Course Description Queuing Analysis This queuing course is a vital tool used in evaluating system performances its application covers A wide spectrum from Bank automated teller machines to transportation and communications data networks This queuing course focuses on queuing modeling techniques and its application in data networks

Queuing definitions Kleinrock Bose Takagi “We study the phenomena of standing, waiting, and serving, and we call this study Queueing Theory.” Bose “The basic phenomenon of queuing arises whenever a shared facility needs to be accessed for service by a large number of jobs or customers.” Takagi “A queue is formed when service requests arrive at a service facility and are forced to wait while the server is busy working on other requests”

Wrap up The study of queuing is the study of waiting Customers may wait on a line Planes may wait in a holding pattern Jobs may wait for the attention of the CPU in an computer Packets may wait in the buffer of a node in a computer network Telephone calls may wait to get through an exchange

History lesson History goes back to primitive man The history of queues goes back to primitive man. If you don’t believe me please take a look at the above presented figure Edward Hicks “Noah’s Ark”

Combinatorics Permutations K-permutation of a set of n elements Combinations K-combination of a set of n elements => k-permutation / k! (where k! is the number of possible ways to permute that combination)

Combinatorics (cont’d) Binomial coefficients Binomial expansion

Poisson distribution Poisson distribution is Associated with the observation of event occurrences If N represents the number of events in T => N/T = average number of events /minute interested in answering the following question How many occurrences of this event take place per minute? The way it has been done Either 0 or 1 event occurrence per minute T=15 min Event#1 Time 7

Poisson distributed random variable A Poisson random variable X Characterizes the number of occurrences of an event Typically an arrival => X = # arrivals per unit time With parameter λ (average # of arrivals per unit time) The value of λ (arrival rate) 8

Poisson distribution: example 1 If number of accidents occurring on a highway per day is a Poisson r.v. with parameter λ = 3, What is the probability that no accidents occur today? Solution 9

Poisson distribution: example 2 Consider an experiment that counts the number of α-particles emitted in a one-second interval by one gram of radioactive material. If we know that ,on average , 3.2 such α-particles are given off what is a good approximation to the probability that no more than 2 α-particles appear? 10

Expectation: example 3 Calculate E[X] For Poisson random variable X with parameter λ 11

Exponential distribution is the foundation of most of the stochastic processes Makes the Markov processes ticks is used to describe the duration of sthg CPU service Telephone call duration Or anything you want to model as a service time 12

Exponential random variable   13

Link between Poisson and Exponential Time Exponentially distributed with 1/ λ If the arrival process is Poisson # arrivals per time unit follows the Poisson distribution With parameter λ => inter-arrival time is exponentially distributed With mean = 1/ λ = average inter-arrival time 14

Proof  

Memoryless Property   16

Example  

Further properties of the exponential distribution      

Further properties of the exponential distribution (ct’d) X1 , X2 , …, Xn independent r.v. Xi follows an exponential distribution with Parameter λi => fXi (t) = λi eλit Define X = min{X1, X2, …, Xn} is also exponentially distributed Proof fX(t) = ?

Queuing system A queuing system is a place where customers arrive According to an “arrival process” To receive service from a service facility Can be broken down into three major components The input process The system structure The output process Customer Population Waiting queue Service facility

Characteristics of the system structure λ: arrival rate μ: service rate λ μ Queue Infinite or finite Service mechanism 1 server or S servers Queuing discipline FIFO, LIFO, priority-aware, or random 21

Queuing systems: examples Multi queue/multi servers Example: Supermarket Multi-server/single queue Bank immigration .

Kendall notation David Kendall A British statistician, developed a shorthand notation To describe a queuing system A/B/X/Y/Z A: Customer arriving pattern B: Service pattern X: Number of parallel servers Y: System capacity Z: Queuing discipline M: Markovian D: constant G: general Cx: coxian

Kendall notation: example M/M/1/infinity A queuing system having one server where Customers arrive according to a Poisson process Exponentially distributed service times M/M/S/K M/M/S/K=0 Erlang loss queue K

Special queuing systems Infinite server queue Machine interference (finite population) λ μ . S repairmen N machines

M/M/1 queue λn = λ, (n >=0); μn = μ (n>=1) λ μ λ: arrival rate μ: service rate λn = λ, (n >=0); μn = μ (n>=1)

Traffic intensity rho = λ/μ It is a measure of the total arrival traffic to the system Also known as offered load Example: λ = 3/hour; 1/μ=15 min = 0.25 h Represents the fraction of time a server is busy In which case it is called the utilization factor Example: rho = 0.75 = % busy

Queuing systems: stability N(t) λ<μ => stable system λ>μ Steady build up of customers => unstable busy idle 1 2 3 1 2 3 4 5 6 7 8 9 10 11 Time N(t) 1 2 3 1 2 3 4 5 6 7 8 9 10 11 Time

Example#1 A communication channel operating at 9600 bps Receives two type of packet streams from a gateway Type A packets have a fixed length format of 48 bits Type B packets have an exponentially distribution length With a mean of 480 bits If on the average there are 20% type A packets and 80% type B packets Calculate the utilization of this channel Assuming the combined arrival rate is 15 packets/s

Performance measures L Lq W Wq Mean # customers in the whole system Mean queue length in the queue space W Mean waiting time in the system Wq Mean waiting time in the queue 30

Mean queue length (M/M/1)

Mean queue length (M/M/1) (cont’d)

Little’s theorem This result The theorem Existed as an empirical rule for many years And was first proved in a formal way by Little in 1961 The theorem Relates the average number of customers L In a steady state queuing system To the product of the average arrival rate (λ) And average waiting time (W) a customer spend in a system

LITTLE’s Formula  

Graphical Proof  

Graphical Proof (continued)

Graphical Proof (continued)  

Mean waiting time (M/M/1) Applying Little’s theorem