Switching stochastic models and applications in retrial queues

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Sociedad de Estadistica e Investigaci6n Operativa Top (1999) Vol. 7, No. 2, pp. 169-186

Switching stochastic models

and applications in retrial queues

V l a d i m i r V. A n i s i m o v

Department of Industrial Engineering

Bilkent University, Bilkent 06533, Ankara, Turkey ~J Department of Applied Statistics

Kiev University, Kiev 17, 252017, Ukraine e-mail: vlanisObilkent.edu.tr

A b s t r a c t

Some special classes of Switching Processes such as Recurrent Processes of a Semi- Markov type and Processes with Semi-Markov Switches are introduced. Limit theorems of Averaging Principle and Diffusion Approximation types are given. Ap- plications to the asymptotic analysis of overloading state-dependent Markov and semi-Markov queueing models MSM,Q/MsM,Q/1/oo and retrial queueing systems M/G/1/w.r in transient conditions are studied.

K e y W o r d s : Markov Process, Semi-Markov Process, Switching Process, Av- eraging Principle, Diffusion Approximation, Queueing Models, Retrial Queueing Systems

A M S s u b j e c t classification: 60K25, 60K15, 60F17

1 I n t r o d u c t i o n

Models of real information and c o m p u t i n g s y s t e m s have as usual a high di- mension and a complex structure. Exact analytic solutions can b e o b t a i n e d only for special rare cases, and m e t h o d s of a direct stochastic simulation work very slow and not efficient. Therefore a s y m p t o t i c m e t h o d s play the basic role at the investigation and a p p r o x i m a t e modelling.

In the p a p e r a new a p p r o a c h is suggested for s t u d y an a s y m p t o t i c behavior of wide classes of overloading queueing models and in particular retrial queueing systems in transient and stable regimes. This a p p r o a c h is b a s e d on limit theorems of Averaging Principle (AP) and Diffusion Ap- p r o x i m a t i o n (DA) types for so called Switching Processes (SP).

The paper was supported by INTAS Project 96-0828

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170 ~adirnir V. Anisimov

SP have a property that the character of their development varies spon- taneously at some epochs of time which can be random functionals of the previous trajectory.

The class of SP was introduced by Anisimov (1975, 1977). SP are the natural generalization of such well-known classes of random processes as Markov processes homogeneous in the 2nd component (E~ov and Sko- rokhod, 1969), processes with independent increments and semi-Markov switches (Anisimov, 1973), piecewise Markov aggregates (Buslenko et. al., 1973), Markov processes with semi-Markov interference of chance (Gikhman and Skorokhod, 1973) and Markov and semi-Markov evolutions (Papani- colaou and Hersh, 1972; Hersh, 1974; Kertz, 1978; Kurtz, 1973; Pinsky, 1975ab; Korolyuk and Swishchuk, 1994).

Theorems, concerning the convergence of the trajectory of SP to a solu- tion of some ordinary differential equation (AP) and the convergence of the normed difference to some diffusion process (DA), were proved by Anisi- mov (1989, 1991, 1992, 1994, 1995ab) and Anisimov and Aliev (1990) for different important subclasses of SP: recurrent processes of a semi-Markov type (RPSM), processes with semi-Markov switches and general SP with feedback between both components. The results of A P type for stochastic differential equations (Khas'minskii, 1968) and for some Markov models in the case of feedback (Skorokhod, 1987) based on a martingale technique also are very closely connected with this direction.

Different asymptotic approaches for various classes of general queueing systems are considered in books of Buslenko et a1.(1973), Basharin et al. (1989), Anisimov et al. (1987), and papers of Chen and Mandelbaum (1994), Harrison (1995), Harrison and Williams (1996), Anisimov (1998), Mandelbaum and Pats (1998), (see also references there). A new approach based on A P and DA type theorems for SP was used by Anisimov (1989, 1995a,b, 1996) and in the book of Anisimov and Lebedev (1992) for study various classes of state-dependent Markov queueing systems and networks in conditions of heavy loading. For non-Markov models some results were obtained by Anisimov (1989, 1995ab, 1996).

Retrial queues is comparatively a new direction in queueing models. Over recent years there were appeared many publications concerning the de- veloping of approximating methods and analysis of steady-state behaviour for different classes of retrial queueing models (see reviews of Yang and Templeton (1987), Falin (1990) and Kulkarni and Liang (1997), papers

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Switching models in retrial queues 171

of Artalejo (1995), Artalejo and Falin (1995,1996), Martin and Artalejo (1995), and book of Falin and Templeton (1997).

T h e 4th section of the paper is devoted to applications of A P and DA type theorems for SP in the asymptotic analysis of some classes of overloading state-dependent Markov and semi-Markov queueing models as well as for retrial queueing systems in transient conditions. Some results in this direction were obtained by Anisimov and Atadzhanov (1991,1994) and Anisimov (1999).

2 S w i t c h i n g S t o c h a s t i c P r o c e s s e s

SP can be described as two-component processes (x(t), ~(t)), t > 0, with the property existing a sequence of epochs tl < t2 ~ " ' " s u c h that on each interval [tk, tk+l), x(t) = x(tk) and the behaviour of the process ~(t) depends on the value (x(tk),~(tk)) only. Epochs tk are switching times and x(t) is the discrete switching component. A general definition of SP was given by Anisimov (1975, 1977). Now we consider some important subclasses of SP which appear in different applications.

2.1 R e c u r r e n t p r o c e s s e s o f a s e m i - M a r k o v t y p e ( R P S M )

Let Fk = {(~k(a),Tk(a)),a E 7~r},k _> 0, be jointly independent families of r a n d o m variables with values in T~ r • [0, oc) and So be independent of Fk, k >_ 0 r a n d o m variable in 7~ r. We assume here (and further) that variables introduced are measurable in the ordinary way in the variable a concerning a-algebra Bzer. Denote to = 0, t k + l = tk

-k Tk(Sk),

S k + l -~-

Sk + ~k(Sk), k > O and p u t

S(t) = S~ as tk ~ t < tk+l, t > O.

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T h e n a process S(t) forms a simple recurrent process of a semi-Markov type (RPSM) (see Anisimov, 1989, Anisimov and Aliev, 1990).

In the case when distributions of variables (~k(a), Tk(a)) do not depend on the parameter k, the process S(t) is a homogeneous semi-Markov process (SMP). If in addition these distributions do not d e p e n d on the parameter a, epochs tk, k _~ 0 form a recurrent flow and S(t) is a generalized renewal

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172 Vladimir V. Anisimov

process. In particular w h e n variables Tk(~) have exponential distributions, a process S(t) is a Markov process (MR).

F u r t h e r let Fk = { ( ~ k ( X , ~ ) , r k ( X , ~ ) ) , x E X , ~ E ~ r } , k >_ 0 be jointly i n d e p e n d e n t families of r a n d o m variables with values in ~ r • [0, c~), and let xt, l > 0 be an i n d e p e n d e n t of [Zk, k > 0 M P with values in X, (x0, So) be an initial value. We p u t to ---- 0, t k + l -~ tk + Tk(Xk, Sk), S k + l ---- S k -~-~k(Xk, Sk) , k ~_ O, a n d d e n o t e

S(t) -= Sk, x(t) = Xk as tk ~_ t < tk+l, t ~ O. (2.2)

T h e n the process ( x ( t ) , S ( t ) ) forms a R P S M with Markov switches. W h e n distributions of variables Tk(X,~) do not d e p e n d on p a r a m e t e r s c~ and k t h e process x(t) is a semi-Markov process (SMP).

2.2 P r o c e s s e s w i t h s e m i - M a r k o v s w i t c h e s ( P S M S )

Now we consider in some sense m o r e general construction w h e n we have an evolution of some r a n d o m process in a semi-Markov environment.

Let Fk = {~k(t,x,(~), t ~_ O, x E X , ~ E T~r}, k > 0 be jointly i n d e p e n d e n t families of r a n d o m processes, where ~k(t, x, ~) at each fixed

x(t), t > o

k, x, ~ be a process with trajectories in Skorokhod space 7)c~,

be an i n d e p e n d e n t of Fk, k > 0 S M P in X and So be an initial value. Denote by 0 = to < tl < -.. t h e epochs of sequential j u m p s for x(.) a n d p u t Xk : x(tk), k ~_ O. We c o n s t r u c t a process with semi-Markov switches (or in a semi-Markov environment) in the following way: put Sk+l = Sk + ~k, where ~k = ~k(Tk, 2Ck, Sk), Tk : t k + l - - tk, and denote

~(t) = Sk + ~k(t -- tk, xk, Sk) as tk ~_ t < tk+l, t > O. (2.3) T h e n a two-component process (x(t), ~(t)), t > 0 is t h e process with semi-Maxkov switches (PSMS). If we introduce also an i m b e d d e d process S ( t ) : Sk as tk ~_ t < tk+l, t ~_ O, t h e n the process (x(t), S(t)) forms a

R P S M with Markov switches.

In t h e case w h e n (~(t, x), t > 0} is t h e family of Markov processes and ~(t, x, c~) denotes the process ~(t, x) w i t h initial value ~, we get t h a t the process (x(t),~(t)) forms a Markov r a n d o m evolution (when t h e process x(t) is a MR) or semi-Markov evolution (when the process x(t) is a SMP)

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Swstching models in retrial queues 173

(see Hersh, 1974; Kurtz, 1973; Kertz, 1978; Pinsky, 1975; Korolyuk and Swishchuk, 1994).

In particular if {~(t, x), t > 0} is the family of processes with independent increments and the process x(t) is a MP, then the pair (x(t), ~(t)), t > 0 forms a M P homogeneous in the 2nd component (see E2ov and Skorokhod, 1969). If x(t) is a S M P then the pair (x(t), ~(t)), t >_ 0 forms a process with independent increments and semi-Markov switches (see Anisimov, 1973).

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/cx3, MSM,Q/MSM,Q/I/k, (MsM,Q/MsM,Q/li/ki) r 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.

E x a m p l e 2.1. Stochastic network

(MsM,Q/MSM,Q/1/oo) r.

We consider an information network consisting of r nodes with one server in each node. Suppose for simplicity that each node has an infinite capacity. Let S M P x(t) with values in some space X, families of nonnegative functions {~(x, q), #i(x, 9), i = i, r, x E X, 9 E R~+ }, independent families of random vectors {~(x,q), x E X, 9 E R~_} and {-~i(x,9), i = 1 , r , x E X , 9 E R~_} with values in R~_ and R~. +1 respectively be given. Denote by Qi(t) the total volume of the information in the i-th node and put Q(t) = (Q1 (t), ..., Qr(t)).

We assume that if x(t) = x, Q(t) = 9 = (ql,..., qr), then with inten- sity )~(x, q) a call of a random size ~(x, q) may enter the system (it means that the i-th component of the vector ~(x,q) enters the i-th node). Corre- spondingly if x(t) = x, Q(t) = q, then with intensity of service #i(x, q) the random portion of the information of the size ~i(x,9) may leave the i-th node, where ~i(x,-~) = ~ j = 0 ~ ~(j)i (x,q), and ~iO)(x,9 ) is j - t h component of the vector gi(x, 9). This means that the value ~I j) (x, q) passes into the j - t h node, j = 1, r, and the portion ~0)(x, q) leaves the network (we can always assume that ~i(x,9) <_ qi).

To describe the process (x(t), Q(t)) in the network as a SP, we introduce independent families of multi-dimensional Markov processes {~k (t, x, q), t _> O, x E X, ~ E R~_}, k _> 0 with values in R~_ in the following way: ~k(O, x, q) =

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174 Vladimir 11". Anisimov

and if ~k(t, x, q) = s then with intensity A(x, ~) = ),(x, ~)

+ Er=l

~ i ( x , 8) the process can get a jump of the size ~(x, ~) where

= + with prob. with prob. i = i l r , /~(x, ~)A(x, ~) -1 , /~(i)(x, ~)h(x, ~)-1,

where we denote for the vector b = (b (i), i = O, r) by [b]0 the vector (b (i), i = 1, r), and ~i is the vector with i-th component equal 1 and others equal O.

Now we introduce the family of processes r x, a) in the following way: ~ k ( t , x , a ) = ~/a(t,x,a) -- ~, t <_ Ta(X).

Then the process (x(t), Q(t)) constructed by introduced families ~a(') and S M P x(.) (see (2.3)) is the process with semi-Markov switches.

By analogy we may also describe the networks ( S M / M s M , Q / 1 / o c ) r with semi-Markov input (calls enter at epochs of jumps of x(.)).

3 A v e r a g i n g p r i n c i p l e a n d d i f f u s i o n a p p r o x i m a t i o n f o r P S M S Consider a triangular scheme. Let for each n > 0 Fnk = {r X, a), t >_ 0, X E X , a E R r } , k >_ 0 be jointly independent families of random processes in D r , Xn(t), t > 0 be an independent of them S M P in X, Sn0 be an initial value. Let also 0 - - - - tno < t n l < "'" be the epochs of sequential jumps of Xn('), Xnk = Xn(tnk), k > O. We con- struct a P S M S according to formulae (2.3): put Snk+l = Snk + ~nk, where ~nk ~-- ~nk(Tnk, Xnk, Snk), Tnk : tnk+l -- tnk, and denote

~ n ( t ) = S n k + ~ n k ( t - - t n k , Xnk, Snk) as tnk ~ _ t < t n k + l , t > O . (3.1) Then the process (xn(t), ~n(t)), t >_ 0 is a P S M S .

At first we study an A P for the switched component ~n('). Consider for simplicity a homogeneous case (distributions of processes ~nk(') do not depend on the index k > 0). Let Tn (X) be a sojourn time in the state x for S M P Xn('). Denote for each x E X , a E R r

= = s u p

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Suppose t h a t M P Xnk , k > 0 has at each n _> 0 a s t a t i o n a r y m e a s u r e rrn(A), A 6 B x and denote ran(x) = Ernl(X), bn(x,a) = E~nX(X, na),

a n ( k ) =

4)

Ib (x, where N; mn = f x m n ( x ) r r n ( d x ) , bn(ot) = /xbn(x,o~)rrn(dx),

sup

IP

{Xni E A, Xni+k 6 B} - P {Xni 9 A} P {Xni+k 6

B}[.

A , B 6 B x ,i>O

T h e o r e m 3.1. (Averaging principle). Suppose that n - l Sno 2+ so and: 1) there exists a sequence of integers rn such that n - l r n -~ 0 and SUPk>_r ~ an(k) -+ O;

2) for any N > O,e > 0 limn-~oosupx,lal<NP{n-lgn(X,a) > e} = 0; 3) limL-~oo lim supn-+oo sup[a[<y supz{Ernx (x)X(Tnl (X) > L) + El~n~(x, na)lX(l~(x, na)l > n)} = 0;

for any x as m a x ( l m l , [a2[) < N

Oq) -- bn(x, or2) [ < C N t a l -- a2[ + a n ( N ) ,

C N are some constants, a n ( N ) --+ 0 uniformly o n [Oq[ < N, [ax[ <

5) there exists a function b(a) and a constant m such that for any ot E R r bn(oL) --+ b(o~), mn -"r m > O.

Then for any T > 0

sup ]n-lr - s(t)] 2+ O, (3.2)

0 < t < T

where

s(0) = so, as(t) = m - l b ( s ( t ) ) d t

(it is supposed that a solution of the equation exists on each interval and is unique).

R e m a r k 3.1. Condition 1) covers also m o r e general situations t h a n only t h e case w h e n the process Xnk is ergodic in t h e limit. For instance state space can form an S-set (see Anisimov, 1973, 1996).

Now let us consider a DA for the sequence of processes on(t) = n-1/2( n(nt) - n s ( t ) ) .

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We introduce a uniformly strong mixing coefficient:

~on(r) = SUPx.y,A IP{xn~ e Alxno = x} - P{xnr e Alxno = Y}I"

Put b~n(c~) = bn(a)m; 1, b(c~) = b(c0m -1, pnk(X,C~) = ~nk(x, nc~) -- b,(x, a) - b(a)(Tnk(X) -- ran(X)), On(x, a) 2 = E p , l(x, a)pnl (x, c0*,

7 . ( x , ~ ) = b~(x, ~ ) - bn(~) - b ( ~ ) ( m n ( ~ ) - r a n )

T h e o r e m 3.2. (Diffusion Approximation). Suppose that On(O) =~ 0o, there exist fixed r > 0 and q E [0, 1) such that ~n(r) <_ q, n > O, conditions of Theorem 3.1 hold where ~/~an(N) --~ O, and ]or any N > O:

1) linvn~ooSuPla[<Nsup~nP{n-1/2gn(x,~ ) > e } = 0 , V~ > 0; 2) limL-,c~ limsuPn_~c ~ suplaI<N SUPx ( ETnl (X)2X(Tnl (X ) > L) + El~nl(X,

n(~)]

> L)} = 0;

3) IDn(x, al)2-Dn(x,(~2)21 < CglC~l--C~21+an(N), as max(Icql, Ic~21) < N, where C~n(N) --+ 0 uniformly in I~11 < N, la21 < N;

4) there exists a function q(a, z) such that for any N in the domain Ic~l < N Iq(c~,z)l < CN(l + lz D and uniformly in I(~1 < N at each fixed

v/n(bn(C~

-'1"-

•--1/2Z)

--

8(0/)) --9"

q(ol, z);

Z

5) there exist ]unctions D(c~) and S(c~) such that for any c~ E R m Dn(a)u = f x n n ( x , c 0 r n ( d x ) -+ n ( c 0 2, S(1)Q~2~-S(2)(c~)2+(B(2)(a)*)2n ~ j - n -+ S ( . ) : , w h e r e B(.1)(.) ~ = f x ' ~ n ( x , ' ~ ) ' ~ " ( ~ , ' ~ ) * ' n ( d x ) , B(.~)(~) ~ =

Ek>t E~(X~o,a)7~(x~k,a),*

at P{x~o e A} = lrn(A), A e B x .

Then the sequence o] processes On(t) weakly converges at each T > 0 in 9 the space T)~ to the diffusion process O(t):

0(0) = 00, d0(t) = q(s(t), O(t)) dt + m-U2(D(s(t)) 2 + B(s(t))2)U2dw(t), where w(t) is a standard Wiener process in R r, and the solution of this equation exists and is unique.

The proof of these theorems can be found in Anisimov, 1994. These results also can be extended on nonhomogeneous in time models (see Anisi- mov, 1995).

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4 D i f f u s i o n a p p r o x i m a t i o n i n q u e u e i n g m o d e l s 4.1 M a r k o v m o d e l s

We consider a general Markov queueing system of the type M - ~ / M - ~ / 1 / ~ which includes state-dependent systems with group input and service, systems with impatient calls, and even Markov networks.

Suppose that characteristics of the system depends on a parameter n , n -+ or Let non-negative functions )~(q),#(q),vi(q),i = 1, m , q E R ~ be given where R ~ = [0, cr m. Let also a(q), 7(q), fli (q), i = 1, m, q E Rr~ be random variables with values in R ~ . T h e system consists of one server and infinitely many places for waiting. Denote by Qn(t) the number of calls in the system at time t, Qn(t) E R ~ . Vector values may denote the different types of calls or different priorities. T h e system is working in the following way: if Qn(t) = nq, then with intensity A(q) (~(q) calls may enter t h e system, and correspondingly with intensity of service #(q) 7(q) calls may finish service. In addition to this each call of the type i independently of others with intensity n - l v i ( q ) may transform into ei +/3i(q) calls, where ei is a vector with i-th component is equal one and other components are equal 0. Vector /~i(q) may have positive and negative elements. All calls that enter the system, stand in the queue, calls after finishing service leave the system.

By the construction the process Qn(t) is a multidimensional MP. We represent it now as RPSM.

Denote by t n k , k >_ 0 epochs of sequential j u m p s of Qn(t) and p u t Tnk = tnk+l --tnk, Qnk = Qn(tnk + 0), ~nk = Qnk+l - Q n k , k > O. Let us introduce jointly independent families of r a n d o m variables Tnk(nq) and ~nk(nq) such t h a t P{rnk(nq) < t} = P{Tnk <: t/Qnk = nq}, P{~nk(nq) e A} =

A/C&k = nq}.

In terms of introduced variables the process Qn(t) is a RPSM. In our case it can be easily seen t h a t the value T,~k(nq) has an exponential distribution with parameter A(q) = A(q) +#(q) +t,(q) where v(q) = ~-]~iml qwi(q), q = (ql, .-., qm), and the value ~nk(nq) can be represented in the form:

{

a(q) with probability ,~(q)A(q) -1, ~nl(nq) = - 7 ( q ) with probability /~i(q)A(q) -1,

~i(q) with probability qivi(q)A(q) -1, i = 1,m. According to the construction the trajectory of R P S M coincides with tra-

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jectory Qn(t) on some interval [0, T] only if its trajectory on this interval completely belongs to the space R ~ .

Let us introduce the following values: m(1)(q) = E a ( q ) , m(2)(q) = ET(q) , ml3)(q) = Ej3/(q), dfl)(q) = E(x(q)~(q)*, d(2)(q)= ET(q)7(q)*, dla)(q) = Efli(q)l~i(q)*, and denote:

b(q) = m(1)(q)A(q) - m(2)(q)#(q) + ~-:~im=i ml3)(q)qivi(q), B2(q) = d(1)(q)A(q) + d(2)(q)#(q) + ~ i m l dl3)(q)qivi(q).

Let also G(q) be the matrix of partial derivatives for b(q) that is limh-~O h - i (b(q + hz) - b(q) ) = G(q)z, z E R TM.

T h e o r e m 4.1. I) Let the variables c~(q),'y(q), ~(q) be uniformly in q inte- grable in each bounded region in R ~ , functions A(q),#(q),vi(q),m~J)(q) be locally Lipschitz, A(q) > O, q E Rr~ and n-lQn(O) P> so.

Then

sup jn-lQn(nt) - s(t)J

P> 0,

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0 < t < T

the function s(t) satisfies the equation: ds(t) = b(s(t))dt, siO ) = so, and T is any positive value such that y(+oc) > T and in each component s(t) > 0 as t < T, where y(t) = fo q(~(u)) - l d u , and the function ~(t) satisfies the equation

r/(0) = so, dr/(t) = b(rl(t))q(rl(t))-ldt

(it is assumed that the solution of this equation ezists and is unique). 2). Suppose that in addition to these conditions variables

I~(q)l 2, h,(q)l 2,

[f~(q)J 2 are integrable uniformly in q in each bounded region in Rr~, func- tions B(q) and G(q) are continuous and n-1/2(Qn(O) - nso) :=~ @.

Then the sequence of processes ~n(t) = n-W2(Qn(nt) - u s ( t ) ) weakly converges in 79~ to a diffusion process ~(t) satisfying the equation:

d~(t) = G(s(t))~(t)dt + B(s(t))dw(t), ~(0) = @.

P r o o f . In our case the process Qn (t) is represented as a simple R P S M and the proof directly follows from Theorems 3.1, 3.2 given in Anisimov (1995a) (see also Anisimov, 1995b).

E x a m p l e 4.1. Consider state-dependent system M Q / M Q / 1 / o o with im- patient calls. It means t h a t if Qn (t) = nq, then the local intensities of input

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Switching models in retrial queues

179

and service are correspondingly A(q) and #(q), and each call in the queue i n d e p e n d e n t l y of others with intensity

n-iv(q)

m a y leave the system. Here we s u p p o s e for simplicity t h a t c~(-) - 1, ~(.) = 1.

T h e n in notations of T h e o r e m 4.1 3 ( a ) = - 1 , A(a) = A(a) + # ( a ) +

av(a), b(a) = A(a) - It(a) - av(a),

B2(a) = )~(a) + It(a) + av(a),

G(a) =

)~(a) -It~ (a) - v ( a ) -av~(a),

and we can write corresponding equations for

s(t)

a n d r

In particular if )~(q) - )~, It(q) -- It, v(q) - v, we get: ds(t) = (), - It -

vs(t))dt, s(O) = so

and d~(t) =

- v ~ ( t ) d t + ()~ + # + vs(t))l/2dw(t).

Solving these equations we find:

s ( t ) = _ It) _ _ It) _ s 0 ) e - V t , r = + w ( r

where r = v-l()~ - It)(e 2tv - 1) - v-l()~ - It -

uso)(e tv -

1).

We mention t h a t if )~ > It t h e n (4.1) is true for any T > 0 and in this case we have a q u a s i - s t a t i o n a r y point s* -- v-l()~ - It).

If )~ < It t h e n (4.1) is true on each interval [0, T] such t h a t T ~ / , I - 1 ln((It - )~ +

Vso)/(It

-

)~)~.

k /

4 . 2 S e m i - M a r k o v m o d e l s

Consider semi-Markov t y p e queueing s y s t e m

SM/MsM,Q/1/oc.

Let x(t), t > 0 b e a

SMP

with values in X such t h a t the e m b e d d e d

MP xk, k > 0

is uniformly ergodic w i t h s t a t i o n a r y m e a s u r e ~r(A),

A E B x .

D e n o t e b y

T(X)

a s o j o u r n time in the s t a t e x, x E X . Let non-negative functions #(x, (~), x E X, c~ > 0, b e given. T h e r e is one server and infinite n u m b e r of places for waiting. Calls enter the s y s t e m one at a time in the epochs of j u m p s tl < t2 < ... of the process

x(t).

If a call enters the s y s t e m at a

time tk

and the n u m b e r of calls in t h e s y s t e m becomes equal to Q, t h e n

the intensity of service on the interval

[tk, tk+l) is

I t ( X k ,

n - l Q ) .

Let

Sno

b e the initial n u m b e r of calls, and

Qn (t)

denote the n u m b e r of calls in the s y s t e m at the time t.

We p u t

m(x)

-~ ET(X), m =

fxm(X)~r(dx), b((~) =

(1 - c ( ( ~ ) ) m -1,

c(a) = f x i t ( x , ~ ) m ( x ) r ( d x ) ,

d 2 ( x ) =

V a r T ( x ) ,

d 2 = f x d 2 ( x ) r ( d x ) ,

el(or)

= f x #2(x, cr)d2(x)Tr(dx),

e2(~)

-= f x #(x,a)d2(x) r(dx),

D2(c0 -- c(c~) + e l ( a ) -t- 2(1 -

c((~))m-le2(~) +

((1 -

c(a))m-t)2d 2,

g(x,a)

= 1 - re(x)(1 - c(c 0 + # ( x , a ) m ) m

-1, G(a) = b'(a).

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180 Vladimir K Anisimov

T h e o r e m 4.2. Suppose that ]unctions #(x, a) are locally Lipschitz with respect to ~ uniformly in x E X , m > O, function c(~) has no more then linear growth and n-lQn(O) P ) 8 0. Then the relation (4.1) holds, where ds(t) = (1 - c(s(t))m-ldt, s(O) = so and T is any positive value such that

s(t) > 0, t [0, T].

If in addition the variables T(X) 2 are uniformly in x integrable, function c(a) is continuously differentiable, n-1/2(Qn(O) - n s o ) =~ ~/0, and there exists a variable B2(a) = E ( g ( x o , a ) 2 + 2 Ek~=l g(xo, a ) g ( x k , a ) ) , where P{xo E A} = 7r(A), A E B x , then the sequence of processes

~n(t) = n - U 2 ( Q n ( n t ) - n s ( t ) ) weakly converges in 7)T to a diffusion process "7(t) such that "7(0) = "Yo and

d'),(t) = G(s(t))7(t)dt + m-1/2(D2(s(t)) + B2(s(t)))l/2dw(t). P r o o f . We m a y represent the process Qn(t) as PSMS. Here times

tk

a r e

switching times and variable ~nk(X, na) can be represented in a form: n . ) = 1 -

where Hx(t) is a Poisson process with p a r a m e t e r ),. It is easy to see t h a t E~l(x, n a ) = 1 - # ( x , a ) m ( x ) and not so difficult to calculate a n o t h e r characteristics a n d apply T h e o r e m s 3.1, 3.2. We m e n t i o n also t h a t due to the condition s(t) > 0, t E [0, T] and Uniform convergence in (4.1) we get that P{Qn(nt) > O, t E [0, T]} -+ 1 t h a t is the t r a j e c t o r y of a constructed R P S M is a s y m p t o t i c a l l y equivalent to t h e t r a j e c t o r y of the queue.

In particular when # ( x , a ) =- a/~, our s y s t e m is equivalent to sys- t e m S M / M / c ~ with semi-Markov input and exponential service. In t h a t case c(~) = ~ t t m , b(~) = 1 ~ m - a # , G(a) = - # , D(c~) 2 = t i m e +

d2/m 2,

g(x, ~) = 1 - m ( x ) / m , a n d we obtain s(t) = ( # m ) -1 -- ((#m) -1 -- so)e -tit.

Some o t h e r examples of non-Maxkov and even non-semi-Maxkov models of the types GQ/MQ/1/oo, SMQ/MQ/1/cx~ a n d (GQ/MQ/1/oo) r are considered by Anisimov (1989, 1995ab).

4.3 Retrial queueing s y s t e m s

In retrial systems customers finding the server busy m a y join the special retrial queue a n d repeat their attempts for service after s o m e r a n d o m time.

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1). System M Q / G / 1 / w . r . Consider a system with one server and infinitely m a n y places for waiting. Calls enter the system one at a time. If the server is free it i m m e d i a t e l y takes the call for service. If the server is busy t h e call will wait in a queue.

Suppose t h a t system characteristics d e p e n d on some p a r a m e t e r n, n --~ oc. Let A(q), v(q), q > 0 be given non-negative functions. Denote by Qn(t) the n u m b e r of calls in the queue at time t. Let t,tl < tn2 < ... be sequential epochs of finishing service, tno = O. Denote Qnk =- Qn(tnk + 0). We assume t h a t on the interval [tnk, tnk+l ) a n input flow is a Poisson one with p a r a m e t e r A ( n - l Q n k ) a n d each call in the queue i n d e p e n d e n t l y of other calls with local intensity n - l v ( n - l Q n k ) m a y re-apply for service. If the server is free it takes the call for service. If the server is busy the call remains in the queue a n d repeats its a t t e m p t s for service later in t h e same way. A service time an d o e s n ' t d e p e n d on that, if a call comes from an input flow or from t h e queue, and has a general distribution function Bn(x) = P ( ~ n < x) with finite m o m e n t s of the first and second order rnn a n d m (2). We denote this s y s t e m as M Q / G / 1 / w . r .

T h e o r e m 4.3. 1). If functions A(q),t,(q) are locally Lipschitz, A(q) > O, r,(q) > 0 , q > 0 , h(q) = A ( q ) + q v ( q ) <_ L ( l + q ) , a s n - ~ oc rnn--+ m, n - l Q n ( O )

e) SO,

and variables t~n are uniformly integrable, then for

any T > 0 p

sup I n - l Q n ( n t ) - s(t)l ) O. (4.2)

O < t < T

Here the function s(t) satisfies the equation:

ds(t) = (A(s(t)) - m ( s ( t ) ) - l ) d t , (4.3) where re(q) = m + (A(q) + qv(q)) -1.

2.) If in addition to these conditions functions A(.) and v(.) are con- tinuously differentiable and as n -+ oa

v ~ ( m n - m) --+ O, rn(n 2) --+ ra (2), n - 1 / 2 ( Q n ( n t - nso) =~ @,

2 uniformly integrable, then the sequence of processes and variables gn are

~n(t) -= n - U 2 ( Q n ( n t ) - n s ( t ) ) , t > 0 weakly converges in E)T for any T > 0 to a diffusion process ~(t):

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Here g(q) = d~(A(q) -- r e ( q ) - 1 ) ,

D2(q) = A(q)m(q) + a(q)2m(q) -2 - 2 A ( q ) m ( q ) -1 ( A ( q ) + qp(q)) -2, a2(q) = a 2 + ( A ( q ) + q u ( q ) ) -~, a 2 = m ( 2 ) - m 2.

R e m a r k 4.1. As the function s(t) has continuous derivative and for any t such t h a t s(t) = 0 we get s'(t) = A(0) - A(0)(1 + A(0)m) -1 > 0, then from the relation so > 0 it follows t h a t a solution of the equation (4.3) is positive on any interval [0, T].

P r o o f . We represent the process Qn(t) as a SP. Let us choose times t n k as switching times. D e n o t e T n k ( n q ) = t n k + l -- t n k given that Q n k -= nq. According to the construction P(Tnk(nq) < X) = P(7/(A(q)) + an < x) where ~/(A(q)) is an i n d e p e n d e n t of an exponentially d i s t r i b u t e d r a n d o m variable with p a r a m e t e r A(q). Further denote ~nk(Qnk) = Qnk+l -- Qnk. T h e n we can write a representation:

P(~nk(Q.k) < x / Q n k = nq, an = z) = ($(q) + qu(q))-I • (qu(q)P(H~(q)(Z) - 1 <_ x) + A(q)P(Hx(q)(Z) <_ x ) ) , \ where we denote by Hb(t) a Poisson process with p a r a m e t e r b.

Now if we introduce an e m b e d d e d process ~)n(t) = Qnk as tnk < t < tnk+l, then the process Qn (t) is a R P S M c o n s t r u c t e d with the help of variables Tnk(nq), ~,k(nq). It is not so difficult to calculate first and second m o m e n t s of these variables and using results of the Paper (Anisimov and Aliev, 1990) to prove T h e o r e m 4.3 for the process Qn(t). Further in our case due to m o n o t o n e b e h a v i o u r of the t r a j e c t o r y of Qn(t) on each interval [tnk, t,~k+l) and the convergence of Qn(t) conditions 2) of T h e o r e m 3.1 and 1) of T h e o r e m 3.2 are a u t o m a t i c a l l y fulfilled. T h a t finally implies the s t a t e m e n t of T h e o r e m 4.3.

In particular if A(q) = A,u(q) = u we get a classical retrial system. In this case if Am < 1 there exists a s t a t i o n a r y point s* of the equation (4.3): s(t) ~ s* = )~2m((1 - Am)u) -1. In s t a t i o n a r y case (so = s*) we get s(t) = s* and the process ~(t) satisfies the equation:

d~(t) = - ( 1 - Am)2vr + XX/Aa 2 + 2m - ),m 2 dw(t), r = @. (4.4)

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Switching models in retrial queues

183

This is Ornstein-Uhlenbeck process and after solving (4.4) we get:

~(t) = e-argo + b

e-a(t-U)dw(u),

where a = (1 -

)~rn)2v,

b 2 = ,~2()~a2 + 2m - Arn2).

We remark that as t --+ oc the d i s t r i b u t i o n of r weakly converges to Gaussian d i s t r i b u t i o n with p a r a m e t e r s (0, b 2 ( 2 a ) - l ) . T h a t is in s t a t i o n a r y case at large n and t we can use an approximation:

Qn(nt) .~ ns + v/-db(2a)-U2Af(O,*

1).

T h e result of T h e o r e m 4.3 is also true if we have d e p e n d e n c e of functions )~(-), v(.) on the current n u m b e r of calls Qn(t).

These results can be e x t e n d e d on the base of T h e o r e m s 3.1, 3.2 on the case when we have additional Markov switches at the epochs

tnk,

and on the case w h e n the server is not reliable (see Anisimov, Atadzhanov, 1994). We mention t h a t in (Anisimov, 1999) analogous results

(AP

and

DA)

were o b t a i n e d for a s y s t e m

M Q / G / 1 / w . r

which is described as a one-server s y s t e m w i t h multiple Poisson input (a call of a t y p e i has an input rate )~i, i = 1, .., r, r < cx)), general service d e p e n d i n g on the t y p e of a call, and intensities of r e p e a t e d calls in the queue {vi(~), ~ E 7~_} depending on the current vector of all waiting calls.

2). S y s t e m

M / M / m / w . r .

Consider now a s y s t e m with m identical servers and intensity of service #. An i n p u t is a Poisson flow of identical calls with p a r a m e t e r )~. Denote by Rn(t) a n u m b e r of b u s y servers at time t. Let families

{pi(s),qi(s),ri(s), i = O, 1 , . . , m } and

{v(s),a(s),g(s)}, s >_ 0

of continuous nonnegative functions b e given. Here

pi(s) +qi(s) +ri(s)

= 1,

o~(s) +g(s)

= 1 for any s _> 0, i = 0, 1 , . . , m . Let Qn (t) denotes the n u m b e r of waiting calls at time t. T h e n if a call enters the s y s t e m at time t and (R~(t),

Qn(t)) = (i, nq) (i <_ r)

then with p r o b a b i l i t y

Pi(q)

it i m m e d i a t e l y takes one of free servers, with probability

qi(q)

the call directly goes to the queue, and w i t h p r o b a b i l i t y

ri(q)

the call gets a refusal and leaves the s y s t e m (if i = m t h e n we p u t

Pro(q)

= 0). If

Qn(t) = nq,

each call in the queue i n d e p e n d e n t l y of others can re-apply for service with local intensity

n-iv(q).

If there is a free server, it i m m e d i a t e l y takes the call to serve. If a call finds all servers busy, then it remains in the queue w i t h p r o b a b i l i t y (~(q) or with p r o b a b i l i t y

g(q)

it leaves the system.

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T h e process (Rn(t), Qn(t)) is a M P a n d we can represent it as a R P S M and using t h e result of Anisimov (1991, 1992) it is possible u n d e r some conditions of regularity to prove t h a t the process n - l Q n (nt) weakly converges in DT to a solution of a differential equation (see Anisimov, 1999).

These results show t h a t a technique based on limit t h e o r e m s of A P a n d DA t y p e s for SP gives us a new effective approach for s t u d y transient and stable regimes of operating r a t h e r complex queueing models u n d e r overloading conditions. It also gives us a possibility instead of direct simulation to use an a p p r o x i m a t e relation Qn(nt) ~ us(t) + v/-~(t) for calculation different reliability and cost functionals of the system.

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Switching stochastic models and applications in retrial queues