Slides12 - 1 TELCOM 2120 Network Performance David Tipper Associate Professor Graduate Telecommunications and Networking Program University of

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Unformatted text preview: 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 2 Nomenclature ♦ Standard notation   mean arrival rate of customers/time unit   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 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 4 M/M/1 Analysis ♦ Consider n(t) behavior over small time interval  t P{ more than one arrival} ≈ P{ more than one arrival} 0, P { more than one service completion} ≈ P{ arrival and a service completion} ≈ Get birth-death state transition diagram and generator matrix              ) (       TELCOM 2120: Network Performance 7                     ) (     Q M/M/1 Queue • From state transition diagram flow balance...
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This note was uploaded on 06/14/2011 for the course DATABASE 101 taught by Professor - during the Spring '11 term at Aarhus Universitet.

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Slides12 - 1 TELCOM 2120 Network Performance David Tipper Associate Professor Graduate Telecommunications and Networking Program University of

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