Presented By : Rajeev N Bharshetty ( USN : 1RV08IS036 )
Queuing Theory...What is it?
“ 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 ” .
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
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).
Kendall Notation 1/2/3(/4/5/6) Six parameters in shorthand First three typically used, unless specified
Number of servers
Total Capacity (infinite if not specified)
Population Size (infinite)
Service Discipline (FCFS/FIFO)
M: stands for "Markovian", implying exponential distribution for service times or inter-arrival times.
D: Deterministic (e.g. fixed constant)
E k : Erlang with parameter k
H k : Hyperexponential with param. k
G: General (anything)
Kendall Notation Examples
Poisson arrivals and exponential service, 1 server, infinite capacity and population, FCFS (FIFO) the simplest ‘realistic’ queue
Same, but M servers
General arrival and service distributions, 3 servers, 17 queue slots (20-3), 1500 total jobs, Shortest Packet First
Analysis of M/M/1 queue
: Arrival rate of jobs (packets on input link)
: Service rate of the server (output link)
L :average number in queuing system
L q :average number in the queue
W :average waiting time in whole system
W q :average waiting time in the queue
M/M/1 queue model
Solving queuing systems
4 unknowns: L, L q W, W q
L q = Wq (steady-state argument)
W = W q + (1/ )
If we know any 1, can find the others
Finding L is hard or easy depending on the type of system. In general:
Solving for P 0 and P n
Step 1 :
Step 2 :
Solving for P 0 and P n
Step 4 :
Solving for L
Solving W, W q and L q
Applications of Queuing Theory
Determining the sequence of computer operations
Predicting computer performance
Health services (eg. control of hospital bed assignments)
Airport traffic, airline ticket sales
Layout of manufacturing systems.
Application in Computer Networking
Open Queueing Network .
Closed Queuing Network
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?
Measurement of a network gateway:
mean arrival rate (l): 125 Packets/s
mean response time (m): 2 ms
Assuming exponential arrivals:
What is the gateway’s utilization?
What is the probability of n packets in the gateway?
mean number of packets in the gateway?
The number of buffers so P(overflow) is <10 -6 ?
Arrival rate λ = 125 pps
Service rate μ = 1/0.002 = 500 pps
Gateway utilization ρ = λ/μ = 0.25
Prob. of n packets in gateway =
Mean number of packets in gateway =
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