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Queuing Analysis

Queuing Analysis. COMT 429. 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. “Call Length”. Service Time, s Service Rate,  =1/s

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Queuing Analysis

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  1. Queuing Analysis COMT 429

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

  4.  =  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. “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. “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. 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. 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. Multi-Server Call Queue Queue “a” Erlangs of Traffic “c” circuits aka servers This system is only stable for “a” less than “c”

  10. 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. 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. Queuing Analysis for Data Traffic COMT 429

  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 C bits/sec circuit speed  packets/sec L bits/packet L sec/message Service Time s = C

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

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

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

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

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

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