Asymptotic analysis of nonhomogenous hierarchical Markov processes and applications in models of the consolidation of queueing systems

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52 V. V. ANISIMOV

3. ASYMPTOTIC CONSOLIDATION OF STATES IN NONHOMOGENEOUS HIER ARCHIC AL MARKOV SYSTEMS

If the state space of an MP may be split up into asymptotically connected subsets such that the transient probabilities are asymptotically negligible, then we prove, under rather general conditions, that the accumulation processes on SP's weakly converge to a SMP with Markov switches and with the state space corresponding to the total number of subsets of the initial process.

First we introduce the notion of a class of nonhomogeneous MP's with characteristics slowly varying in some scale of time.

3.1. Quasi-ergodic Markov processes. For the sake of simplicity we consider the case of a finite number of states. Let _Xn(t), t 2 0, be an MP with the state space

X

=

{1, 2, ... , r} generated by the family of instantaneous intensities { an(i, j, t), i, j E

X, i -:j:. j, t 2 O} of the transient probabilities. Denote by r.pn(u, T) the uniformly strong

mixing coefficient (USMC) for a process Xn(t) on an interval [O, knT]:

(3.1) :Pn(u, T)

=

o::;;t::;;knT i,JEX,sup .. max ACX IP{xn(t + u) E A/xn(t) � i}

-_P{xn(t + u) E A/xn(t)

=

j}I.

Assume that there exists a family of continuous functions {a0(i,j,v),i,j E X,i -:j:. j, v 2 0} and a sequence of natural numbers kn --+ oo such that

(3.2) lim sup lan(i,j, knv) -ao(i,j, v)I

=

0

n-->oo v::;;T

for all i, _jE X, i -:j:. j, and T 2 0. For any fixed v 2 0 we denote by X6v) _{( ·) an auxiliary MP}

with the state space X generated by the family of intensities { a0(i, j, v), i,j E X, i -:j:. j}. Similarly to (3.1) we introduce the USMC r.p(v) ₍_u)_{for this process. Suppose that there}

exists q, 0 :::; q < l, and that, for all T > 0, there exists r(T) > 0 such that

(3.3) r.p(v_{l(r(T)):::; q}

for all v:::; T.

Assertion 3.1. Let conditions (3.2) and (3.3) be satisfied. Then for all v > 0

(3.4) j EX,

as n --+ oo, where 1r(v)_{(j), j} E _{X, is the stationary distribution (existing if} (3.3) _is

satisfied) for the MP X6v\·). Also there exists q, 0 :::; q < l, and for all T > 0 there exists r(T) > 0 _{such that}

(3.5) <{)n(r(T), T) :::; q.

An MP satisfying (3.4) is called a quasi-ergodic MP.

Proof. Denote by Xn(t), t 2 0, an MP generated at time t by the instantaneous transient

intensities _{{ao(i,j, t/k}n), i,j EX, i-:/- j, u 2 O} and such that Xn(O)

= X

n(O). Let

Pn(i,j, u)

=

P{xn(knu)

=

j/xn(O)

=

i}, Pn(i,j, u)

=

P{xn(knu)

=

j/xn(O)

=

i}.

By conditions (3.2) and (3.3), for all T we have

sup max lf>n(i,j, u) -Pn(i,j, u)I--+ 0 u::;;T i,J

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ASYMPTOTICS OF HIERARCHICAL MARKOV PROCESSES 53 s n _, 00 according to results of [6, Chapter 2, Section 1] (see also [7]) and relation (3.5)

� satisfied for the process Xn(t). Further, since functions ao(i,j,u) are continuous, anal

ogously to [6] and [7] we obtain for all u > 0 that lfin(i,j,u) - 1r<ul(j)I-+ 0 in view of

condition (3.3). This proves the statement of Assertion 3.1. D 3.2. Asymptotic behavior of the first exit time from a subset of states. Let Xn(t), t 2: 0, be a nonhomogeneous MP with a finite state space X

=

{O, 1, ... , d} generated by a family of instantaneous transient intensities { an ( i, l, t), i, l E X, i =/- l,

t 2: o}. Denote by

(3.6) On(io)

=

inf{ t: t > 0, Xn(t)

=

0, given Xn(O)

=

io }, io

=

1, ... , d,

the first exit time from the subset Xo

=

{1, 2, ... , d}. Let us investigate the behavior of

On(io) if the set {1, 2, ... , d} forms a single quasi-ergodic class as n-+ oo.

Theorem 3.1. Let there exist a sequence kn -+ oo satisfying condition (3.2) for all

i, l E Xo, i =/- l. Further, let the auxiliary homogeneous MP X6v\ ·) generated by the

intensities {ao(i,l,v),i,l E Xo,i =/- l} satisfy relation (3.3), and for all T > 0

(3.7) lim sup max sup knan(i,O,knu)

<Cr<

oo. n--->oo iEXo u<T

Then for all io E Xo

(3.8) where

lim sup IP{On(io) > knu} - exp{-An(u)}I

=

0, n--->oo u2:0

An(u)

=

kn 1 u

(

L

1f(vl_(i)a_n_{(i, 0, k}_n_{v)) dv,} O iEXo

and 1r<v)(i), i E X_o, is the stationary distribution for the MP X6v)_(,).

Remark. Under the same conditions we obtain in the homogeneous case (i.e. an(i, l, t)

=

an(i,l)) that An(u)

=

uknL_iEXo 1r(i)an(i,O), which means the exponential approxima

tion for On(io).

Proof. Denote by Xn(t) the auxiliary nonhomogeneous MP with the state space Xo

=

{1,2, ... ,d} and intensities of transitions {an(i,l,t),i,l E X0,i =/- l,t 2: O}. Further,

we denote by (xn(t), IIn(t)), t 2: 0, a two-component MP such that IIn(t) is a Poisson

process switched by Xn(t) and having instantaneous intensity of a jump an(xn(t), 0, t) at

the moment

t.

Put

Dn(io)

=

inf { t:t > 0, ITn(t) 2: 1, given Xn(O)

=

io}, io E Xo.

It is not hard to prove (see [6]) that for all io E X0 random variables On(io) (see (3.6))

and Dn(io) have the same distribution. According to relations (3.2) and (3.3), Xn(·) is a

quasi-ergodic MP and Assertion 3.1 holds. Now we use the representation

(3.9) P{On(io) > knu}

=

Eexp{ -1 knu

an(Xn(t),O,t) dt}.

Put An( u) = E f_oknu an(xn(t), 0, t) dt. Using the inequality le°'-e.8-e.B(a-,8)1 :S

!

la-,812,

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58 V. V. ANISIMOV

Assertion 3.2. Under the above assumptions and condition (3.22) the process Qn(t) J-converges on any finite interval to the process Q(t).

Remark. This means that the size of a queue in the initial system may be approximated

by the size of a queue in the limit system with averaged characteristics.

Proof. Consider the MP (Qn(t), Xn(t)) and describe it as a SP. In this case the component Qn(·) is the environment and Xn(·) is a process of Markov type switched by Qn(·).

Therefore the statement directly follows from Theorem 3.2. O 2) Consolidation of states of the environment. Now we consider the preceding sys tem in the case where the process Xn ( ·) admits the asymptotic consolidation of states.

Let families of continuous nonnegative functions {>,(i, t, q), µ(i, t, q), i, l E X, i =I- l,

q E {O, 1, 2, ... }} be given, where X

=

{1, 2, ... , r }. We suppose that representa tion (3.1 2) is valid (it is possible to consider the case where different q correspond to differ ent partitions) and families of continuous nonnegative functions {aUl_{(i, l,}

t,

_{q), i, l E X}

1,

i =/- l, bUl_{(i, k,}t, _{q), i E X}

j, k (j. Xj, j E Y, t � 0, q E {0, 1, 2, ... }} are given.

Let us describe the evolution of the system. We assume that calls enter the system one by one. If the total number of calls in the system at the moment t is Q and Xn(t)

=

i E Xj,

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

to a state l E Xj with intensities naUl(i, l, t, Q), l E Xj, or it may jump to a state k E Xm, m =I- j, with intensities bUl(i, k, t, Q), k (j. Xj.

For every fixed (j,v,q) consider the auxiliary homogeneous MP x(u,j,v,q), u � 0,

assuming values in Xj and generated by transient intensities {aUl(i,l,v,q), i,l E Xj,

i =I- l}. Let r.p(u,j,v,q) be its USMC (see (3.1)). Assume that the USMC satisfies

condition (3.2 2) for all j E Y. Denote by {n(i,j,v,q),i E Xj} the stationary distribution

of the process x(u,j,v,q), u � 0, and put

>-u,

v, q)

=

L

>-c i, v, q)n( i, j, v, q), fl(j, v, q)

=

L

µ(i, v, q)n(i,j, v, q),

(3.2 4) iEXj iEXj

b(j,m,v,q)

=

L

n(i,j,v,q)

L

bUl_(i,k,v,q).

iEX_j kEX_.,,,

Also denote by Qn(t) the size of a queue in the system at the moment t and put Yn(t)

=

K(xn(t)).

Introduce the system MM,Q/MM,Q/s/m switched by the process y(·) and described in the following way: if the size of a queue at the moment t is Q(t)

=

Q and y(t)

=

j,

then the instantaneous intensity of the input flow is >.(j, t, Q), the instantaneous intensity of the service for any busy server is µ(j, t, Q), and the intensity of the transition of the process y( ·) from a state j to a state m is b(j, m, t, Q) (note that the process y( ·) in the

general case is not Markov, since its transient intensities depend also on the current size of a queue). Assume that the process (y(t), Q(t)) is regular.

Assertion 3.3. If under the above conditions x_n(O)

=

io E X_j

0, then the process (Yn(t), Qn(t)) J-converges on any finite interval to the process (y(t), Q(t)), where y( O)

=

Jo.

Remark. In this case the limit system operates in an environment with a consolidated

state space and with averaged characteristics in every asymptotically connected subset.

3.4.2. Analysis of losses in the system MM,Q/MM,Q/s/m. As another example we

consider the same system operating in the same fast time scale as the environment. Let the system be described in the same way as that in Section 3.4.1 2) with the only

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60 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. V. V. ANISIMOV

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CHAIR OF APPLIED STATISTICS, DEPARTMENT OF CYBERNETICS, KYIV NATIONAL T. SHEVCHENKO UNI VERSITY, 2 ACADEMICIAN GLUSHKOV AVENUE, 252127 KYIV, UKRAINE & VISITING PROFESSOR, BILKENT UNIVERSITY, ANKARA, TURKEY

Current address: Department of Industrial Engineering, Bilkent University, Bilkent 06533, Ankara,

Turkey

Received 20/MAR/98