# Slides12 - TELCOM 2120 Network Performance David Tipper...

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1 TELCOM 2120 Network Performance David Tipper Associate Professor Graduate Telecommunications and Networking Program University of Pittsburgh Slides 12 Nomenclature of a Queueing System The input process – how customers arrive The system structure – waiting space – number of servers, etc. The service process Kendall’s Notation 1/2/3/4/5/6 Shorthand notation to describe a queueing system 1 : Customer arriving pattern (Interarrival times distribution). TELCOM 2110 2 2 : Service pattern (Service-times distribution). 3 : Number of parallel servers. 4 : System capacity. 5 : Queueing discipline. 6: Customer Population

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2 Nomenclature Standard notation mean arrival rate of customers/time unit i t i t /ti it mean service rate in customers/time unit n(t) – number of customers in the system at time t π i = lim t P{n(t) = i}    is server utilization remember    for stability L – Average number of customers in systems L q - Average number of customers in the queues know L = L q + W – Average delay in system (includes server + queue) W q – Average delay in queue know W = W q + 1/ Little’s Law L = W TELCOM 2120: Network Performance 3 Nomenclature Standard notation - relationships TELCOM 2120: Network Performance 4
3 Single Queue Analysis Consider single queue case – G/G/C doesn’t have a closed form solution - will consider approximations later First focus on basic models widely used in network performance analysis Data networks and database systems M/M/1 M/M/1/K T l h TELCOM 2110 5 Telephony M/M/C Erlang C M/M/C/C Erlang B All are Markovian queues, study using Birth Death process CTMC Single Queue Analysis (M/M/1) Most basic Markovian queue is the M/M/1/ /FIFO/ queue Customers arrive according to a Poisson process with exponentially distributed interarrival times (IAT) P{ IAT t} = 1 – e - t , mean interarrival time = 1/ Customers are served by a single server with exponential service time distribution P(service time < t ) = 1 – e - t TELCOM 2120: Network Performance 6 mean service time = 1/ The arrival rate ( ) and service rate ( ) do not depend upon the number of customers in the system or time Consider behavior of n(t) – number of customers in the system at time t forms a Markov Process

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