The M/M/ N / N Queue etc COMP5416 Advanced Network Technologies

School of Information Technologies M/M/ N / N 01-2 Link Models for circuit switching Telephone Traffic Call initiation.Poisson Process Length of calls.Negative exponential Number of circuits.Fixed, N Traffic arrival Poisson, mean arrival rate of  customers/sec Mean conversation time h seconds. Offered traffic is A =  h Erlangs => an M/M/N/N queue.

School of Information Technologies M/M/ N / N 01-3 Erlang Loss Function N circuits on a single link, arrivals Poisson at rate  customers per second, conversation time neg exp, mean h seconds, neg exp parameter  = 1/h, loss system no repeat attempts

School of Information Technologies M/M/ N / N 01-4 Let random variable R represent the number of customers currently in the system, R is between 0 and N. => state-dependent queue, arrival rate (independent of the state), service rate i  when the system is in state R=i. i.e. an M/M/N/N queue

School of Information Technologies M/M/ N / N 01-5 Let p i = Pr{R = i } In statistical equilibrium:

School of Information Technologies M/M/ N / N 01-6 p 0 is determined by the normalising condition:

School of Information Technologies M/M/ N / N 01-7 I.e. a truncated Poisson distribution

School of Information Technologies M/M/ N / N 01-8 Blocking Probability

School of Information Technologies M/M/ N / N 01-9 Example of the use of the Erlang Loss function Forecast offered traffic, 59.0 Erlangs. Specified blocking probability, better than 1.0%. Find n such that E n (A)<0.01 E 73 (59.0)= and E 74 (59.0)= Need 74 circuits.

School of Information Technologies M/M/ N / N Properties Example 59 Erlangs offered traffic and 74 circuits. Carried traffic is 59.0 × ( ) = Erlangs. Lost traffic is 59.0–58.42 Erlangs = 0.48 Erlangs

School of Information Technologies M/M/ N / N Time Congestion, Call Congestion E N (A) represents the proportion of the time that all circuits are busy, and is therefore called the time congestion. Call congestion is defined as the proportion of calls that find the system busy (i.e. the grade of service). Time congestion is congestion observed by the system. Call congestion is the congestion seen by customers.

School of Information Technologies M/M/ N / N PASTA For Poisson arrivals, time congestion = call congestion. PASTA theorem. (Poisson Arrivals See Time Averages.)

School of Information Technologies M/M/ N / N Validity of the Erlang Loss Function Valid for non–negative exponential distributions. The Poisson approximation for new call attempts has been validated by measurements. It is not valid if repeat attempts are significant It is not valid for overflow traffic.

School of Information Technologies M/M/ N / N M/G/1 Queue In practice, exponential service time is not very common => need more general –And it is shown first two moments of of service time is enough to characterise such a queueing system Let Q be queue length at departure times Use Embedded Markov Chain – queue lengths at departure times only Service time has second moment –Pollaczek-Khinchin formula Note: Second moment = mean 2 + variance

School of Information Technologies M/M/ N / N M/G/1 Using PASTA, Using Little’s formula

School of Information Technologies M/M/ N / N Example: M/G/1 This is exactly the equation that we derived earlier for the M/M/1 queue! If we assume the service distribution is exponentially distributed, the following holds:

School of Information Technologies M/M/ N / N Example: M/D/1 (constant service time) This is exactly the HALF the waiting time of the M/M/1 queue If service distribution is constant:

School of Information Technologies M/M/ N / N Squared Coefficient of Variation A critical factor in deciding the type of queueing system to use A measure of variability M/M/1: M/D/1: Note: Stallings uses

School of Information Technologies M/M/ N / N M/G/1 Mean Queue Length