Mohammad Khalily Islamic Azad University.  Usually buffer size is finite  Interarrival time and service times are independent  State of the system.

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Presentation transcript:

Mohammad Khalily Islamic Azad University

 Usually buffer size is finite  Interarrival time and service times are independent  State of the system depends on : 1. Packet arrival process, (Poisson, deterministic, etc) 2. Packet length distribution 3. The service discipline (FCFS, LCFS, priority, etc) 4. # of Server, service process

Little’s Formula

E[N] : Average number of customers in a system : Average arrival rate E[T] : Average time spent in the system

Mean Number in Queue

Server Utilization

a/b/m/K a: Type of arrival process b: Service time distribution m: Number of servers K: Maximum number of customer allowed in the system Kendall’s Notation for describing a queueing system

Arrival process : Poisson process of rate Interarrival time : independent and identically distributed (i.i.d.) exponential random variable with mean 1/ Service time : iid exponential random variable with mean 1/ 

 The time until next arrival is exponential random variable is independent of the service times of customers already in the system.  The memoryless property of the exponential random variable implies that this interarrival time is independent of the present and past history of N(t).  Also on departure time

Transition rate diagram of M/M/1 Queue

The M/M/1 Queue

The M/M/1 Queue The Mean Number of Customers in the System

The M/M/1 Queue The Mean Total Customer Delay in the System

The M/M/1 Queue The Mean Waiting Time in Queue

The M/M/1 Queue The Mean Number in Queue

The M/M/1 Queue The Server Utilization

The Mean number of customer in the system versus utilization for M/M/1 Queue

The total customer delay versus utilization for M/M/1 Queue

The M/M/1 System with Finite Capacity The M/M/1/K System

The M/M/1/K System

The M/M/1/K System The Mean Number of Customers in the System

Typical pmf’s for N(t) of M/M/1/K System

The M/M/1/K System The Mean Total Time Spent By Customers in the System

The M/M/1/K System The Offered Load (Traffic intensity) and Carried Load

Carried Load versus Offered Load for M/M/1/K System with K = 2

Mean Customer Delay versus Offered Load for M/M/1/K System with K = 2 and K=10

The M/M/c System

         

         

The M/M/c Queueing System Erlang C Formula

The M/M/c System

M/M/1 Versus M/M/c

M/M/1 SystemM/M/c System M/M/1 Versus M/M/c

The M/M/c/c Queueing System

The M/M/c/c Queueing System Erlang B Formula     

Erlang B Formula (Example)

The M/M/  Queueing System

The number of customers in the system is a Poisson random variable

M/G/1 Queueing System The state of an M/G/1 system at time t is specified by N(t) together with remaining (“residual”) service time of the customer being served at time t.

 M/G/1 Queueing System?  G-Network?  Queueing simulations? Arena

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