Asymptotic analysis of stochastic models of hierarchic structure and applications in queueing models

Asymptotic Analysis of Stochastic Models of

Hierarchic Structure and Applications in Queueing

Models

V. V. Anisimov

Bilkent University, Bilkent 06533, Ankara, Turkey & Kiev University, Kiev, Ukraine

vlanis@bilkent.edu. tr

Abstract: A class of switching stochastic systems with hierarchic state space work­ ing in different scales of time ( slow and fast) that are adequate mathematical models at the analysis and modelling of various classes of computing systems and networks of a complex stochastic structure is studied. Models of asymptotic decreasing di­ mension and enlargement ( merging) of state space for general switching systems and nonhomogeneous Markov systems are considered. Applications to approxima­ tive analytic modelling of queueing systems with hierarchic state space switched by some Markov environment are studied.

1. INTRODUCTION

In various models appearing at studying of complex stochastic systems �.uch as state-dependent queueing systems and networks, communication systems, storage, production and manufacturing systems, etc., we come to a necessity to consider systems working in different scales of time ( slow and fast) and such that their local transition characteristics can be dependent on a current value of some another stochastic process ( external random environment, discrete interference of chance, stochastic failures or in general some stochastic functional on a trajectory of the system itself).

Taking into account a complex structure of real systems only in special rare cases it is possible to get an analytic representation for different characteristics. Therefore asymptotic methods play the basic role at the investigation and approximative modelling. Different asymptotic approaches for various classes of these systems are considered in the books [6,8,9] and papers [10,13,14] (see also references there).

In the paper we study models of a hierarchic structure and with different scales of time that allows to study asymptotic merging of states and decreasing dimension for complex stochastic systems and also investigate the behavior of additive func­ tionals on trajectories. We consider systems which can be described in terms of switching stochastic systems. A special class of stochastic processes named switch­ ing processes can be an adequate mathematical model at the analytical and ap­ proximative modelling for these systems.

The general description of switching stochastic systems is given and models of asymptotic decreasing dimension and merging of the state space for general switch­ ing systems and also for homogeneous and nonhomogeneo1:1s hierarchic Markov systems in a triangular scheme are studied.

Applications to the asymptotic analysis of state-dependent queueing systems and networks with hierarchic state space, working in different scales of time and under the influence of some external Markov environment admitting an asymptotic merging of its state space, are considered. These results extend approximative methods of analysis and allow to obtain explicit analytic formulas for different asymptotic characteristics of systems with complex stochastic structure.

2. SWITCHING STOCHASTIC SYSTEMS

The main property of switching processes (

SP)

is that the character of thei1 development varies spontaneously (switches) at certain epochs of time which can be random functionals of the previous trajectory.

SP

are described as two-component processes (x(t), ((t)), t � 0, with the prop­ erty existing a sequence of epochs t1

<

t2

<

···such that on each interval [tk, tk+i}, x(t)

=

x(tk) and the behavior of the process ((t) depends on the value (x(tk), ((tk)) only. The epochs tk are switching times and x(t) is the discrete switching compo­ nent. In various applications the component x( ·) usually is considered like some external random environment or a regime of operation of a system.

Special classes of these processes were studied by different authors: Markov processes homogeneous on the 2-nd component ([11)); processes with independent increments and semi-Markov switches ([1,3]); piecewise Markov aggregates ([9]); Markov processes with semi-Markov interference of chance ([12]), Markov and semi­ .Markov evolutions ([15,16,18,20,21]), and switching processes ([2-4] ).

In this section we consider some important classes of switching stochastic mod­ els which appear in various applications.

2.1. Recurrent process of a semi-Markov type.

Let

Fk

=

{(tk(x,a),rk(x,a)),x E X,a E Rr},k

2

0

be jointly independent families of random variables with values in Rr x [O, oo ), and let x1,

l

2

0

be an independent of Fk,

k

2

0

Markov process

(MP)

with values in

X,

(x0, S0) be an initial value. We put

to·= o, tk+l

=

tk

+

rk(Xk, Sk), sk+l

=

sk

+

tk(Xk, Sk), k

2

0, (2.1)

S(t)

=

Sk, x(t)

=

xk as tk � t

<

tk+i, t

2

o.

(2.2) Then the process (x(t), S(t))forms a recurrent process of a semi-Markov type

(RPSM)

with independent Markov switches (or in a semi-Markov environment). We assume that

RPSMis

regular, i.e. the component x(t) has a finite number of jumps on each finite interval with probability one.

We mention that this type processes usually appear when we consider input flows under the influence of a semi-Markov environment. In particular when distri­ butions of families Fk do not depend on the parameter. k, the process (x(t), S(t)) is a homogeneous

SMP

(semi-Markov process). If also distributions of families Fk

do not depend on parameters a, x, epochs tk, k

2

0 form a recurrent flow and S(t) is a generalized renewal process (batch recurrent input flow). When distributions of variables rk(x,a) do not depend on parameters a and k the process _x(t) itself is

aSMP.

In particular when variables rk(a, x) have exponential distributions, then the process (x(t), S(t)) is a

MP

and the process S(t) can be considered as a batch Markov arrival process.

2.2. General case of RPSM.

Let

Fk

=

{(tk(x,a),rk(x,a),/3k(x,a)),x E X,a E Rr},k

2

0

be jointly independent families of random variables taking values in the space Rr x [O, oo) x

X, X

be some measurable space, (x0, S0) be an initial value. We put

to =

0,

tk+i

=

tk

+

rk(xk, Sk),

Sk+1 = Sk

+

tk(xk, Sk), xk+1 = /3k(xk, Sk), k

2

0,

S(t)

=

sk, x(t)

=

Xk as tk � t

<

tk+i, t

>

0.

(2.3)

(2.4)

Then the pair (x(t), S(t)), t

2'.

0 forms a general RPSM with feedback be­ tween both components, i.e. here we have a mutual influence between the random environment x(t) and the trajectory of a system S(t).

In particular when distributions of the variables f3k(

x, a)

do not depend on the parameter a, the sequence Xk forms a MP and we obtain the previous case .

2.3. Processes in semi-Markov environment.

NaN we consider an operation of some random process in a semi-Markov en­ vironment. Let Fk

=

{(k(t,x,a), t

2'.

0, x EX, a E Rr_},

k

2'.

_{0 be jointly} independent parametric families of random processes, where (k(

t, x, a)

for each fixed k, x, a be a random process with trajectories belonging to Skorokhod space D':x,, x(t), t

2'.

0 be an independent of Fk, k

2'.

0 SMP with values in X and

S0 be an initial value. We suppose that variables introduced are measurable in the ordinary way in the pair

(x,a)

concerning er-algebra Bx x BR·· Denote by 0

=

t0 < t1 < · · · the epochs of sequential jumps of x( · ), Xk

=

x(tk),

k

2'.

0. We

construct a process in a semi-Markov environment ( or with semi-Markov switches) in the following way:

put Sk+1

=

Sk

+

fk, where fk

=

(k( Tk, xk, Sk), Tk

=

tk+i -tk, and denote

(2.5) Then a two-component process (x(t), ((t)), t

2'.

0 is the process with semi-Markov switches and the component x( t) itself is a SMP. Let us introduce also an imbedded process

(2.6) Then the process (x(t), S(t)) forms a RPSMwith additional Markov switches.

If {((t,x), t

2'.

O} is a family of Markov processes and ((t,x,a) denotes the process ((t, x) with initial value a, then the process (x(t), ((t)) forms a Markov (when the process x(t) is a MP) or semi-Markov (when the process x(t) is a SMP) random evolution (see [15,16,19,20,21]).

We mention that this type processes are mathematical models for service pro­ cesses in Markov type queueing networks operating in semi-Markov environment.

2.4. Switching processes.

be jointly independent parametric families where (A:(t, x, a) for each fixed k, x, a be a random process belonging to Skorokhod space D₀,/ and rk(x,a),,6k(x,a) be

possibly dependent on (1c(·,x,a) random variables, rk(·)

>

0,,6k(·) EX. Let also (x₀,

S0

) be an initial value. We put

Xk+l

=

.6k(Xk, Sk), k

2:

o,

(2.7)

where lk(x,a)

=

(k(rk(x,a),x,a), and set

Then a two-component process (x(t), ((t)), t

2:

0 is called a switching process (SP). Now we consider as examples some special subclasses of SP.

2.4.1. Processes with independent increments and semi-Markov switches.

Let x(t), t

2:

0 be a SMP with state space X and let {l(kl(t, x), t

2:

0, x E X}, k

2:

0 and {'y(k)( x ), x E X}, k

2:

1 be independent on x( ·) and jointly indepen­ dent at different k families of homogeneous processes with independent increments and correspondingly random variables with values in Rr_{. Suppose for simplicity}

that distributions of processes (Ck)( t, x) and variables ,(A:)( x) do not depend on the index

k.

Denote

'1/1(8,x)

=

�lnEexp{i(B,l(ll_{(t,x))}, cp(8,x)}

=

_Eexp{i(e,,< 1 l_{(x))},x E X,8 E R}r_.

t

Suppose that functions '1/J( 8, x ), cp( 8, x) at each fixed 8 are measurable in the variable x concerning u-algebra Bx.

We construct the two-component process (x(t), ((t)), t

2:

0 with values in (X, Rr_{) in the following way. Let (x}

0, (0) be some initial value. Denote by O

=

t

₀

<

t1

< ...

the epochs of successive jumps for x(t) and let Xk

=

x(tk), k

2:

0 be the

embedded MP. Then we put ((0)

= (0

_{and on each interval [tk, t}A:+1)

((t)

=

((t,.)

+

e(kl(t, xk) - e<kl(tk, xk),

(2.9) It means that on the interval [t,., t1c+1) the process ((t) takes increments of the process e<k_{l(t, x}

,.) and in the epoch t,. it has a jump of the size ,(kl(x,.), k

>

0.

By the construction at the fixed trajectory of x(t) the process ((t) is a nonhomo­ geneous process with independent increments, instantaneous value .of the cumulant

'ifJ(B, x(t)) and with additional jumps in epochs tk,

k

>

_{0 of sizes ,(k)(xk), If (}0

=

0,

then the characteristic function of a one-dimensional distribution of the process

((t)

is given by the expression:

Eexp{i(B,((t))}

=

Eexp{ht

'1/J(B, x(u))du

+

I::

lnrp(B,xk)},

O _O<tk9 (2.10)

Here the averaging is made by all possible trajectories x( u ), 0 ::; u ::;

t.

By analogy a multi-dimensional characteristic function can be written.

If processes t<k)(

t,

x) are nonhomogeneous in time and distributions of variables

,(k)( x,

t)

depend also on

t

then with the help of these families it is possible by

analogy to construct the process (x(t), ((t)), t � 0 which is nonhomogeneous in time.

In particular if the process x(·) is a Markov process then the pair ( x(t), ((t)), t � 0 forms a process with independent increments and Markov switches (see [11]).

2.4.2. Switching Poisson process.

Now we consider a special construction of a Poisson type process with switches. Let the family of non-negative functions a(x), x E X and a random process x(t) with values in X be given. Denote .\(t)

=

a(x(t)), t � 0. Suppose that .\(t) is the instantaneous intensity for the process of Poisson type TI>.(t)(t). It means that at the fixed trajectory of x(t, w) the process TI>.(t,w)(t) is developing like a nonhomogeneous Poisson process with the instantaneous intensity a(x(t,w)).

In that case TI>.(t)(t) is a Poisson type process with the parameter which is switched by the process x(t). In particular if x(t) is a MP then the process TI>.(t)(t) is a Poisson process with Markov switches ( or Markov arrival process), correspond­ ingly if x(t) is a SMP then the process TI>.(t)(t) is a Poisson process with semi­ Markov switches.

The joint distributions on non-overlapping intervals [ai, bj],

j =

1, r are

calcu-lated by the formula

. - r .· J\:(ai, b;j"' P{TI>.(b;)(bi) - TI>.(a;)(ai)

=

ni, i

=

1, r}

=

E

f1

exp{-A(ai, bi)} ., ,

i=l •· ·

n,.

where

A(a, b)

=

J;a(x(u))du.

We mention that such type processes arises at studying input Poisson processes for queueing models in a random environment.

In terms of SP it is possible to describe various classes of stochastic models such as state-dependent queueing systems and networks of the types SMQ/MQ/1/oo,

MsM,Q/MsM,Q/l/k, (MsM,Q/MsM,Q/li/kiY with batch Markov or semi-Markov input, finite number of nodes, different types of calls (impatient calls and possibly of a random size - volume of information or necessary job) and batch state-dependent service, switched by some external semi-Markov environment and current values of queues.

Let us consider some examples.

Example 2.1: System MsM,Q/MsM,Q/1/V.

We consider a system in a semi-Markov environment with input calls and service portions of a random size depending on the current state of the system ( queue size or total volume of the information in the system etc.) and on the current state of the environment.

Let

x(t), t

2::

0 be a SMP with state space X given by the embedded Markov process x1

,l

2::

0 and the family of sojourn times {r(x),x E X}. Let also jointly independent parametric families of nonnegative random variables {17(x,a),x E X,a

2::

O} and {K(x,a),x E X,cx.

2::

O} and families of nonnegative functions {>.(x,cx.),µ(x,cx.),r(x,cx.), x E X,cx.

2::

O} be given where O � r(x,a) � 1.

An input flow is a Poisson one switched by SMP

x(t)

and the current state of a system. The calls have a random size which can be interpreted as a volume of information, necessary job, etc. The input flow is constructed in the following way. Denote by Q(t) the total volume of the information in the system at the time

t.

If x(t)

=

x and Q(t)

=

q

then with instantaneous intensity >.(x,

q)

the call of the size 77( x,

q)

can enter the system. This size is added to the total volume of the information in the system with probability 1 -

r( x,

q)

and with probability

r( x,

q)

it is lost. At the fixed state of a system the sizes of a different calls are jointly independent.

The system has one server with instantaneous intensity of service µ(x(t), Q(t)) and with capacity

V.

Each time when the service is finished the random portion of the information of the size min(Q(t), K(x(t), Q(t)) leaves the system.

To describe the process (x(t), Q(t)), t

2::

0 as a process with semi-Markov switches we introduce jointly independent families of decreasing step-wise MP { 'Yk(t, x, a), t

2::

0, x E X, a

>

O}, k

2::

0, with values in [O, oo) such that 'Yk(O,x,cx.)

=

a for each x,cx.,k and if 'Yk(t,x,cx.)

=

s then with instantaneous

intensity >.( x, s)

+

µ( x, s) the process can get a jump of the size

{3(

x, s) where

/3(

x,s ) _ { - -min{K(x,s),s} with probability µ(x,min{77(x, s),

V

- s} with probability >.(x, s)(>.(x, s)

+

µ(x, s)t1.

s)(>.(x,s)

+

µ(x,s)t1 •

We introduce families of processes (k(

t,

x, a):

Then the process (x(t), Q(t)) constructed as a

SP

according to formulas (2.5) by introduced families (k(·) and

SMP

x(·) is equivalent to the process (x(t), Q(t)) and is the process with semi-Markov switches.

Example

2.2:

Stochastic networks.

In the same way it is possible to describe state-dependent queueing networks with input calls and service portions of a random size controlled by some semi­ Markov environment.

Markov models of the type (MM,Q/MM,Q/mdv;Y with exponential intensities of input and service depending also on current volumes of an information in the nodes and a state of some external Markov environment are described as multidi­ mensional Markov processes that is a particular case of

MP

with Markov switches. We consider more complicated non-Markov network of the type (MsM,Q/MsM,Q/ 1 / oo

y

which is the analog to the system described in Example 2.1. The network consists of

r

nodes with one server in each node. Suppose for simplicity that each node has an infinite capacity.

Let

SMP

x(t) with values in some space X, parametric families of nonnegative functions {>..(x,q),µi(x,q),Pij(x,q), i = 1,r,j = O,r,x E X,ij ER�}, independent parametric families of random vectors {rj(x, q), x E X, q E R'.i.} and random vari­ ables {x:i(x,ij),

i

=

1,r,x E X,q ER�} with values in R� and R+ respectively be

given. Here Lj=oPij(x,q)

=

1 for any i,x,q. Denote by Qi(t) the total volume of

the information in the i-th node at the time t and put Q(t)

=

(Q1(t), ... , Qr(t)).

Then, on the interval where x(t)

=

x, Q(t)

=

ij the input flow is a multidimen­ sional Poisson one with parameter >..( x, q) and call size rj( x, q) (it means that the i-th component of the vector rj(x,q) enter the i-th node). The intensity of a service on this interval for the i-th node is µi(x, q). After finishing the service the random portion of the information of the size Ki(x,q)

=

min{x:i(x,q),q(i)} leaves the i-th node, where q(i) means the i-th component of the vector ij, and then this portion with probability Pij( x, q) passes into the j-th node,

j

=

1, r, and with probability

Pio( x,

q) leaves the network.

To describe the process (x(t), Q(t)) in the network as a

SP,

we introduce in­ dependent families of multi-dimensional Markov processes {1k(t, x, q), t � 0, x E X, q E R'.i.}, k � 0 with values in R+ in the following way: 1k(O, x, q)

=

q and if 'Yk(t,x,ij)

=

s then with instantaneous intensity A(x,s)

=

>..(x,s)

+

Li=1 µ(i)(x,s)

the process can get a jump of the size

/3(

x,

s)

with probability with probability with probability

i

=

1, r, >..(x, s)A(x, s)-1, (i)( -) (

-)A(

-)-1

µ

_{X' s Pij X' s} X' s '

where we denote by ei a column-vector in Rr _{which i-th component is 1 and the}

other components are 0.

Now the process

(x(t), Q(t))

can be described as the process with semi-Markov switches with help of processes 1k(t,

x,

q) in the same way as it was shown in Example 2. 1 .

B y analogy we can also describe the network ( SM/MQ / 1 / oo Y with semi­ Markov input ( calls enter the system in epochs of jumps of some

SMP).

These examples show that different types of stochastic queueing networks op­ erating in a random environment can be describes in terms of

SP.

3. ASYMPTOTIC MERGING OF STATES IN SWITCHING SYSTEMS

3.1. Convergence of switching processes.

In this section we consider general models of an asymptotic merging ( enlarge­ ment ) of states in the class of

SP.

An asymptotic merging property means that the parameter space of the initial

SP

can be divided on regions such that character­ istics of the limiting

SP

depend on the less number of parameters each of which corresponds to some region of the original parameter space.

Suppose that for each

n

>

0 we have jointly independent families

Fnk

=

{((

n

k(t, x, a), 'T

n

k(x, a), ,6

n

k(x, a)),

X E

X, a

E Rr},

k

2:

0

and the initial value

(x

n

o, S

n

o), n

=

1 , 2, . . . . This set for each

n

>

0 defines a

SP

(x

n

(t), (

n

(t)), t

2:

0 according to relations (2.7),(2.8).

Let

K(

· ) :

X

---*

Y

be a measurable map where Y is some metric space. Assume that

(y( · ), (( ·))

is some regular

SP

with values in

(Y,

Rr_{) determined by the families}

Fk

=

{((k(t, y, a), fk(y, a), ,6k(y, a)), t

2:

O, y

E

Y, a

E Rr}, k

2:

O and the initial value

(Yo, So).

We study general conditions fo r the convergence of the sequence o f merged in the first component

SP (K(x

n

(·)), (

n

(-)) (n

=

1 , 2, . . . ) to the limiting

SP (y(·), ((·)).

Definition 3. 1 :

We say that the sequence of random processes (

n

(·) J-converges

to

the process

(( ·)

on some interval

[O, T]

at n

---* oo

if the sequence of measures

generated by the sequence of processes (

n

(

·)

in Skorokhod space Dfo,

T]

weakly

converges

to

the corresponding measure generated by the process

(l · ) .

Denote

j

E exp{i I)>.i,

(

n

k(ti, x , a)) +

i(>.o,

t

n

k(x, a))

-

8T

n

k(x,

a)}

f(.6

n

k(x, a)),

(3. 1)

i=l

where

t

n

k(x, a)

=

(

n

k(T�k(x, a), x, a), >.i

E

R

r

, l

=

O,j, 0 �

t1 � ... �

tk,

e �

0,

f(·)

is a continuous function in

X.

Suppose that

X

is also some metric space. Let the function

'iflk(>.

0 , • • • ,

>.;,

t

_{1, . . . ,}

t;,

8,

f(·),

y, a)

be determined by the expres­ sion of the form (3.1) for the families

A,

k �

0.

Theorem

3.1 :

Let the following conditions hold:

1 . (K(x

n

o),

Sno) ::";

(Yo,

So).

(symbol ::"; means the weak convergence of distributions);

2. Given families of sets B

m

E By and D

m

E BR• ,

m �

0 such that for any

m

>

0,

u

0 E Dm,

9o

E

B

m

, for any sequences U

n -+

u

0

and V

n E

X such

that K(v

n

) -+ 9o and for any k �

0,

j �

0, >.o, . . . ,>.;,

t1 , ... , t;,

8, f(·) the

following relation is true:

lim V'nk(>.o, ... ,

A;,

t1 , .. , t; ,

8, f(K(·)),

Vn,

U

n

) =

n-+oo

'if/k(>.o,

... ,

>.;,

t1 , . . , t; ,

e,

/(·), go , uo),

and also there exists the sequence of intervals [O, Ti], where Ti

-+ oo

as

l

-+ oo,

such that the sequence of measures generated by the sequence

of processes (

n

k(t, V

n

, U

n

) is weakly compact in Skorokhod space Dfo,T,] on

each interval [O, Ti];

3.

P{fk(y,a) > O} = l,

y E Y, a E R\ k � O;

4.

P{yk

E

Bk}

=

1,

P{Sk

E

Dk}

=

1,

k �

0,

where sequences Yk , Sk, k �

O

are

constructed by SP (y(t), ((t)) (see {2. 7), (2. 8 )).

Then the sequence of merged processes (K(x

n

(·)), (

n

(·)) as n -+

oo

J-converges

to SP

_{(y(·), ((·))}

on each interval [O,

Ti] .

The proof of Theorem 3.1 is given in [3J. In most applications a limiting process has much more simple structure (for instance process with independent increments with Markov or semi-Markov switches). Therefore Theorem 3.1 gives us a new ap­ proach in asymptotic decreasing dimension at studying complex stochastic systems.

4. ASYMPTOTIC MERGING OF STATES IN MARKOV SYSTEMS

Here we consider applications of Theorem 3. 1 to models of an asymptotic merg­

ing (enlargement) of a state space for hierarchic nonhomogeneous in time Markov

systems. We prove that if the state space of the process can be divided on regions

such that transition probabilities between them are small in some sense then under

rather general conditions additive functionals on this process can be weakly ap­

proximated by processes with independent increments in Markov or semi-Markov

environment with number of states equal to the number of regions.

4. 1 . Quasi-ergodic Markov processes.

At first we introduce a special subclass of nonhomogeneous in time MP which

intensities are slow varying in time. For simplicity we consider finite space state.

Let X

n

(t), t � 0 be a nonhomogeneous MP with finite state space

X

= { 1 , 2, . . . , d}

given by the family of instantaneous intensities { a

n

( i, j, t), i, j E X, i # j, t � O}

of transition probabilities. Suppose that there exists a family of continuous func­

tions {a

0

(i, j, v), i, J. E X, i # j, v � O} and a sequence k

n

--t oo such that for any

i, j E X, i # j and any fixed

T

>

0

lim sup Ja

n

(i, j, k

n

v) - ao(i, j, v)I = 0.

n-+oo v< T

( 4. 1 )

For each fixed v � 0 denote by x�

v

)(· ) an auxiliary homogeneous MP with state

space

X

given by the family of intensities

{a

0(

i, j,

v

), i, j E X,

i

# j} and let rp(v)(.)

be its uniformly strong mixing coefficient:

<p(

v

)(u) = sup . _ max JP{x�

_{t�o i,3 EX,} v

)(t

+

u) E A/x�

v

)(t) =

i}-ACX

P{x�

v

) (t

+

u) E A/x�

v

)(t)

=

j} J .

(4.2)

Suppose that there exists q, 0

:S

q

<

1 and for any

T

>

0 there exists a constant

r(T)

>

0 such that for any v

:S

T

<p(

v

)(r(T)) :S q.

(4.3)

Statement 4. 1 : Let conditions

(4. 1), (4-- 3)

hold. Then for any v

>

0 as n --t oo

P{x

n

(k

n

v)

=

j} --t 7r

(v)

(j), j E X,

(4.4)

where 7r

(v)

_(j)

1

j E

X

is the stationary distribution of the MP x�

v

\ ·) which

exists under the assumption

(4 . 3).

4 . 2 . Asymptotic behaviour of the first exit time from the subset of states.

Now we consider the asymptotic behaviour of the time of the first exit from the subset of states for a nonhomogeneous MP. In most applications this time is usually interpreted as the time of the first failure of a system.

Let xn(t), t

2::

0 be a nonhomogeneous MP with finite state space X

=

{O, 1, 2, .. .. ,d} given by the family of intensities {an(i, l, t), i, l E

X, i #

l, t

2::

O}.

Denote by

On(io)

=

inf{t : t

>

0, Xn(t)

=

0 given that Xn(O)

=

io}, io

=

1,d, (4.5) the time of the first exit from the subset

X

₀

=

{1, 2, .. d}.

We study the asymptotic behaviour of the variable On( i0) at the assumptions

that the intensities of the exit tend to O and the subset {1, 2, .. d} forms in a limit a one connected subset.

Theorem 4.1: Suppose that there exists such s equence kn -+ oo that relation

(4 . 1)

holds, for any i, l E

X

0 , i # l, the auxiliary homogeneous MP x�v)(·) given

by the intensities {a0(i, l, v), i, l E

X

0, i # l} satisfies relation

(4. 3)

and for any

T > O

lim sup �ax supknan(i,O,knu)

< GT <

oo.

n-HX> •EXo u< T

Then for any io E Xo

where

lim sup IP{On(io)

>

knt} -exp{-An(t)}J

=

0,

n-+oo _t�O

and 7r(v) ( i), i E

X

₀ is the stationary distribution of the MP x�v)( ·)

( 4.6)

(4.7)

Remark: In homogeneous case (an( i, l, t)

=

an(

i,

l)

i.

e., intensities do not

depend on the time t) under the same assumptions we get

An(t)

=

t

kn

L

1r(

i)an( i, 0)

iEXo

that means an exponential approximation for the variable On(i0).

Proof. Let us denote by Xn( ·) an auxiliary nonhomogeneous MP with state space

by (xn(t), ITn(t)), t 2:

0

the two-component MP such that ITn(t) is a Poisson process

switched by the process Xn(t) with instantaneous intensity of a jump at the time

t an(xn(t), 0, t). Put

On(io)

=

inf{t : t

>

0, 11n(t) 2: 1, given that Xn(O)

=

io}, io E Xo.

By the construction it's not so difficult to prove that at each i0 E

X

0 the

variables On(io) (see (4.5)) and On(io) have the same distribution.

Then according to relations ( 4. 1),( 4.3) the process xn( ·) is a quasi-ergodic pro­

cess and Statement 4. 1 holds.

Now we can write a representation:

P{On(io)

>

knt}

=

Eexp{- i k.. t

an(xn(u), O, u)du}.

H k., t

H

Denote An(t)

=

E fo an(xn(u), 0, u)du.

( 4.8)

Using the inequality lea _{- e}/3 _{- e}f3_{(a - ,6) 1 � ! la - ,61}2 _{that is true at a,,6 �}

₀

we get from ( 4.8)

H 1 Ik..t H

IP{On( io)

>

knt} - exp{-An(t)}I �

2

EI

_Jo

an( Xn( U

),

0, U )du - An(t) l2. ( 4.9)

Denote by 'Pn( v) the uniformly strong mixing coefficient for the process Xn( ·) (by

analogy to (4.2)). From conditions (4. 1) and (4.3) it follows that for any T

>

0

there exists qi, q

<

qi

<

1 and r(T) such that

'Pn(r(T)) � qi, (4.10)

Taking into account the known inequality

Elan(Xn(u), 0, u)an(Xn(v), 0, v) - Ean(Xn(u), 0, u)Ean(xn(v), 0, v) I �

sup qn(x, 0, s)rpn(v - u)

that is true at any u

<

v, it's not difficult to get that El font qn(xn(v), v)dv - An(t) l2 --t 0.

But the relation (4.4) together with (4.6) implies An(t) - An(t) --t 0. This fact

finally proves our statement.

We mention that Theorem 4. 1 generalizes some results devoted to the asymp­ totic behaviour of the time of the first exit out of a subset of states studied for homogeneous MP and SMP independently in [1,3] and [17,19].

In terms of queueing systems On( i0 ) is equivalent to the moment of the first

loss of a call ( or overfilling of the system). For homogeneous in time models some applications to the analysis of the behaviour of complex renewable systems with quick repair were given in [5,7,22].

4.3. Asymptotic merging of states in nonhomogeneous Markov systems. Now we consider applications of Theorems 3.1, 4.1 to models of asymptotic merging (enlargement) of states in nonhomogeneous Markov systems. For simplic­ ity we consider a finite state space.

Let for each

n

>

0 Xn(t), t

2:

0 be a nonhomogeneous

MP

with state space

X

=

{1, 2, ... , d}, given by the family of instantaneous intensities of transition prob­ abilities an(i,l,t),i,l

=

1,d,i -f:.

l.

We suppose that the state space

X

for :Z:n(·) can

be represented in the form:

X

=

LJ

X

1, where

X

11

n

Xii

=

0 at J1

-f:.

J2.

jE Y

We introduce a map

K(

·) from

X

on

Y

such that K( i)

=

j

for any i E

X

1

and consider a merged process

K(:z:n(t))

=

j

at Xn(t) E

X

₁, t

2:

0.

(4. 1 1 )

It is well-known that the merged process K(xn(t)) in general is not a

MP.

Here we study conditions of its convergence to some

MP

under assumptions that intensities of transitions between regions

X

1 are small in some sense.

Suppose that intensities of transition probabilities are represented in the form an( i, l,

t)

=

a�0l( i, l, t)

+

_.!:_bn( i,

l,

t), i,

l

=

1, d,

n

where for any T

>

0

lim sup max sup lbn( i, l, t)I

<

Gr

<

oo,

n-+oo 1,I t< _{n T}

and for any

j

E Y, t

>

0

(0_{l( .}

l

_{) -} .

X l

_d

X

an i , , t

=

0 at i E j , _{.,:. j •}

( 4.12)

( 4.13)

( 4.14) Let functions a�0_{l( i, l,}

t)

_{regularly depend on the parameter}

t

_{in the following way:}

there exists a family of continuous functions {a0 ( i, l, u ), i,

l

=

1, d, i

-f:.

l,

u

2:

O}

such that for any

j

E Y and T

>

0

lim sup la�0l(i,l,nu)-ao(i,l,u)I

=

0, i,l E

X

1.

For each

J

E Y and fixed

v

2:

0 let us denote by x�>(t,

v

), t

2:

0 an auxiliary homogeneous MP with state space

X;

and intensities of transition probabilities

a

0( i,

l,

v

), i,

l

E

X;, i

=f

l,

and let us introduce a uniformly strong mixing coefficient

<p�\u,v)

= . .

_{11 ,,2 EX,,Acx,} max IP {x�>(u,v) E A/x�)(O,v)

=

i1}-( 4.16) Suppose that there exists q, 0 � q

<

1 and for any T

>

0 there exists a value r(T) such that for any

J

E

Y,

v

<

T

<p�)(r(T), v) � q. ( 4.17)

We mention that conditions ( 4.15) ( 4.17) mean that each subset

X;

forms a quasi­ ergodic set.

Further for each v

2:

0 denote by 1r�i)( i, v), i E X; a stationary distribution of a

MP x�)(t, v) (it exists under the assumption (4.17)). For any J E Y, m E Y,J

i-

m

we put

CLn(j,m,v)

=

L

1r;,i)(i,v)

L

bn(i,l,nv).

Suppose that for any

J,

m

E Y,

J

=f.

m

and any

t

>

0 there exist the following limits:

A(j, m,

t)

=

lim

r

t o.n(j, m, u )du

n-+oo

lo

where functions A(j, m,

t)

are represented in the form:

rt

A

A(j, m,

t)

=

_lo

>.0(j, m, u)du, and �0

(j,

m,

t)

are some continuous functions.

Theorem 4. 2 : Suppose that conditions

(4 . 1 1)-(4. 1 9)

hold and

K(xn(O)) � Jo.

( 4.18)

( 4.19)

Then the sequence of merged processes K( Xn( nt)) ]-converges on any in­

terval [O, TJ to nonhomogeneous MP y(t) with state space

Y,

initial state

Jo, given by the family of instantaneous intensities of transition probabilities

�o(J, m,

t),

J, m E Y,J

i-

m, and also for any i E X; and t

>

0

lim P {xn(nt)

=

i}

=

'lr�i)(i,t)P{y(t)

=

J}.

Now we consider the convergence of additive functionals of an integral type on the trajectory of the process Xn(t). Let {f( i, t), i E X, t

2

O} be a family of continuous functions. We consider a functional

Sn(t)

=

k

t

f(xn(nu), u) du. Denote j(j, t)

=

L

7r�j) (

i,

t)f(

i,

t), j

E

Y, t

2

_0.

iEXj

Theorem

4.3: Suppose that conditions of Theorem 4 . 2 hold. Then the se­ quence of processes (K(xn( nt) ) , Sn(t)) ]- converges on any interval

[O, T]

to

the process (y(t) , S(t)) where y(t) is defined in Theorem 4 . 2 and

S(t)

=

lot f(y(u), u)du. _(4.20)

It is possible to study also stochastic additive functionals. For example we consider a flow of Poisson type events.

Let {µn( i,

t),

i

E X, t

2

O} be a family of continuous nonnegative functions. Suppose that we have a Poisson type process switched by the trajectory of Xn(

·)

constructed in the following way: if Xn(t)

=

i

then an instantaneous intensity of a jump is µn(

i,

t). Denote by ITn(t) the total number of jumps on the interval [O,

t]

and put

Theorem

4.4: Suppose that conditions of Theorem 4 . 2 hold, for any T

>

0

lim sup max sup nµn(i,nu)

<

Cr

<

oo,

n-+oo 1EX u< T

and there exists a family of continuous functions {µ(j, v

),

j

E

Y, v �

O}

such that for any u

>

0

J�� AWl(u)

=

fou µ(j, v)dv.

Then the s equence of processes (K(xn(nt)'), ITn(nt)) ]- converges on any inter­

val

[O, T]

to the process (y(t) , IT(t) ) where MP y(t) is defined in Theorem 4 . 2 and

IT(

t) is a Poisson process switched b y y (

·)

(if at the time t y ( t )

=

j then the local intensity of a jump is

µ(j,

t)) .

Proof of Theorems 4.2-4.4. At first we represent the process K(xn(t)), Sn(t))

as a SP. In this case switching times are the times of transitions between dif­ ferent regions X_i and corresponding process (n(t,j,a) is constructed as an addi­ tive functional on the auxiliary process xWl(-) in the region Xj. Let us defined

it more formally. For any

j

E Y, l E Xj denote by xWl(t, l) an auxiliary MP

with state space Xi, initial state l given by intensities of transition probabilities

an(i, k, t), i,

k

E X₃, i -=p

k.

Also let rrWl(t) be a nonhomogeneous generalized Pois­

son type process switched by xWl(t,

l)

with the instantaneous intensity of a jump �n the �tate i at the time t n-1_{bWl( i, t), where bWl ( i, t)}

=

_{EulX; bWl( i, l, t), and the} Jump size

Ul( . t) _ {

l

with probability bWl( i, tt1bWl( i, l, t),

Kn i, -

_l

_<t

_X

i·

We consider a two-component process (xWl(t, Z), rrWl (t)). Denote by u + r,Vl( u,

l)

the time of the first jump of the process rrWl(t)) on the interval [u, oo) and by

f3!fl(

u,

l)

its size. Let us construct a SP Yn(t) with values in

Y

using to families of variables

{ r,Vl ( u, l),

f3�l(

u,

l), l

E X₃}, j E

Y.

Suppose that i0 E X_jJ is an initial value. Put

tno

=

0,

t nk+ l

=

t nk Tn

+

(.ii.l(t · ) . nk1 ink , lnk+ l

=

1-'A(.ini.l(t . ) . n nk1 Znk , )nk

=

_{K( . ) k} lnk , _

>

0 1

and denote

Yn(t)

=

)nk at tnk ::; t

<

tnk+ l , t � 0.

Then by the construction the process Yn(·) is equivalent to the process K(xn(·)).

Denote by rrUl(t) a generalized Poisson process with instantaneous intensity of a jump >.0(j,t)

=

Emh >.0(j,m,t) (see (4.18),(4.19)) and the size of the jump

(j) ( . _{t) _ {}

l

with probability >.0(j, m, t)>.0(j, tt1,

Ko i , -

_l

_<t

_X

j

.

If n -+ oo then according to Theorem 4.1 the distribution of the variable r,Vl(u,

l)

for any

l

E X_j, u � 0 weakly converges to the value: the time of the first jump of the process rrUl(

t)

- rrUl( u ),

t

>

u minus u, and correspondingly the distribution of a variable

f3�l(

u,

l)

weakly converges to the distribution of the size of the jump. But the SP constructed in this way is equivalent to the MP y(t) given in Theorem 4.2. Finally using the result of Theorem 3.1 we get the statement of Theorem 4.2. In the case of Theorem 4.3 we put (n(t,j,i)

= J;

J(xWl(nu,i),u) du. Then the

process (K(xn(t)), Sn(t)) by analogy can be represented as a SP using previous

satisfies the uniformly strong mixing condition, then the process (n(t, j, i) for any

i E X1 uniformly converges to the deterministic function

Ji

j(j, u )du and the lim­

iting process constructed with the help of limiting variables corresponds to the expression ( 4.20). Analogous conclusions are made at the proof of Theorem 4.4.

4.4. Asymptotic merging of states in queueing models.

Now we consider applications of Theorems 3.1, 4.4 to the asymptotic merging of states for some classes of queueing models.

4.4. 1 . System MM,Q /MM,Q /s/m in a fast environment.

1) A veraging of states of an environment.

Let

X

=

{1, 2, .. . , r} and let the families of continuous in argument

t

non-negative functions {a(i,l,t,q),>..(i,t,q),µ(i,t,q), i,l E X,i

-:f.

l,q E {0,1,2, .. . }} be given. System consists from s identical servers and

m

places for waiting. Denote by Xn(t) a

random process which forms a random environment for the system. Let us describe both the evolution of the system and the environment.

Calls enter the system one at a time. If at the time t the total number of calls in the system is Q, and Xn(t)

=

i, then the instantaneous intensity of an input is

>..( i, t, Q), the instantaneous intensity of a service for each busy server is µ( i, t, Q) and the process Xn(t) may jump from the state i to

l

with instantaneous intensity

na( i, l, t, Q) ( n is a scaling parameter and n -t oo ). After finishing service the call leaves the system.

Let us consider at each fixed ( v, q) an auxiliary homogeneous

MP

x ( u, v, q), u

2:

0 with values in

X

given by intensities of transition probabilities {a(i, l, v, q), i,

l

E

X,i

-:f.

l}. Let <p(u,v,q) be its uniformly strong mixing coefficient (see (4.2)).

Suppose that there exists d, 0 � d

<

1 and for any T

>

0 there exists a value r(T)

>

0 such that for any v � T and any

q

<p(r(T),v,q) � d. ( 4.21)

Denote by {1r( i, v, q), i E X} the stationary distribution for the process x( u, v, q), u

2:

0 and put

�(v,q)

=

L >..(i,v,q)1r(i,v,q), fl(v,q)

=

L µ(i,v,q)1r(i,v,q).

iEX iEX

Denote also by Qn(t) the total number of calls in the system at the time

t.

( 4.22)

Let M Q / M Q / s /

m

be a state-dependent queueing system such that if the total

intensity of the input is �(t,

Q)

and the instantaneous intensity of the service for each busy server is

µ(t,

Q)

(for this system the environment is absent). Suppose that the process Q(t) is regular (finite on each finite interval).

Statement

4.2:

If at our assumptions the condition (4 . 21) holds then the pro­ cess Qn(t) ]- converges on each finite interval to the process Q(t) .

Remark: This means that the queue for the initial system can be approximated by the queue for the system with averaging characteristics.

Proof. We consider a two-component MP (Qn(t) , Xn(t)) and describe it as a SP. In

this case the component Qn( · ) plays the role of an environment, Xn( · ) is a process

of Markov type switched by Qn( ·) and the proof of Statement directly follows from

Theorem 4.2.

2) Merging of states of an environment.

Now we consider the previous system ir1 the case when the process Xn( ·) allows an

asymptotic merging of its state space. Consider for s�.mplicity a homogeneous case. Let for each n

>

0 xn(t) , t

2:

0 be a homogeneous MP with state space

X

=

{1, 2, ... , r}, given by the family of intensities of transition probabilities {nan(i, l),

i, l E

X,

i _{-:/= l}. Let also the family of non-negative variables {).( i),}

µ(

i), i E X} be given.

The system is switched by the process Xn( · ) in such a way that if Xn(t)

=

i then

the instantaneous intensity of the input is >.( i) and the instantaneous intensity of the service for each busy server is

µ(

i).

Suppose that the state space X can be represented in the form ( 4.11) and values

an( i,

l)

are represented in the form

an( i,

l)

=

ao( i,

l)

+

_n

!_b(

i,

l),

i,

l

E X, i -:/=

l,

( 4.23)

where for any

j

E

Y

a0

(i,

l,

t)

=

0 at i E Xj,

Z (/.

Xj,

For each

j

E Y denote by x�) (t) , t

2:

0 an auxiliary MP with state space Xj

and intensities of transition probabilities {a0(i, Z),

i,

l E X_j,

i

-:/= l}. Suppose that

for each

j

this process is irreducible and denote by { 7rU) _{( i), i E X}

j} its stationary

distribution. Let us devote by y(t) a MP with values in Y given by intensities

{ci(j,

m), j -:/= m}, where

ci(j,

m)

=

I: I:

b(i,

l).

iEX1 IEXm

Denote by Qn(t) the total number of calls in the system at the time t.

Let MM/ MM/ s / m be a queueing system switched by the process y( ·) in such a way that if y(t)

= j

then the intensity of the input is

>.(j)

and the intensity of the service for each busy server is µ(j ) . Suppose that the process Q(t) is regular.

Statement

4.3: Under our assumptions the process Qn(t) ]- converges on each

finite interval to the process Q(t) .

Remark :

In this case a limiting queueing system is operating in the environ­ ment with merged space state and averaging of local characteristics

is

made in each asymptotically connected subset of this environment.

4.4.2.

System

MM/MM/s/m

with losses.

Now we consider a queueing system MM/M/s/m with losses in the case when environment is operating in the same scale of time as service processes in the system. Here we study the behavior on the large interval of time under the assumption that the environment admits an asymptotic merging of its state space.

Suppose that the family of non-negative functions A( i),

µ(

i), i E / and MP

xn(t) described above such that its state space can be represented in the form

( 4.11) and the condition ( 4.23) is satisfied, are given. The system has s identical servers and m places for waiting. The calls enter the system one at a time and if

xn(t)

=

i then the instantaneous intensity of an input is A( i) and the instantaneous

intensity of a service for each busy server is

µ(

i).

Denote by Pn(t) the probability of a loss for the call entered the system at the

time t. Let Zn(t) be the total number of calls lost on the interval [O, t] .

For the analysis of this system let us consider at first an auxiliary MP xUl_(t)

with state space X1 given by intensities {

a

0( i, Z), i, l E X1, i =I= l} and introduce

an auxiliary queueing system MZ,) /

M

/

s

/ m with losses constructed by the process xUl_{(t) and families of functions {,>i(i),µ(i), i E X}

1} as described above.

Denote for this system by q(j) a stationary probability for a call to be lost. Let y(t) be a process introduced in the previous subsection.

Statement 4.4: If Xn(O)

=

io E Xia then under our assumptions for any t

>

0

Pn(nt) --+ E q(y(t)), and for any T

>

0 the process � Zn(nt) weakly converges in

Skorokhod space _{D[o, TJ} to the process

Ji

q(y( u) )du.

Now let us consider the case when the system MM/ M / s / m works in conditions of " quick" service. It means that for any i µ( i)

=

µn( i) and µn( i) --+ oo. For

Suppose that we have the same assumptions about the process xn( · ). Denote

by 1rCil(i),

i

E _X1 the stationary distribution for the auxiliary _{MP x}Ci l_(t) given

above and let

A (j)

=

_I 1 m

L

7rCil(

i)>.(

i)(

>.(

�) y+m '

S . S _iEI,

c(

i)

Statement 4.5: Under our assumptions for any T

>

0 the process Zn( nt)

weakly converges in Skorokhod space D_[o, T] to a process of Poisson type switched

by the process y(t) introduced in the previous subsection (if y(t)

=

j then the local intensity of a jump is ACil) _{and also for any t}

>

0 _npn(nt) -+ _{E g(y(t))}

where

g(j)

= -/-

I:

1rul_{( i)(}

>-(

�) y+m .

s.sm

iEI, c(i)

These results mean that if parameters of a system depend on some Markov environment admitting an asymptotic merging of its state space, then under ap­ propriate scaling of the time the behavior of the system depends on another slowly varying Markov environment that is constructed by merging of the state space of the initial environment and by averaging of its transition probabilities.

Results given for queueing systems can be extended using the same technique on state-dependent and also nonhomogeneous in time queueing networks and give us a new approach in analytic modelling of wide classes of queueing models with hierarchic stochastic structure.

REFERENCES.

[1] V.V. Anisimov (1973): Asymptotic consolidation of the states of random pro­ cesses, Cybernetics, 9, 3, 494-504.

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