Queueig Aalysis
Outlie M/M/ Queue (ifiite buffer M/M//N (fiite buffer M/M// (Erlag s B forula M/M/ (Erlag s C forula Networks of M/M/ Queues M/G/ Priority Queue
M/M/ M: Markovia/Meoryless Arrival process is Poisso Departure process is Expoetial Sigle Server Ifiite buffer for waitig custoers server Poisso arrivals with rate queue Expoetial service with rate μ
Poisso Process Arrival process, Δt is a very sall segeted tie ad is the average arrival rate P arrival i t, t+δt Δt P o arrival i t, t+δt -Δt P ore tha arrivals i t, t+δt Joit or split Poisso process is also Poisso process
Markov Property Meoryless: P (T>t + t T>t P (T> t P(X(t + x + X(t x, P(X(t - x -. X(t x P(X(t + x + X(t x The future is idepedet of the past!
M/M/ queueig odel State trasitio diagra - t -(+μ t -(+μ t -(+μ t -(+μ t -(+μ t -(+μ t μ t μ t μ t μ t μ t μ t μ t μ t 3 - + t t t t t t t t
M/M/: balace equatios Equilibriu or steady state aalysis: P + μ t P t P μ P P μ P P μ P > 3.. P - μ P P /μ P P P /μ P P P 3 /μ P 3 P.. P /μ P - P /μ is called utilizatio factor or traffic itesity < or <μ <> Syste is stable!
M/M/: fid P P P p P (,,, Geoetric distributio!
Little s law The log ter average uber of custoers i a queueig syste is EN ET, where ET is the average tie a custoer speds i the syste, is the average arrival rate of custoers.
M/M/: syste perforace The average uber of custoers i the queueig syste EN The average waitig tie i the queueig syste (usig Little s Law E T E N μ μ
M/M/: queue perforace The average uber of custoers i the queue E Nq The average waitig tie i the queue (or usig Little s Law E T q E T E Ts ( μ μ
M/M//N queueig syste (fiite buffer (-P B Queue Server N P B N: The syste ca hold at ost N custoers (icludig the oe i service
N B N N N P P p p N p p + +, 3 N- N- N t t t t t t t μ t μ t μ t μ t μ t μ t μ t - t -(+μ t -(+μ t -(+μ t -(+μ t -μ t -(+μ t Blockig probability: the probability that the syste is full M/M//N : blockig probability
M/M// queueig odel (-P B 3 P B Servers - t -(+μ t -(+μ t -(+3μ t μ t μ t 3μ t 4μ t (-μ t -(+(- μ t -(+(-μ t (- μ t μ t -μ t 3 - - t t t t t t t
M/M//: blockig probability,,,, - μ μ, Blockig probability P B (fid all servers are busy Erlag s B forula B i P P P p p i P (! (! (!! ( ( μ μ μ μ μ
M/M/ queueig odel: parallel servers with a ifiite queue Queue 3 Servers Perforace etric: the probability that a arrival has to wait i the queue (P Q
M/M/: state trasitio diagra - t -(+μ t -(+μ t -(+3μ t -(+(- μ t -(+μ t μ t μ t 3μ t 4μ t (-μ t μ t μ t -(+ μ t μ t 3 - + t t t t t t t t μ( μ, μ, >
M/M/: fid P! ( ( p p! p p μ!(! ( + p P
M/M/:PQ All the servers i this syste are busy Erlag s C forula + Q!(! (!(!( P p P
Networks of M/M/ queues L M/M/ queues for a syste, ad i ad μi are Poisso arrival ad departure rates of a queue, Qi. The average uber of custoers i each M/M/ queue(ode is E Ni i i μi i i
The total uber of custoers i the syste The average delay per packet (little s law L i i i i L i Ni E N E μ S k k L i i i i N E T E γ μ, is the average arrival rate of the custoers eterig the syste
M/G/ queueig odel Poisso arrivals with rate Arbitrary service tie (ay distributio EX/μ first oet (of service tie EX Var(X+E X secod oet
M/G/: syste perforace The average uber of custoers i the queueig syste The average waitig tie i the syste ( μ + X E N E ( μ + + X E E T E T T E q s
M/G/: queue perforace The average uber of custoers i the queue The average waitig tie i the queue ( X E N E q ( X E T E q
Priority queueig (M/G/ queue Two priority classes case: delay sesitive ad best effort Type packets are give service priority i that service will ever begi o Type packets if a Type packet is waitig If Type packet is beig served ad a Type packet arrives, the service of the Type packet is cotiued util copletio (there is ot pre-eptio
The average delay per Type packet is give by E T + μ E T q + μ ie T ( The average delay per Type packet is give by E T + μ E T q + μ i i si ( ( ie T si