Queuing theory with Applications <ul>Presented By : Rajeev N Bharshetty ( USN : 1RV08IS036 ) </ul>
Queuing Theory...What is it? <ul><li>“ The study of the waiting times, lengths, and other properties of queues ” .
“ The theoretical study of waiting lines, expressed in mathematical terms--including components such as number of waiting lines, number of servers, average wait time, number of queues or lines, and probabilities of queue times' either increasing or decreasing ” . </li></ul>
Why queues form? Queues or waiting lines arise when the demand for a service facility exceeds the capacity of that facility, that is, the customers do not get service immediately upon request but must wait, or the service facilities stand idle and wait for customers. Some customers wait when the total number of customers requiring service exceeds the number of service facilities, some service facilities stand idle when the total number of service facilities exceeds the number of customers requiring service.
A Queueing System : An Example <ul><li>The simplest queueing system consists of two components (the queue and the server) and two attributes (the inter-arrival time, i and the service time, t). </li></ul><ul>Server System </ul><ul>Queuing System </ul><ul>Queue </ul><ul>Server </ul><ul>Queuing System </ul>
Kendall Notation 1/2/3(/4/5/6) Six parameters in shorthand First three typically used, unless specified <ul><li>Arrival Distribution
Kendall Notation Examples <ul><li>M/M/1: </li></ul>Poisson arrivals and exponential service, 1 server, infinite capacity and population, FCFS (FIFO) the simplest ‘realistic’ queue <ul><li>M/M/m </li></ul>Same, but M servers <ul><li>G/G/3/20/1500/SPF </li></ul>General arrival and service distributions, 3 servers, 17 queue slots (20-3), 1500 total jobs, Shortest Packet First
<ul>Analysis of M/M/1 queue </ul><ul><li>Given: </li><ul><li> : Arrival rate of jobs (packets on input link)
: Service rate of the server (output link) </li></ul><li>Solve: </li><ul><li>L :average number in queuing system
Sample Problem <ul><li>On a network gateway, measurements show that the packets arrive at a mean rate of 125 packets per second (pps) and the gateway takes about 2 milliseconds to forward them. Assuming an M/M/1 model, what is the probability of buffer overflow if the gateway had only 13 buffers. How many buffers are needed to keep packet loss below one packet per million? </li></ul>
Problem Analysis <ul><li>Measurement of a network gateway: </li><ul><li>mean arrival rate (l): 125 Packets/s
mean response time (m): 2 ms </li></ul><li>Assuming exponential arrivals: </li><ul><li>What is the gateway’s utilization?
What is the probability of n packets in the gateway?
Mean number of packets in gateway = </li></ul>
Solution... Probability of buffer overflow: = P(more than 13 packets in gateway) = ρ 13 = 0.25 13 = 1.49x10 -8 = 15 packets per billion packets To limit the probability of loss to less than 10 -6 :
Solution... To limit the probability of loss to less than 10 -6 : or = 9.96