COMT 4291 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

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

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 *

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

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

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

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)

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”

COMT 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

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

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

COMT 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  = 

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

COMT 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

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

COMT 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 