20. Extinction Probability for Queues and Martingales ( Refer to - PowerPoint PPT Presentation

20. Extinction Probability for Queues and Martingales ( Refer to section 15.6 in text ( Branching processes) for discussion on the extinction probability). 20.1 Extinction Probability for Queues: • A customer arrives at an empty server and immediately goes for service initiating a busy period. During that service period, other customers may arrive and if so they wait for service. The server continues to be busy till the last waiting customer completes service which indicates the end of a busy period. An interesting question is whether the busy periods are bound to terminate at some point ? Are they ? 1 PILLAI

Do busy periods continue forever? Or do such queues come to an end sooner or later? If so, how ? • Slow Traffic ( ) ρ ≤ 1 Steady state solutions exist and the probability of extinction equals 1. (Busy periods are bound to terminate with probability 1. Follows from sec 15.6, theorem 15-9.) • Heavy Traffic ( ) ρ > 1 Steady state solutions do not exist, and such queues can be characterized by their probability of extinction. ρ < 1 . •Steady state solutions exist if the traffic rate Thus { } = = exists if ρ < p lim P X nT ( ) k 1. k →∞ n •What if too many customers rush in, and/or the service ρ ≥ rate is slow ( ) ? How to characterize such queues ? 1 2 PILLAI

Extinction Probability for Population Models π ( ) 0 X = X = 1 1 0 0 ( ) ( ) 2 Y 2 ( ) Y 2 X = Y 3 2 3 1 1 X = 9 2 X = 26 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 3 3 3 3 3 3 3 3 3 3 Y Y Y Y Y Y Y Y Y � 1 2 3 4 5 6 7 8 9 Fig 20.1 3 PILLAI

Queues and Population Models • Population models : Size of the n th generation X n : Number of offspring for the i th member of Y ( ) n i the n th generation. From Eq.(15-287), Text X = ∑ n ( n ) X Y + n 1 k = k 1 Let ∆ = ( ) n a = { P Y k } k i • Offspring moment generating function: P ( z ) 1 ∞ ∑ = a k P ( z ) a k z 0 (20-1) z = k 0 1 4 Fig 20.2 PILLAI

∞ ∑ = = = X k P ( ) z P X { k z } E z { } + n 1 + + n 1 n 1 = k 0 { }   ∑ j Y = = = = X E E z ( { X j }) E E z X j i   + = n 1 i 1 n n   ∑ j = = = = j E {[ ( )] } P z { ( )} P z P X { j } P P z ( ( )) (20-2) n n j = = P ( z ) P ( P ( z )) P ( P ( z )) (20-3) + n 1 n n = = P X { k } ? n Extinction probability = = = π = lim { P X 0} ? n o →∞ n π = Extinction probability satisfies the equation P ( z ) z 0 which can be solved iteratively as follows: ∆ = = z P (0) a (20-4) 0 0 and = = (20-5) z P ( z ) , k 1, 2, � 5 − k k 1 PILLAI

• Review Theorem 15-9 (Text) ∞ ∞ ∑ ∑ ′ ρ = = = = = > Let E Y ( ) P (1) k P Y { k } ka 0 (20-6) i i k = = k 0 k 0 ( ) ( ) P z P z 1 a 0 a 0 Fig 20.3 π 1 1 0 ρ ≤ 1 ρ > 1 (a) (b) ρ ≤ ⇒ π = 1 1  0  ρ > ⇒ π is the unique solution of P z = (20-7) 1 ( ) z ,  0  < π < a 1  0 0 • Left to themselves, in the long run, populations either π 0 die out completely with probability , or explode with π . probability 1 - ( Both unpleasant conclusions ). 6 0 PILLAI

Queues : Customers that arrive X during the first service time X 2 First X 3 customer 1 X Y Y Y 2 3 0 1 s s s s s 2 1 3 4 k Service time of Service time of first customer second customer X Fig 20.4 = ∑ n X Y + n 1 i = i 1 Busy period t π 0 P  k arrivals between two  = = = ≥ a P Y ( k ) 0.   k i successive departures   { } Note that the statistics depends both on the arrival as well a k as the service phenomena. 7 PILLAI

∞ ∑ = k P ( z ) a k z • : Inter-departure statistics generated by arrivals = k 0 ∞ ∑ ′ ρ = = P ( 1 ) ka • : Traffic Intensity Steady state } ≤ 1 k ⇒ = k 1 > Heavy traffic 1 • Termination of busy periods corresponds to extinction of queues. From the analogy with population models the π extinction probability is the unique root of the equation 0 = P ( z ) z ρ ≤ ⇒ π 0 = • Slow Traffic : 1 1 ρ > ⇒ < π 0 < Heavy Traffic : 1 0 1 ρ > i.e., unstable queues either terminate their busy periods ( 1) π 0 < with probability , or they will continue to be busy with 1 π probability 1- . Interestingly, there is a finite probability of 0 busy period termination even for unstable queues. π : Measure of stability for unstable queues. 0 8 PILLAI

Example 20.1 : M/M/ 1 queue From (15-221), text,we have k k µ  λ   ρ  1 = = =     a , k 0, 1, 2, � (20-8) k λ + µ λ + µ + ρ + ρ 1 1     ( λ P ) t ( µ P ) Number of arrivals between any two departures follows a geometric random variable. λ 1 = ρ = P ( z ) , (20-9) + ρ − µ [ 1 ( 1 z )] = ⇒ ρ − ρ + + = − ρ = 2 P ( z ) z z ( 1 ) z 1 ( z 1 ) ( z - 1) 0 π 1 ρ ≤ 0  1 , 1 ρ  Fig 20.5 π =  1 (20-10) ρ > 0 , 1  ρ 9  ρ PILLAI 1

Example 20.2 : Bulk Arrivals M [x] / M/ 1 queue Compound Poisson Arrivals : Departures are exponential µ . random variables as in a Poisson process with parameter λ Similarly arrivals are Poisson with parameter However each . arrival can contain multiple jobs. ( λ P ) A 2 A 1 Fig 20.6 t t t 2 1 ( µ P ) A : Number of items arriving at instant i t i Let = = = P { A k} k 0 , 1, 2, � c i k , ∞ ∑ and = = represents the bulk arrival statistics. A k C ( z ) E { z } c z i k = k 0 10 PILLAI

Inter-departure Statistics of Arrivals ∞ ∑ = = = k Y P z ( ) P Y { k} z E z { } = k 0 ∞ ∞ ∑∫ + + + arrivals in arrivals in = ( A A � A ) E z { n (0, )} { t P n (0, )} (t) t f dt 1 2 n s 0 = n 0 λ λ n ∞ ( t ) ∞ ∑∫ − λ − λ = µ ρ = A n t t [ { E z }] e e dt , i µ n ! 0 = n 0 λ n ∞ [ t C z ( )] 1 ∞ ∑ ∫ ( ) − λ + µ = µ t = + e (20-11) ρ − n ! 1 {1 C z ( )} 0 = n 0 − = + ρ − = − α α = 1 k P z ( ) [1 {1 C z ( )}] , Let c (1 ) , k 0, 1, 2, � k − α − α 1 1 z = = C z ( ) , ( ) P z (20-12) − α + αρ − α + ρ 1 z 1 (1 ) z αρ Traffic Rate ′ = = > P ( 1 ) 1 − α 1 = ⇒ α + ρ = π = α + ρ 11 P ( z ) z (z - 1) [ (1 )z - 1] 0 1 /( ( 1 )) 0 PILLAI

Bulk Arrivals (contd) • Compound Poisson arrivals with geometric rate M / M /1 π 0 − α 1 1 π = ρ > , . (20-13) 0 α + ρ α (1 ) ρ 2 α = − α For , we obatain 1 (a) 1 α 3 π 0 3 1 1 π = ρ > , (20-14) 0 + ρ 2 (1 ) 2 ρ 1 (b) 2 Fig 20.7 • Doubly Poisson arrivals gives ⇒ = e µ − − (1 z ) ( ) C z 1 = = P z ( ) z (20-15) + ρ − 1 [1 C z ( )] 12 PILLAI

Example 20.3 : M/E n / 1 queue ( n -phase exponential service) From (16-213) − n ρ   = + − (20-16) P ( z ) 1 ( 1 z )    n  1   n − n n 1 = ⇒ + + + − = =   P ( z ) z x x � x 0 , x z n ρ   2  − + + ρ  1 1 8 ρ = = ⇒ π = ρ >   2 n 2 , 1 0 2   π 0 = → n = 0 . 5 M / M / 1 1 − π ≈ + ρ n >> (1 / ) n , n 1 (20-17) 0 π 0 = → n = 0 . 38 M / E / 1 2 2 Example 20.4 : M/D/ 1 queue m → ∞ Letting in (16.213),text, we obtain = − ρ − ( 1 z ) P ( z ) e , so that − ρ π ≈ − ρ − ρ > ( 1 e ) e , 1 . 13 (20-18) 0 PILLAI

20.2 Martingales Martingales refer to a specific class of stochastic processes that maintain a form of “stability” in an overall sense. Let X i ≥ refer to a discrete time stochastic process. If n refers { , 0} i to the present instant, then in any realization the random variables are known, and the future values X , X , � , X 0 1 n X , X + � , are unknown. The process is “stable” in the sense + n 1 n 2 that conditioned on the available information (past and present), no change is expected on the average for the future values, and hence the conditional expectation of the immediate future value is the same as that of the present value. Thus, if = E X { | X , X , � , X , X } X (20-19) + − n 1 n n 1 1 0 n 14 PILLAI