1. COMT 4291 Queuing Analysis COMT 429

2. 2 Call/Packet Arrival Arrival Rate, Inter-arrival Time, 1/ Arrival Rate measures the number of customer arrivals per time unit, e.g –Calls per hour –Packets per second

3. COMT 4293 “Call Length” Service Time, s Service Rate,  =1/s In circuit switched networks, the service time is the average call length.

4. COMT 4294 Utilization Measures the Arrival Rate relative to the Service Rate The queue becomes congested if the utilization is larger than the number of servers  =   = s *

5. COMT 4295 “Poisson Arrival” Describes random call or packet arrival Measured over a short time, the probability of call arrival is proportional to, the arrival rate Over long times, the probabilities of x call arrivals follow the Poisson Distribution

6. COMT 4296 “Exponential Service Time” Despite the different name, this assumption means that calls “leave” the system as randomly as they entered. It assumes that services times are random, and not related from call to call

7. COMT 4297 Queue Type: Kendall Notation Arrival/Service/#Servers/Queue Slots M/M/n or M/M/n/∞ –Poisson Arrival, Exponential Services time, n Servers M/G/n –General services times M/D/n –Fixed (Deterministic) service times

8. COMT 4298 Queueing in Circuit Switching Remember that Utilization is defined as the (Service Time) * (Calls/Time Unit) Service Time in a circuit switching environment is identical to call length Utilization is therefore the same as our definition of total traffic (in Erlangs)

9. COMT 4299 Multi-Server Call Queue “c” circuits aka servers “a” Erlangs of Traffic Queue This system is only stable for “a” less than “c”

10. COMT 42910 Erlang C (M/M/c) Recall that the blocking probability for “a” Erlangs offered on “c” circuits is given by Erlang-B: E’(c,a) The probabilty of delay is Erlang-C In Erlang C Tables, delay is given in units of h, not including the service time

11. COMT 42911 Systems with Queuing and Blocking M/M/c/K Queues “c” servers in the system Only “K” calls can be active in the system

12. COMT 42912 Queuing Analysis for Data Traffic COMT 429

13. 13 Service Time for Data Traffic In packet networks, the service time is computed from the packet length and the bit rate of the circuit. Buffer packets/sec L bits/packet C bits/sec circuit speed Service Time s = L C sec/message

14. COMT 42914 Summary Buffer messages/sec L bits/message C bits/sec circuit speed Service Time s = L C sec/message Service Rate  = 1 / s messages/sec Utilization  = 

15. COMT 42915 Queuing Formula M/M/1 Queue Average Queue Length E(n) =  1 -  messages queued

16. COMT 42916 Queue Delay M/M/1 Queue Average Delay E(T) = s 1 -  Including the service time for the call or packet In general, E(T) * = E(n), the queue delay times the arrival rate equals the queue length

17. COMT 42917 Delay Example M/M/1 200 characters/message 8 bits per character 9600 bits/sec line

18. COMT 42918 M/D/1 Queue results Average Queue Length E(n) =  (1 -  messages queued Average Delay E(T) = s (1 -  Including the service time for the call or packet 