Quasi-Stationary Distributions and One-Dimensional Circuit-Switched NetworksAuthor(s): Ilze ZiedinsSource: Journal of Applied Probability, Vol. 24, No. 4 (Dec., 1987), pp. 965-977Published by: Applied Probability TrustStable URL: http://www.jstor.org/stable/3214219Accessed: 03/06/2010 21:24

Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available athttp://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unlessyou have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and youmay use content in the JSTOR archive only for your personal, non-commercial use.

Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained athttp://www.jstor.org/action/showPublisher?publisherCode=apt.

Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printedpage of such transmission.

JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact support@jstor.org.

Applied Probability Trust is collaborating with JSTOR to digitize, preserve and extend access to Journal ofApplied Probability.

http://www.jstor.org

J. Appl. Prob. 24, 965-977 (1987) Printed in Israel

© Applied Probability Trust 1987

QUASI-STATIONARY DISTRIBUTIONS AND ONE-DIMENSIONAL CIRCUIT-SWITCHED NETWORKS

ILZE ZIEDINS,* University of Cambridge

Abstract

We discuss the quasi-stationary distribution obtained when a simple birth and death process is conditioned on never exceeding K. An application of this model to one-dimensional circuit-switched communication networks is des- cribed, and some special cases examined.

BIRTH AND DEATH PROCESSES; COMMUNICATION NETWORKS

1. Introduction

Let {N(t), 0 t <o} be the birth and death process on state space N = (0, 1, 2, * * *) with upward rates A and downward rates from state n of n#. Now suppose that the process N(t) is restricted to the set of states (0, 1, 2, .. , K + 1) with K + 1 an absorbing state, and call this absorbing process {NK(t), t > 0}. Let pij(t) be its transition probability from state i to state j in time t. Define the conditioned process {NK(t), - oo < t < a } to be the process with transition probabilities from state i to statej in time z given by

pf)(T) = lim lim P(N(t + T)= j IN(t)= i, N(r)< K, 0 < r s). t-oo s- oo

That is, we condition on the process N(t) never exceeding K. We call the conditioned process NK(t) the quasi-stationary process for K. The process NK(t) has a stationary distribution {(,K(i), 0 < i < K} which we shall refer to as the quasi-stationary distribution for K.

Darroch and Seneta [1], [2] discuss the concept of quasi-stationary distribu- tions for both discrete- and continuous-time finite Markov chains. They were looking for distributions that would describe the transient behaviour of an absorbing process, hence the term quasi-stationary. In [2], two possible quasi- stationary distributions for a continuous-time Markov chain are described.

Received 3 March 1986; revision received 28 July 1986. * Postal address: Department of Actuarial Mathematics and Statistics, Heriot-Watt University,

Riccarton, Edinburgh EH14 4A5, U.K.

965

ILZE ZIEDINS

For the process N(t), they are, for T= {, 1, 2, * -, K}, jE T, and K + 1 absorbing,

(K') = lim lim P(N(t) in statej at time s < t t s-oC t-'Or

(1) absorption has not occurred by time t)

fi Pij(s)(1 -Pj,K+ ,(t -S)) = lim lim iET

- t- flz i(1 - Pi,K+(t)) iET

and

rK(J) = lim P(N(t) in state at time t |

(2) absorption has not occurred by time t)

fl PiP(t)

t- f ,i( - Pi,K+1(t)) iET

where {fi, 0 <i _ K} is any initial distribution. Note that {7K(i)} can be viewed in two ways - as the stationary distribution of the conditioned process NK(t), and as the quasi-stationary distribution of the absorbing process NK(t). Both {7K(i)} and {(K(i)} have been studied in the literature, although more

commonly in a discrete-time setting (see e.g. [3], [7], [8]). This paper will deal solely with the distribution {rK(i)} and we describe below a model where it is appropriate to use it. Henceforth, when talking of 'the quasi-stationary distribution' we will be referring to {(K(i)}.

In Section 2, exact expressions are found for the quasi-stationary distribu- tion and the transition rates of the quasi-stationary process NK(t). It is also shown that the probability that the quasi-stationary process NK(t) does not exceed K - 1 for a time z decays approximately exponentially as z -> 0, with a

given decay parameter. Examples are discussed in Sections 3, 4, and 5. The motivation for this work arose from the following model of a one-

dimensional circuit-switched communication network. Suppose that there are K circuits of infinite length along a line. Calls can begin and terminate at any point along the line. To describe the arrival process of the calls, we suppose that the line has been marked off at unit intervals. Requests for calls arrive at each interval as a Poisson process of rate A, and the left-hand end of the call is then assigned randomly within the interval (i.e. following a U(O, 1) distribu- tion). The value of A is assumed to be the same in each interval. From its left- hand end, the call requests a length of circuit that is exponentially distributed with mean 1/u, independently of its arrival time and position. If there is a

966

Quasi-stationary distributions and one-dimensional circuit-switched networks

circuit free for that length, or if it can be made free by rearranging the circuits used by other calls, then the call will be accepted. Otherwise it will be lost. Note that we could just as well have assumed that right-hand ends of calls arrive as a Poisson process and worked from there - it makes no difference in the final model. The holding periods of the calls are exponentially distributed, with mean 1. Holding periods, arrival times and lengths of calls are distributed independently of one another.

Associated with this model is the process {NK(t,s), 0 < t <oo, - oo < s < oo , where NK(t, s) gives the number of calls in progress at time t at the point s along the line. We claim that, as T - oo, the cross-section of this process at time z will look like a sample path of the conditioned birth and death process NK(t). It is not the intention of the author in this paper to make a rigorous connection between the conditioned birth and death process and the interpretation of it as a one-dimensional network. However, to support our claim, note that the model described is the natural continuous analogue of the discrete model of Kelly [4]. In [4], calls are assumed to begin and end at discrete fixed points along the line; arrival rates of left-hand endpoints of calls at each point are Poisson and decay geometrically with increasing length. The holding periods of calls are identically and independently distributed. Kelly shows that the number of circuits busy on successive links of the network at a fixed point in time is a Markov chain, with a stationary distribution and transition probabilities given by the discrete analogue of the quasi-stationary process NK(t). Our model is then a particular limiting case of that described by Kelly. The main advantage of discussing the continuous model of the one- dimensional network, rather than the discrete, is that analysis of the system is considerably simplified.

The examples discussed in Sections 3, 4 and 5 are particularly interesting in the context of this interpretation. In Section 3, the single circuit network is analysed in some detail. Section 4 looks at the limiting regime as i -- 0, first with A held fixed and then with A# = v held fixed. This corresponds to a situation where the length of the offered calls is uniformly distributed. In Section 5, the special case A = (K + 1 ) is addressed. In all three cases explicit expressions for the equilibrium distribution and the transition rates of the quasi-stationary process are given. The decay rate for the acceptance of calls as their length increases is approximately exponential, and in each case the decay parameter is obtained.

2. The quasi-stationary distribution

Let Q be the matrix of transition rates for N(t), with qnn = - (A + n), n,n + I = )A, n,n - I = nq, qij = 0 if I i -j I > 1. Define the (K + l)th section of Q

to be the (K + 1) X (K + 1) matrix in the upper left-hand corner of the matrix Q, i.e. its (K + 1) X (K + 1) northwest corer truncation, and call this QK+ 1.

967

ILZE ZIEDINS

-A A 0 .... . 0

/ -(A +i) A

0 2 -(A + 2,u) A *

QK+ I ..

_0 . .. K -(A.+Ku)_

We shall need the following lemma.

Lemma 2.1. Let aK+ I(t) = 1 IO - QK+ 1 be the characteristic polynomial of the (K + 1)th section. Then, for K > 1,

aK+ (O)= E( ) K+I n i (o + i). n -=0 nt i=0

Proof. The proof is by induction. We have immediately a1(5) = A + 5 and a2(0) = A2 + 2A)k + /4u + 4). Suppose the hypothesis is true for all i< n + 1 and observe that the following recursive relation holds:

(3) a + 1() = (A + n# + O)a,(¢) - nia,n_ -().

The term in A"-m +1, 1 m _ n, on the right-hand side is, by hypothesis,

/n\ mr- /n m-2 A ) n-m n (ok+ i#)+(ny + ) ( I n-m+' (o + i#)

\m/ ,=o m - 1 (=o

m I i=o

which reduces to

,n-m+1 ( + (k+ (n ( + i)m, m i-O

the desired result. It is easily seen that the terms in An + and A° also have the

required form, and the result follows.

The theorem giving the quasi-stationary distribution can now be stated and proved.

Theorem 2.2. Let ¢f+ ' be the eigenvalue of least modulus (it is real and strictly negative) of the matrix QK+I. Then the quasi-stationary probability distribution {rlt(i) for the process NK(t) is, for 1 _ i _ K

968

Quasi-stationary distributions and one-dimensional circuit-switched networks

Proof. From Darroch and Seneta [2],

K(i) = WiVi7tK(O), 1 < i < K

where v and w are the positive left and right eigenvectors corresponding to K+ i'. Without loss of generality assume that w0 = vo 1. Then

Vi =-- i! #

and

This is because the wi must satisfy

-n#w_l + (n+n + ; + n#)wn - A;W+ = 0.

That is,

- n#fa_1 ({K+' + A + n+)an _ Aa + + I 0,

A,n-I , n An+1

which is true if and only if

- n;a, l + (u +' + + na)a, - an1 = 0.

But this is just (3). Similarly, the vi must satisfy

- _ + (K +(A + +un)v, -(n + 1)uv,+1 = 0

and again, from (3), we see that they do satisfy these equations.

It is also possible to find the transition rates of the quasi-stationary process.

Theorem 2.3. The quasi-stationary process NK(t) has upward transition rates from state n

;tK(n) = n0 <n < K - 1

and downward transition rates from state n

#K(n) = n;# n

I1 _-< n -< K, lwhere we define a'(s+l)= 1.

where we define ao(OK' 1) = 1.

969

ILZE ZIEDINS

Proof. Consider the downward transition rates first:

P(N(t + St) = n | N(t) = n + 1, N(t + s) < K)

Pn + 1,n (t)(l - Pn,K+ 1(S - 0t))

1 - Pn+1,K+ (S)

u(n + 1)(St + o(St))(1 - Pn,K +(s - 6t))

1 - Pn+l,K+1(S)

Dividing by 6t and taking the limit as 6t - 0 we get

~(4) U/1+(n + 1)(1 - Pn,K+1i())

1 - Pn + K+i(S)

But from Darroch and Seneta [2], K

1 - P,,K+(s) = c E (exp(ps)wvi + O(exp(p's))) i=0

where p = K+ 1, p' <p, and c is a normalizing constant. Thus the limit as s - x of (4), after dividing top and bottom by exp(ps) is

,/(n + 1)w. (n + 1)na,,(fI+l)

Wn + an+1(¢l)

Similar arguments give the upward transition rates.

The following relationships exist between the arrival and departure rates of the quasi-stationary process.

Lemma 2.4. Let K((n), #K(n) be as defined in Theorem 2.3. Then

(i) ; > ;K(0)> K(1)> ...

and

(ii) #K(n) > nu, uK(n + 1)- U(n)> .

Proof. The proof of (i) follows by induction, using the recursive relation-

ship (3). The proof of (ii) follows upon applying the results of (i).

The polynomial aK+ 1(¢) and its root of least modulus figure prominently in the results above. It is therefore useful to investigate some of their properties. Let K+1, 1 <j < K + 1} be the eigenvalues of the (K + l)th section, and

suppose that they are ordered so that j:+1 < K+ 1 . Ledermann and Reuter [6] have shown that for a birth and death process, the eigenvalues of the Kth section will be distinct, and ifA0 > 0 (as it is in this paper) then the eigenvalues will all be strictly negative. Furthermore, the eigenvalues of the Kth section separate those of the (K + 1)th section:

(5)

970

,OK++l < f < K+ + 1, 1 < i < K.

Quasi-stationary distributions and one-dimensional circuit-switched networks

Thus if A and , are held fixed then the 4K are strictly increasing in K, and bounded above by 0; therefore the 0K must tend to a limit as K-- o0. But Keilson and Ramaswamy [3] have shown that as K-- oo, {7tK(i)} converges in distribution, and hence elementwise, to the stationary distribution {(e } of the process N(t). Now ei = eo(i/#)ii!. From Darroch and Seneta [2], tK(l) =

(vl/#),tK(O). So el = (A/u)eo, and *K(l) - el, implying that v, = i + K+ 1 -*Aas K - o0. Thus O K+ 1 I 0 as K - oo. Observe also that ai(0) > 0 and therefore

(6) a(x) >0 forx> .

In particular, from (5), this implies that ai(fK) > 0 for 0 < i < K. To conclude, we show that the probability that the quasi-stationary process

does not exceed K- 1 for a time T decays approximately exponentially as T o0.

Theorem 2.5. Let the conditions of Theorem 2.3 and Lemma 2.4 hold. Then

PK-1(T) = P(NK(t) - K- 1, s _ t < + r)

- c exp(xKTr) as r - oo

where c is a constant, and XK is the eigenvalue of least modulus of the Kth section of QK, the Q-matrix of the quasi-stationary process for K.

Proof. Let qiK() = P(NK(t)- K- 1, s _ t < s + NK(s) = i). Then

K-i

PK-1(T)= E 7K(i)q'K(T) i=O

K-1 K-1 = K(i) BK(exp(X% T)oiVj + O(exp(xr)))

i=0 j=0

with v and to the left and right positive eigenvectors corresponding to IX, X < X and BK a normalizing constant. Hence the result.

In terms of our interpretation of the quasi-stationary process as modelling the behaviour of a one-dimensional network, Theorem 2.5 says that the probability of a call being accepted decreases at an approximately exponential rate (with parameter XK) as the length of the call increases.

3. The single-circuit communication system

IfK = 1, then it is possible to calculate exactly the quasi-stationary distribu- tion for all values of u and A.

Applying the results of Section 2, we have

971

ILZE ZIEDINS

2 - (2A + s) + v\(4A/ + #2) 4l=1 2

2l( + - + /(4#u + #2)

2A-

- + 2(4~# + #2)

MA - + (4 + #2)

2

This immediately presents us with at least one rather surprising fact. For observe that 71(1) < 7r1(0), no matter what the relationship of i to ,. Also

lim.o0 r1(O) = lim_.0 7r,(1)= ½. Kelly [4] finds a similar result for a discrete network. In terms of our network model this is saying that, on average, the line will be no more than half full, no matter how high the arrival rate of calls! Furthermore, it can be shown that if u == A/c, then for c large 1/,ul(l) /c/lA. Again, in terms of our model, if the mean length of the offered traffic is clA then the mean length of the accepted traffic will be approximately /c/Al as c - co.

4. Uniform call distribution

We consider now the form of the limiting distribution as , - 0. In our

interpretation of the model this corresponds to a situation where the likelihood of a call being requested does not depend on its length - that is, the line-length of requested calls is approximately uniformly distributed. Lagarias et al. [5] analyse the discrete analogue of such a system. They look at a discrete finite one-dimensional network under the assumptions that no more than one call can connect any two points, and that given this, all configurations are equally likely. They are then able to derive an asymptotic expression for the capacity of the system.

The case K = 1 has already been dealt with in Section 3 above. By analogy with that case, consider a possible solution to aK+ 1() = 0 of the form

K+1 = - (A + yK+11) + V\(XK+ 14)

where XK+ 1 and YK+ 1 are constants to be determined, and depend on the value of K + 1 under consideration. Note that because of the form of the recursive relation

a+1( + 1) = (+I + K + nu)an (K + 1,)- nA a_-I(-K )+ 1)

= ((n - K+ 1)I + v/(XK+ 1,))a (K+ 1) - n._an - 1(K+ 1)

a solution of this form ensures that all the terms in aci(K+ 1) containing a factor Ai, with j > i/2, will have coefficient 0, no matter what the values of XK + and

972

Quasi-stationary distributions and one-dimensional circuit-switched networks

YK+ . Furthermore, the coefficient of (A#)i2 in ai(rK+ 1) depends on ,K+ only through XK+1. We denote the coefficient of (AiU)"2 in ai(K+ ), i < K + 1, by Si(XK+ 1) to emphasize this dependence. Thus, if we can find an XK+1 and YK+ such that I (K+ - OK+ = 0(u), we will have in the limit

UK(i)= nK(O)(si (XK

+1) ), 0 < iK.

We claim that if XK + = SK+1 where SK+ is the maximal positive solution to

SK+ (XK+ 1) = 0 then K+ 1 will satisfy I K+ - o+ 11 = O(u). The argument is inductive - we give a brief outline. It is easy to verify the hypothesis for K = 1. Supose now that, for 2 _ i _ K, si is the maximal positive solution to Si(x) = 0 and that I [ - O\ I = 0(u). Let SK+ be the maximal positive solution to SK+ 1(XK+ 1) = 0. We show first that SK < SK+ . Note that the Si(x) satisfy the

following recursion:

So(x)= 1

(7) Sl(x)= /x

Si + l(X) = /X Si(x) - iSi -_ (x), i 1 .

Hence we have SK+ (K) = - KSK_ 1(K) < 0, since (6) implies that for u small

enough aK- (SK)> 0. Thus the coefficient SK-I(SK) of the leading term in A must be positive. But also, limx_, ac(x) > 0 implies that limx_-, SK+ 1(x) > 0 and hence there exists at least one root of SK+ l(x) = 0 that is greater than SK

and so the maximal root SK+I must certainly be greater than SK. To see that I fK+1 --K+ II = 0(u), let the coefficient of u(u)(i -1)/2 in ai(K+ 1) be Ri and observe that the Ri satisfy the recursion

Ro = 0

RI = -YK+I

Ri+1 = (i -YK + )Si + /(XK + )Ri - iRi- 1, i > 1.

We can then show by induction that the Ri are linear in YK+ 1:

Ri= c, + dyK+l

with ci and di constants possibly depending on XK+ 1, but not on ,u. So, given XK+ = SK+ , choose rK+ such that RK+ = 0. Set ,K+1 = -

( + rK+ 1i) +

V(SK+ 1)A). Then for all J > 0 there exists f such that, for 0 < u <

aOK + 1(K + 1 -4 ) < O <a KK + 1( + i+4 ).

That is, I (K+ I - K+ I < 2< u for some j, and u small enough. Now SK+ I > SK

implies that K+ 1 > .K for small enough u. And (5), together with the initial

973

ILZE ZIEDINS

assumption that I K - OKI = O(u), implies that 4jK+1 < K for j > 1 and small enough u, and so j must equal 1. Thus

CTi(K + 1) = (i) i/2(Si(SK + 1) +- O((, ) 12), i < K.

The quasi-stationary distributions for K = 1, 2, 3, 4 are displayed in Table 1 (ignore the last column for now). For K = 1, 2, SK+ 1(X) is linear in x and for K = 3, 4, SK+ I(x) is quadratic in x, hence the simplicity of SK+ 1. For K _ 5, it is necessary to solve at least a cubic to find sK+1, and the solution to these polynomials is not so easily obtainable.

TABLE 1

Quasi-stationary distributions for 1 _ K _ 4

K SK+I UK(O) 7rK(1) 7K(2) 7K(3) 7K(4) Xl

1 1 0.5 0.5 - Vv 2 3 0.1666 0.5 0.3333 - 0.7321A/v 3 3 + /6 0.0459 0.25 0.4541 0.25 - 0.6024A/v 4 5 + v/10 0.0113 0.0919 0.2887 0.4081 0.2 - 0.5226A/v

Note that for 1 < K < 4, rK(K) = 1/(K + 1). In fact it is not difficult to show that this is true for all K. For

rK(K) = K+I

(SK(sK+ ))2 K (S (SK+ 1))2 1

(K)! / =o i! K+

(SK(SK+ 1))2 K (Si(SK+ 1))2 - E = 0.

(K-1)! i=o i!

But making use of the recursion (7), it can be verified by induction that

(SK(SK + ))2 K-I (Si(SK + 1))2 SK + I(SK + )SK- i(SK + 1)

(K - 1)! o i! (K - 1)!

and SK+1(SK+I) = 0 by definition. We conjecture that, as was shown for K = 1, 7rK(K) _ 1 /(K + 1) for all ,u _ 0;

and that the probability distribution will in general have the shape of the distribution at K = 4 - i.e. with a maximum away from K.

The results of this section can be extended to a more general model. Let A,u = v, v > 0 a constant, and pu - 0. Then, as above, a solution of the form

rK = - (A + Yu) + -v(XK/#) will ensure that all terms in ari(K) containing a

974

Quasi-stationary distributions and one-dimensional circuit-switched networks

factor Ai, with j > i/2, will have coefficient 0. Exactly the same reasoning as above leads to the conclusion that in this case

i(l +1) = Vil2(Si(SK+) + O(U)), O i < K

and so the quasi-stationary distributions do not vary with v and are exactly the same as if A were held fixed. However, the transition rates of the quasi- stationary process do vary with v. From Theorem 2.3 they are given by

S+l(SK+lI) iK(n)= /(v) S, (SK + 1)

and

S- I(SK+ 1) /IK(n)- nf((v)

Sn (SK+ 1)

Theorem 2.5 can now be applied to find the decay rate for the acceptance of calls as their length increases. Let PK(X) be the characteristic polynomial of the Kth section of QK, the Q-matrix for the quasi-stationary process obtained under the limiting regime 2.u = v, u - 0. Then

(8) Ai + 1() = (X + AK(i) + PK(i))fi(X) - PK(i)PKf(i - 1)A i-1()

But Kf(i) + K(i) = V/(VSK+i) and uK(i)K(i - 1) = iv so that (8) simplifies to

fi+ l(x) = (x + V/(VSK+ ))13i(X) iv- i-1i()

Thus J?K(X) is a homogeneous polynomial of degree Kin X and Vv. So if we set X = zvv then PK(X)= vK'2PK(z, SK+ ) where PK(Z, SK+1) is a polynomial of degree K in z. Let zK be the solution of least modulus to PK(Z, SK+ ) = 0, and observe that it will be strictly negative. Then, using the notation of Theorem 2.5, XK = zK\/v. And if z = Vx - V/SK+I then it can be shown by induction that Pi(z, SK+ 1) = Si(x), where Si(x) is given by (7). It follows immediately that z VK= /SK- /SK+I. Thus we have shown that under the limiting regime i# = v, -u 0 the decay rate for the acceptance of calls is approximately exponential with parameter equal to (/SK - VSK+ 1)/\V. The last column of Table 1 gives the value of this parameter for 1 _ K < 4.

As an aside, observe that SK+ I(XK+ ) is a polynomial of degree K + 1 in V/(XK+ 1) Let {f+l , 1 _ i K + 1} be the K+ 1 roots of SK+ 1(VK+ 1), ordered so that y VK++ < / iK+ 1. Thus SK+ = (ylf+ 1)2. Then arguments similar to those above give f+l = - 1 + /iK+ l/(.) + 0(u).

Note that if A -u 0 as u - 0 then K(i)O 0, 0 _ i _ K - 1, K(i) --O, 1 < i _ K, and Xf - 0. In particular, this will be true for A fixed.

975

ILZE ZIEDINS

5. Balanced traffic

The third and final example sets A = (K + 1)/, that is, the arrival rate of calls is proportional to the number of possible states of the quasi-stationary process. This is not quite balanced traffic. Balanced traffic would have the arrival rate proportional to the number of circuits on the line, i.e. A = Ku. However, if

= (K + 1)1, then the quasi-stationary distribution is algebraically tractable; and for K large, we would expect the results to be almost the same as those obtained for balanced traffic, since then the percentage change in A is small.

To simplify the equations in this example, the calculations will be in terms of the process NK-_(t) rather than the process NK(t).

If A = Ku then

aK(-) E= () (K) K- f (q + i,u) n=0 i=0

=( + #) KKK-' + (K ( )n + i/). n =2 i=2

So aK( -) = 0. But if 4 > -u then

IK\ K-n n-I KK! KK-"n K-1

(n (K)K n[l ( +

i-) > (K-n)! n(n -1)

KK#K-1

n(n- 1)

So, for > -

)K-1 1

aK()) > ( + )K)K # -1 I i(i-- > 0. i=- i(i + 1).

Thus the eigenvalue of least modulus is -,u. This allows a great simplification in the quasi-stationary distribution since

now

an( K) = (K#u)"-'(K - n)

and so

Kn - 2(K- n)2 n K 7K- l(n)= 7tK-I(O) , 1 < n <K, n!

AK_-,(n)=K K K-(n +

) 0<n-<K-2, K- n'='

976

Quasi-stationary distributions and one-dimensional circuit-switched networks

and

UK-,(n)- n +1 n nK -1. K-n

Let /K- 1(X) be the characteristic polynomial of QK 1. The form of K-_ I(n) and /1K- ,(n) implies that PK- I(x) is a homogeneous polynomial of degree K - 1 in X and ,u. Thus, from Theorem 2.5, the decay rate is approximately exponential with parameter XK-' - ctu, where c < 0 is a constant.

Acknowledgements

I should like to express my gratitude to Dr F. P. Kelly for some very stimulating and rewarding discussions on the material contained in this paper and the many helpful suggestions he made. I am also grateful to Dr P. Pollett for a careful reading and discussion of an earlier draft, and to the referee for his helpful comments and suggestions. This research was supported by grants from the Association of Commonwealth Universities and Christ's College, Cambridge.

References

[1] DARROCH, J. N. AND SENETA, E. (1965) On quasi-stationary distributions in absorbing discrete-time finite Markov chains. J. Appl. Prob. 2, 88-100.

[2] DARROCH, J. N. AND SENETA, E. (1967) On quasi-stationary distributions in absorbing continuous-time finite Markov chains. J. Appl. Prob. 4, 192-196.

[3] KEILSON, J. AND RAMASWAMY, R. (1984) Convergence of quasi-stationary distributions in birth-death processes. Stoch. Proc. Appl. 18, 301-312.

[4] KELLY, F. P. (1986) One-dimensional circuit-switched networks. Ann. Prob. 14, [5] LAGARIAS, J. C., ODLYZKO, A, M. AND ZAGIER, D. B. (1985) Realizable traffic patterns

and capacity of disjointly shared networks. Computer Networks. [6] LEDERMANN, W. AND REUTER, G. E. (1954) Spectral theory for the differential equations

of simple birth and death processes. Phil. Trans. R. Soc. London A 246, 321-369. [7] SENETA, E. (1966) Quasi-stationary distributions and time-reversion in genetics. J. R.

Statist. Soc. B 28, 253-277. [8] SENETA, E. AND TWEEDIE, R. L. (1985) Moments for stationary and quasi-stationary

distributions of Markov chains. J. Appl. Prob. 22, 148-155.

977