supp_queue - Lecture Set 3, EE 679, Copyright 2000, Sanjay...

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Lecture Set 3, EE 679, Copyright 2000, Sanjay K. Bose 1 Basic Queuing Theory Kendall's Notation for Queues: This is a useful way to represent different types of queues in a compact and easily understood fashion. Kendall's Notation describes the nature of the arrival process to the queue, the nature of the service process (or service time), the number of servers, maximum number in the queue and some basic queuing disciplines. The notation has been considerably extended to allow it to represent a wide variety of queues. We give here a very basic description of this notation. Following this representation, a queue is represented by a sequence A/B/C/D/E with the following meaning attached to the letters A to E A This symbolically represents the nature of the arrival process to the queue. Special letters are used to symbolize the nature of the inter-arrival time distribution as follows - M Exponentially distributed inter-arrival times (Poisson Process) D Deterministic (fixed) inter-arrival times E k Erlang distribution of order k for the inter-arrival times H k Hyper-exponential distribution of order k for the inter-arrival times G General (any!) distribution for the inter-arrival times etc. B This symbolically represents the nature of the service time distribution for the customers getting served in the queue. The same letters as the ones above are used to describe the nature of the service time distribution C Number of servers in the queue D Maximum Number of jobs/customers that can be there in the queue - this includes both the ones currently being served and the ones waiting for service. Note that the default is infinity ( ) which is assumed when this is omitted E Queuing Discipline such as - FCFS First Come First Served LCFS Last Come First Served SIRO Service In Random Order etc. This may also be omitted if the queue is FCFS in nature (default)/ Examples: M/M/1 or M/M/1/ Poisson Arrivals, Exponential Service Time Distribution, Single Server, Infinite Number of Waiting Positions M/E 2 /2/K Poisson Arrivals, Erlangian of order-2 Service Time Distribution, 2 Severs, Maximum Number K in queue G/M/2 Generalized Arrivals, Exponential Service Time Distribution, 2 Servers, Infinite Number of Waiting Positions
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Lecture Set 3, EE 679, Copyright 2000, Sanjay K. Bose 2 Equilibrium (Steady State) Solutions for Birth-Death Processes For state k, birth rate is l k and death rate is m k . Note that this would correspond to an exponential distribution for the inter-arrival and service times with means 1/ l k and 1/ m k ., respectively. To obtain the equilibrium solution to this, we set 0 = dt dp k for k=0, 1, 2, 3, . ............ This leads to the following equations for the steady state solutions { p k } 1 1 1 1 ) ( 0 + + - - + + + - = k k k k k k k p p p m l m l where we assume further that 0 = = = - - - i i i p m l for i=1, 2, . ....... and
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supp_queue - Lecture Set 3, EE 679, Copyright 2000, Sanjay...

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