = 1 + (0.5) E  ½ E = 1  E = 2 10 QUESTIONS ABOUT CPU DISK EXAMPLE

11. 11 MULTI USER MULTI RESOURCES

12. 12 MULTI PROCESS MULTI DISKS EXAMPLE

13. 13 MEMORY HIT RATIO EXAMPLE

14. 14 BIRTH DEATH PROCESSES

15.  The rate of moving from state 2 into state 3 should be the same as the rate from moving from state 3 into state 2. Over the course of a day if the queuing system moved from state 2 to 3 5000/day then the rate that the system goes from state 3 to state 2 must also be 5000/day(or maybe 4999/day).  The probability of being in one state i is P. The probability of being in state i and going to state i+1 is equal to the probability of doing from state i+1 and going to state i. 15 THOUGHTS ON BIRTH DEATH PROCESSES

16. 16 BIRTH DEATH CONTINUED

17. 17 STATE PROBABILITIES

18. 18 EXPECTED NUMBER IN STATE

19. 19 M/M/1 EXAMPLE

20. 20 TIME IN SYSTEM EXAMPLE

21. 21 TIME IN SYSTEM EXAMPLE CONT.