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

• f 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)

consider an irreducible, positive recurrent, and

aperiodic QBD process, in discrete-time, with infinitely many phase (background) states. More specifically.

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ = O O O A B C A B C A B C A B P

S = {(0, j): j∈S0} ∪{(n, j): n = 1,2,…, j =0, 1,2,…}

Partition the transition matrix according to the level State spaces S0 is a countable set n is called the level variable and j is called the background phase variable A, B and C are matrices of infinite dimension

Introduction (Stationary Vector)

,...) ,..., , (

1 n

π π π π =

1 ,

1 1

≥ =

−

n R n

n

π π

C R RB A R

2

+ + =

1 e e ) (

1 1 1 1

= ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∑ + + + = + =

∞ = n n

R RC B A C B π π π π π π π π

,...), ,..., , (

, 1 , , j n n n n

π π π = π

Partitioned according to the level

) (

, S j j ∈

= π π

Matrix-geometric solution

Introduction (Issues of Interests)

Characterization of tail asymptotics of both the joint distribution along direction n and the marginal distribution as n →∞

Exactly geometric decay rate Light tail behaviour without a geometric

decay

Upper and lower bounds (not in this talk)

j n,

π e πn

Introduction (Selected Literature Review)

Complex analysis (uniformization method, analytic continuation

and analysis of singularities)

Flatto and McKean, (1977, 1984), Leeuwarden, 2005

Probabilistic method (large deviations-like)

McDonald (99), Foley and McDonald (2001, 2004, 2004)

Matrix-analytic methods

Takahaashi, Fujimoto and Makimoto (2001) (QBD) Haque (2003), Haque, Liu and Zhao (2005) (QBD) Miyazawa (2004) (M/G/1) Miyazawa and Zhao (2004) (GI/G/1) Kroese, Scheinhardt and Taylor (2004), (QBD) Li, Miyazawa and Zhao (2007), Motyer and Taylor (2007) (QBD)

In literature, focus has been on

• 1. the joint

distribution

• 2. exactly geometric

decay along level direction

• 3. R is 1/α-positive

for some 0< α<1.

• 4. R is irreducible

The parallel queues feeded by arrivals with two types of demand and joint-the-shortest-queue The tandem queue with coupled processors Generalized joint-the-shortest-queue Modified Jackson network

Main Results (Exact Geometric Decay)

If the following conditions are satisfied, the joint distribution has an exactly geometric decay as n→∞

j n,

π ∞ < < ∞ < = = < < + + M x R R

i i , 1

(ii) and ; (i) , such that tor column vec positive a and vector row positive a , 1 an exists There ) 3 ( arithmetic

• 1

is C} B, {A, (2) ; aperiodic and e irreducibl is C B A ) 1 ( π α α α xy y y x x y x j n j n n

cx =

∞ →

α π , lim

∑ = ∞ < ∑

∞ = ∞ → ∞ = 1 1

lim . rate decay same decay with geometric exactly an has also marginal the , addition, in If

j j n n n j j

x c x α α α e π

R is 1/α-positive

Main Results (Exact Geometric Decay)

If the following conditions are satisfied, both the joint distribution and the marginal distribution have exactly geometric decay as n →∞

j n,

π

e πn

. , lim ) 3 ( ; ) 2 ( ; lim lim ) 1 ( such that ,...) x , (x vector row positive a and , 1 , an exist There

, 1 ) ( , 1

∞ < < = = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = = = < <

∞ → ∞ → ∞ →

c c x R r R

i i i n n j i n n n n

π α α α α α x x x

j n j n n

cx =

∞ →

α π

,

lim

∑ =

∞ = ∞ → 1

lim

j j n n n

x c α α e π

R may not be irreducible R is not 1/α -positive Application to the generalized joint shortest queue in which the difference of the two queues is taken as the level variable n and the minimum of two queues is background state j.

Main Results (Exact Geometric Decay)

If 0<c<∞, the marginal distribution has exactly geometric decay as n →∞ e πn

∞ < ∑ = = = < <

∞ = ∞ → ∞ → i i ,i i i n n n

y π c y R R

1

and , 1 lim ) 3 ( ; ) 2 ( ; lim ) 1 ( such that tor column vec positive a , 1 exists there If y y y α α α

∑ =

∞ = ∞ → 1 , 1

lim then

i i i n n n

y c π α α e π

Main Results (Light tail without a geometric

decay)

true. is following the

• f
• ne

if rate decay with light tail a has but decay, geometric exact an have not does α π n,j : Theorem

j, each for if , 1 , rate decay with light tail a has say We : < <α α πn,j Definition

α η η π α η η π < ∞ = ≥ =

∞ → ∞ →

all for lim and all for lim ) (

, , n j n n n j n n

i

α η η π α η η π ≤ ∞ = > =

∞ → ∞ →

all for lim and all for lim ) (

, , n j n n n j n n

ii α π log log lim

,

=

∞ →

n

j n n

Main Results (Light tail without a geometric

decay)

If α = γ, where γ is the convergence norm of R, then the joint distribution does not have exactly geometric decay as n →∞. That is,

j n

π

,

, lim ) 3 ( ; ) 2 ( ; lim ) 1 ( such that vector row positive a , 1 exists there If

, 1 =

= = < <

∞ → ∞ → i i i n n n

x R R π α α α x x x

lim then

,

=

∞ → n j n n

π α

γ π log log lim

, = ∞ →

n

j n n

lim

,

=

∞ → n j n n

γ π

⎭ ⎬ ⎫ ⎩ ⎨ ⎧ ∞ < ∑ =

∞ = ≥ ) ( , ,

: sup

n n n j i z j i

z r z ξ , ,

inf { }

i j i j

ξ ξ =

1 γ ξ =

Main Results (Light tail without a geometric

decay)

. rate decay with light tail a has but decay geometric exact have not does

• n

distributi marginal the satisfied, are conditions

• f

sets two following the

• f

either If . lim Assume

n

γ γ e π =

∞ → n n n

R

∞ < ∑ = ≤

∞ = ∞ → , 1

) ( lim ) ( such that vector row positive a exists There

i i i i i

x iii x ii R (i) π γx x x ∞ < ∑ = ≤

∞ = ∞ → i i i i i

y iii y ii R i

, 1 1

) ( 1 lim ) ( ) ( such that vector column positive a exists There π γy y y

Main Results (Light tail without a geometric

decay) lim =

∞ → n n n n

e γ π

R n n

n e γ π log log lim =

∞ →

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 λ1, λ2, μ1 and μ2, respectively.

Application (Polling system)

• q1(t) be the queue length of type 1 customers in the system at time t;
• q2(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

, , 1 2

lim { ( ) , ( ) , ( ) }

n i j t

P q t n S t i q t j π

→∞

= = = =

0 when server is idle, ( ) 1 when server is serving type 1 customers, 2 when server is serving type 2 customers. S t ⎧ ⎪ = ⎨ ⎪ ⎩

,1 ,2

[ , ]

n n n

= π π π

,1 ,1,0 ,1,1 ,1,2 ,1, ,

( , , , )

n n n n n j

π π π π =

π

L L

,2 ,2,1 ,2,2 ,2,3 ,2,

( , , , , )

n n n n n k

π π π π =

π

L L

Application (Polling system)

α π α π λ μ λ λ α π log log lim and lim ly, specifical More . n as ) ( rate decay with light tail a has but decay, geometric exactly an have not does

• n

distributi joint The

, 2 , n , 2 , n 2 2 2 1 1 , 2 ,

= = ∞ → − + =

∞ → ∞ →

n

j n n j n j n

Application (Polling system)

α α λ μ λ λ α log log lim and lim ly, specifical More . n as ) ( rate decay with light tail a has but decay, geometric exactly an have not does

• n

distributi marginal The

2 , n 2 , n 2 2 2 1 1 2 ,

= = ∞ → − + =

∞ → ∞ →

n

n n n n

e π e π e π

Applications (Gated Random Order

Service Queue)

Consider an M/M/1 queue with a service room and a waiting room Upon an arrival, if the service room is empty, the arriving

customer goes directly into service room and receives its service immediately.

however, if upon an arrival, the service room is nonempty, the

arriving customer has to enter the waiting room and waits until the service room becomes empty;

• nce the service room becomes empty, all customers in the

waiting room are instantaneously transferred into service room in a random order in which they will receive their services.

μ λ Service room Waiting room

X1(t) = the number of customers in the waiting room at time t; (Level variable) X2(t)= the number of customers in the service room at time t, (Background phase)

Applications (Gated Random Order

Service Queue)

α π α π μ λ α π log log lim and lim ly, specifical More . n as rate decay with light tail a has but decay, geometric exactly an have not does

• n

distributi joint The

, n , n ,

= = ∞ → =

∞ → ∞ →

n

j n n j n j n

Application (Gated Random Order

Service Queue)

α α μ λ α log log lim and lim ly, specifical More . n as rate decay with light tail a has but decay, geometric exactly an have not does

• n

distributi marginal The

n n 2 ,

= = ∞ → =

∞ → ∞ →

n

n n n n

e π e π e π