Light Tailed Behaviour and Decay Rate for a General Type of - PowerPoint PPT Presentation

Light Tailed Behaviour and Decay Rate for a General Type of Two-Dimensional Random Walk with Complex Boundaries Yiqiang Q. Zhao School of Mathematics and Statistics Carleton University (Canada-China Workshop on Industrial Mathematics, BIRS) August 5—10, 2007 This talk is based on joint work with: Q.M. He, Dalhousie University � Hui Li, Mount Saint Vincent University � M. Miyzawa, Tokya University of Science �

Abstract Motivated by characterizing properties of rare events in stochastic models such as telecommunications systems, insurance policies, etc, in this talk, we present some key results for a general type of two- dimensional random walk with boundaries. This type of random walk can be modeled as a quasi-birth-and- death process with countably many background (phase) states. By using the matrix-analytic method, combined with probabilistic arguments, conditions for exactly geometric decay and for light-tailed but not exactly geometric decay are obtained.

Outline � Introduction � QBD Process with Countably Many Phases � Issues of Interest � Selected Literature Review � Main Results � Applications to Queueing Models � Polling System � Gated Service Queues

Introduction (QBD process with countably many phase states) c onsider an irreducible, positive recurrent, and aperiodic QBD process, in discrete-time, with infinitely many phase (background) states. More specifically. State spaces S = {(0, j): j ∈ S 0 } ∪ {(n, j): n = 1,2,…, j =0, 1,2,…} S 0 is a countable set Partition the ⎛ ⎞ B A ⎜ ⎟ transition 0 0 C B A ⎜ ⎟ matrix n is called the level 0 ⎜ ⎟ according to = variable and P C B A ⎜ ⎟ the level j is called the ⎜ ⎟ C B A background phase ⎜ ⎟ O O O ⎝ ⎠ variable A, B and C are matrices of infinite dimension

Introduction (Stationary Vector) Partitioned according to π = π π π the level ( , ,..., ,...) n 0 1 = π π π = π π π ( , ,..., ,...), ( ) ∈ j j S n n n n j 0 0 , , 0 , 1 , 0 Matrix-geometric solution = + π π B π C 0 0 0 1 0 − π = π ≥ R n n 1 , 1 = + + π π A π B RC ( ) n 1 1 0 0 1 ⎛ ∑ ∞ ⎞ n + = = + + ⎜ ⎟ π π R R A RB R C 2 e e 1 0 1 ⎝ ⎠ = n 0

Introduction (Issues of Interests) Characterization of tail asymptotics of π both the joint distribution along n , j direction n and the marginal distribution as n →∞ π n e � Exactly geometric decay rate � Light tail behaviour without a geometric decay � Upper and lower bounds (not in this talk)

Introduction (Selected Literature Review) � Complex analysis (uniformization method, analytic continuation and analysis of singularities) The parallel queues feeded by arrivals with two In literature, focus has types of demand and joint-the-shortest-queue � Flatto and McKean, (1977, 1984), Leeuwarden, 2005 been on � Probabilistic method (large deviations-like) The tandem queue with coupled processors 1. the joint distribution � McDonald (99), Foley and McDonald (2001, 2004, 2004) Generalized joint-the-shortest-queue 2. exactly geometric � Matrix-analytic methods Modified Jackson network decay along level � Takahaashi, Fujimoto and Makimoto (2001) (QBD) direction � Haque (2003), Haque, Liu and Zhao (2005) (QBD) � Miyazawa (2004) (M/G/1) 3. R is 1/ α -positive � Miyazawa and Zhao (2004) (GI/G/1) for some 0< α <1. � Kroese, Scheinhardt and Taylor (2004), (QBD) � Li, Miyazawa and Zhao (2007), Motyer and Taylor (2007) (QBD) 4. R is irreducible

Main Results (Exact Geometric Decay) ∞ < ∞ ∑ x If in addition, , the marginal also has an j = j 1 If the following conditions are satisfied, the joint α π distribution has an exactly geometric decay as n →∞ exactly geometric decay with same decay rate . n , j + + ( 1 ) A B C is irreducibl e and aperiodic ; ∞ π e c n = ∑ x lim (2) {A, B, C} is 1 - arithmetic j n α α → ∞ n = j < α < 1 x ( 3 ) There exists an 0 1 , a positive row vector R is 1/ α -positive y and a positive column vec tor such that , = α = α < ∞ π , x R x R y y xy (i) ; and n j = cx π lim j i n 1 , < < ∞ α M → ∞ (ii) n x i

Main Results (Exact Geometric Decay) If the following conditions are satisfied, both the joint Application to the generalized joint shortest queue R may not be irreducible π π n e distribution and the marginal distribution have in which the difference of the two queues is taken n , j R is not 1/ α -positive as the level variable n and the minimum of two exactly geometric decay as n →∞ queues is background state j. α < α < = x There exist an , 0 1 , and a positive row vector (x , x ,...) 0 1 such that π n j , = cx ⎛ ⎞ n lim n ( ) r R j ⎜ ⎟ n i j α , = = → ∞ n ( 1 ) lim 0 lim 0 ; ⎜ ⎟ n n α α → ∞ → ∞ n n ⎝ ⎠ ∞ π e c = α x R x ( 2 ) ; n = ∑ x lim j π n α α → ∞ n = i j 1 , = < < ∞ c c 1 ( 3 ) lim , 0 . x → ∞ i i

Main Results (Exact Geometric Decay) < α < y If there exists 0 1 , a positive column vec tor such that n then R = ( 1 ) lim 0 ; n α ∞ → ∞ n π e c n = π ∑ y lim = α R y y i i ( 2 ) ; 1 , n α α → ∞ n = i 1 ∞ 1 = < ∞ ∑ c π y ( 3 ) lim , and ,i i 1 y → ∞ i = i i 0 π n e If 0<c< ∞ , the marginal distribution has exactly geometric decay as n →∞

Main Results (Light tail without a geometric decay) π n,j α < α < Definition : We say has a light tail with decay rate , 0 1 , π log if for each j, n j , = α lim log n → ∞ n π n,j Theorem : does not have an exact geometric decay, but has a α light tail with decay rate if one of the following is true. π π n j n j , , = η ≥ α = ∞ η < α i ( ) lim 0 for all and lim for all n n η η → ∞ → ∞ n n π π n j n j , , = η > α = ∞ η ≤ α ii ( ) lim 0 for all and lim for all n n η η → ∞ → ∞ n n

Main Results (Light tail without a geometric decay) If α = γ , where γ is the convergence < α < If there exists 0 1 , π norm of R, then the joint distribution n j x , a positive row vector does not have exactly geometric decay as n →∞ . That is, such that n R = π π , = ( 1 ) lim 0 ; log n j n j , n = α γ → ∞ n lim 0 lim log n γ → ∞ n n → ∞ n = α x R x ( 2 ) ; π 1 = i , ⎧ ∞ ⎫ ( 3 ) lim 0 , n n ξ = < ∞ ( ) z ∑ r z ⎨ ⎬ sup : x ≥ → ∞ i j z i j i , 0 , ⎩ ⎭ i = n 0 π n j 1 ξ = ξ , = γ = inf { } then lim 0 i j i j ξ , , n α → ∞ n

Main Results (Light tail without a geometric decay) n R = Assume lim 0 . If either of the following two sets of n γ → ∞ n π e conditions are satisfied, the marginal distributi on does not n γ have exact geometric decay but has a light tail with decay rate . There exists a positive column There exists a positive row x y vector such that vector such that ≤ γ x ≤ γ y (i) x R i R y ( ) π 1 i = 1 , = ii ii ( ) lim 0 ( ) lim 0 → ∞ i y x → ∞ i i i ∞ ∞ π < ∞ < ∞ ∑ iii ∑ x iii y ( ) ( ) i i i 1 , = = i i 1 0

Main Results (Light tail without a geometric decay) π e n = lim 0 n γ → ∞ n n π e log n = γ lim log R n → ∞ n

Application (Polling system) • Consider an exhaustive polling system with one server switching between two waiting lines that contain type 1 and type 2 customers, respectively. • There is no switching time • At any time, if the server is serving a type k customer, k =1, 2, it will keep serving type k customers, and switch over to serving another type only as the line of the type k customers becomes empty. • The server goes into idle state only there are no customers in the system; and it becomes activated immediately upon a new arrival. • Assume that the arrival processes for both types of customers are Poisson and the service times are exponential with rates λ 1, λ 2, μ 1 and μ 2, respectively.

Application (Polling system) q 1( t ) be the queue length of type 1 customers in the system at time t ; � q 2( t ) be the queue length of type 2 customers in the system at time t ; � S ( t ) be the status of the server at any time t , where � ⎧ 0 when server is idle, ⎪ = ⎨ S t ( ) 1 when server is serving type 1 customers, ⎪ ⎩ 2 when server is serving type 2 customers. = π π π π = = = = P q t n S t i q t j [ , ] lim { ( ) , ( ) , ( ) } n n n n i j , , 1 2 ,1 ,2 →∞ t π = π π π π L L ( , , , ) n n n n n j ,1 ,1,0 ,1,1 ,1,2 ,1, , π = π π π π L L ( , , , , ) n n n n n k ,2 ,2,1 ,2,2 ,2,3 ,2,