AppliED& MATH EIMAT ICS 50 D _% . Ap 4Ai0

STANFORD UNIVERSITY

4r, A

STER'-S MARKOV CHAIN THEOREMS IN CONTINUOUS ItNIE

RUPERT G. M!LLER, JR.

IE -CMMNICAL REPORT NO. 88

PRCPARE1 UNDER .. r.... '5Z2)

ýFOROFFICE O;, NAVAL RESE-ARCf,14

I

FOSTER'S MARKOV CHAIN THEOREMS IN CONTINUOUS TMI

by

Rupert G. Miller, Jr.

TECHNICAL REPORT NO. 88

April 19, 1963

PREPARED UNDER CONTRACT Nonr-225(52)

(NR-342-022) A S T I A

OFFICE OF NAVAL RESEARCH APR 29ý 103 il

TISIA U

Reproduction in Whole or in Part is Permitted forany Purpose of the United States Government

APPLIED MATHEMATICS AND STATISTICS LABORATORIES

STANFORD UNIVERSITY

STANFORD, CALIFORNIA

FOSTER'S MA=KDV CHAIN THEORW4 IN CONTISWXUS TIME

by

Rupert G. Miller, Jr.

1. Introduction

Let (Xt), t e T a [0,-), be an irreducible Markov chain in contin-

uous time with state space I = (0,1,2,...) . The stationary transition

probability matrix P(t) =(pij(t)) is assumed to be measurable and

satisfy

pij(t) 0> O EPij(t) <ý 1, i, j e Ia

(1.1)

P(t+s) = P(t) P(s), P(O+) = I

for all t, s c T. In addition, the states are assumed to be stable;

i.e.,pii(t)-i

0> '(0) lim = >" iq >11 >4i t qii

(1.2)

0 <PijI1() ur = qij < + , iGJoIt4o

The matrix q = (i) is called the Q-matrix or infinitesimal generator

matrix of the process, and it is assumed to be conservative, i.e.,

L qi = 0, i e I. For simplicity, this type of Markov chain will bejreferred to as a simple continuous time Markov chain (SCMC). A thorough

treatise on the properties of a SCMC is contained in (1].

In (8], [9] the solutions to the equations y Q = 0 were inves-

tigated. These stationarity equations are obtained by setting the

derivatives equal to zero in the forward Kolmogorov equations P'(t) -

"P(t)Qp and are the continuous time analog of the stationarity equations

xP a x for a discrete time Markov chain (with stationary one-step

transition probability mtrix P = (pij)). In particular, it was shown

that, under the minimality assumption described below, a NSC for the

SCMC to be positive recurrent is for the equations yQ = 0 to have a

convergent, positive solution y = (yo,yl,y2 ,...). The solution is

unique (up to a multiplicative constant):

(1 3)Yl a=Xi a lim Pii(t) ,i e I .t +0

In [9] yQ - 0 was also shown to have a unique, positive solution in a

null recurrent chain, and for any recurrent chain (positive or null) the

relationship between the unique stationary measures of the SCMC and its

imbedded discrete time Markov chain was obtained.

The analogous stationarity theorem for positive recurrent chains

in discrete time is due to Foster [4] with an earlier, less general

version being given by Feller [3], p. 325 (see also Chung (1], p. 33).

Foster also gives three additional theorems on conditions for recurrence,

ergodicity, etc. in a discrete time chain. The purpose of this paper

is to extend these additional theorems to continuous time.

The minimality assumption referredý to above which is necessary for

the validity of the preceding theorems in continuous time is:

Minimality Assumption: The SCNC is uniquely defined by its Q-matrixj

i.e., the minimal process is an honest process (see [1] for details).

It will be necessary to impose this assumption in Theorem 2 below but

2

will not be needed in Theorem 1. It excludes from consideration those

" processes which can explode to + - in finite time. Various necessary

and sufficient conditions on the Q-matrix for the minimulity assumption

to hold have been derived and can be found elsewhere. For reference,

see Chung (1] and Reuter [10].

2. Results

In the proofs of this section it will be necessary to refer to the

imbedded chain of the SCMC. This is the irreducible (but possibly

periodic) discrete time Markov chain (Xn), n = 0,1,2, ..- , with

stationary transition probability matrix P = (pij) where

pij-- q i/qi c(2.1)

pii = 0i I

The imbedded Markov chain (X n) simply records the sequence of states

through which the SCMC passes without regard to the amount of time re-

quired for the transitions.

Theorem 1: (a) The SCMC (Xt) is recurrent if there exists a sequence

z = (zZl 1,Z 2 ,...) such that (i) zn-*+o as n-4+w and (ii)

Qz < 0 except for the first coordinate.

(b) A NSC for the SCMC (Xt) to be non-recurrent is that

there exist a bounded non-constant sequence z = (zo, zl,,z2 ,...) such

that Qz = 0 except for the first coordinate.

Proof: The proofs of (a) and (b) are Immediate and can be given together.

The system of inequalities or equalities Qz < 0, = 0 can be written as

!.3

(2.2) qlj _, - Z , 0GI

F Division by q yields

(2.3) p P1jZ3 < z = , i VO e I

where P = (pij) is the transition matrix of the imbedded chain. Under

the conditions on z in (a) the system of inequalities (2.3) implies

recurrence for the imbedded chain by Theorem 5 of [4]. Similarly, under

the conditions on z in (b) the system of equations (2.3) is a NSC for

the transience of the imbedded chain by Theorem 4 of (4]. But the

recurrence or transience of the imbedded chain is identical to the re-

currence or non-recurrence, respectively, of the SCMC. Recurrence is

independent of the time component. fj

The term "non-recurrent" rather than "transient" is used here in

dealing with a SCl. because of the two possible types of path function

behavior. A SCMC can be non-explosive (i.e., satisfy the minimality

assumption) but have transient states in the sense that a return to

each ha probability less than one, or it can be explosive and reach

+ c in finite time with positive probability. In both cases the states

of the imbedded chain are transient.

Note that it is not necessary to impose the minimality assumption

in Theorem 1. The fact that the imbedded chain has not been defined

beyond the first infinity does not cause any difficulty. Should Qz = 0

not havesa bounded non-constant solution or Qz < 0 have a solution

whose coordinates tend to + c, the imbedded chain is recurrent, and

4

Iby necessity the BSCC is uniquely defined. Should Qz - 0 have a

T bounded non-constant solution, the imbedded chain is transient; the SCMC

is then either explosive or non-explosive and transient.

This theorem, particularly part (b), is motivated by the following

consideration. Let f be the probability that if the SCMC (or itsioimbedded chain) starts in state i, it reaches state 0 eventually.

If fo0 is defined to be 1, then the fio satisfy the equations

00

(2.4) Z pijufno0 o = io0CIj-0

In a recurrent chain fo m 1, but in a transient chain f 1, so

the fio constitute a bounded, non-constant solution to (2.4).

In an earlier paper [6] Karlin and McGregor extended Ibster's

Theorems 4 and 5 to birth and death processes. Utilizing the special

structure of these processes they also established necessity in part

(a). This does not seem to be true in general (see (4]).

Theorem 2: Under the minimality assumption a NSC that the SCMC be positive

recurrent is that the inequalities

(2.5) q 1 <1 , ip0"eI1=0

(i.e., Qz < - 1 except for the first coordinate) have a non-negative

solution z which satisfies

(2.6) j'j < +"J-)

ie, (Qz) 0 < + s)

5

Proof: (Necessity) Let mO be the expected time it takes the SCMC

to reach state 0 from state i ' 0c iIj mo 0. For a positive

recurrent SCMC miO < + c. These expected first-passage times satisfy

the equations

(2.7) min 1 O i 0eI

where the first term on the right is the expected length of time spent

in state i and the second term is the expected time to reach 0 after

the process leaves state i. Multiplication of (2.7) by qi and

rearrangement of terms yields (2.5) with equality for z = mjo.

Since the chain is positive recurrent, the mean recurrence time to

state 0 is finite; i.e.,•" 00

(2.8) + Z P0 3 mJ<+oO <qO j.0oj

which implies (2.6).

(Sufficiency) Rearrangement of terms in (2.5) gives

(2.9) E p ,jzj i q 0 e IJ=O

Without loss of generality assume z0 = 0. Consider the iterative

inequality obtained by applying pn = (pi)) to z:

O (n) O (n-l) -0E Pui Zj E Pik Z kzJ=O kuO J-0

6

!

f - ki JQ • oOJzJ

P (n-1) 0 (n-i) 1 (ni) +- - k zk Zk Pi k p1 0 ~ (-+

i i

where X- Z Pjzj < + c by (2.6).J,,o

The (n-l)-fold interation of this inequality produces the following

inequality:

(2.11) 0 < pij )Z3 Z, z P (v 1" (-0 -0 nlk=0 k()

n-i v+(q + )v=lPiO~v

The series 1 E P (v'/q is divergent by the minimality assumptionv-i ks e uk

since it is the expected time required to make an infinite number of

transitions after leaving state i. For a rigorous proof see Theorem

II. 19.1, Corollary 1, of (1]. But this means that 1:1 (v) must be•iio

divergent in order to preserve the non-negativity in (2.11). Hence,

the SCMC is recurrent.

To establish positive recurrence sum the inequality (2.10) for

n = 2,...,N+l.

N+( N w i(n) - N (n)(2.12 E• - pin z <n k•= inz k " ný k= Pi~kn q•

n=2J-0 L 1 kuO 0 ~ k-0

1N (on)+ (L + X0n• Pifl •

7

IRearrangement and cancellation produces

T~~ ~ ( nip.n)).< PZ-ku ( + (L)x N (n)

k-O k= - q~O

(2.13)

-= PikZk + (X Nnk) Pi '

Divide both sides of (2.13) by 1 (n) (which is positive for Nn=l

sufficiently large) for any hel. As N -+= Fatou's lemma gives

~N) NI P (n)\ i P o(n)

(2.14) ni l < 1+ ) lim nlk- P (n) T N* P" (n)

n~l ih /n=l

the right hand side reducing to a single term since pi n +n=l

in a recurrent chain. These limits exist by the Doeblin ratio limit

theorem and have been evaluated by Chung. For a recurrent chain

N (n)n)l

(2.15) lim =

(n) 1h

n=l ih

for any I e I, where jp*'k is the expected number of visits to state

k between visits to state i( p*j1 = 1). (For reference see (1],

Sec. 1.9). From (2.14) and (2.15)

(2.16) L* < (L + -k 1 k qO Ap*O

i8

'IDerman (2] shoved that for a recurrent chain the jp*jjt ± - 0,1,2,...,

f constitute the unique (except for a multiplicative constant) positive

solution of the equations xP a x. In a recurrent chain the unique

solutions of yQ - 0 and xP - x are related by yt = xt/qi (see

Theorem 3 of [9]). Thus, by (2.16) yi a jp*ji/qi is a positive,

convergent solution to yQ = 0 so the SCMC is positive recurrent (by

Theorem 1 of [9]).

The motivation for this theorem is clearly contained in the necessity

part of the proof where (2.5) holds with equality for z = m o. That

the equalities can be replaced with inequalities in the sufficiency

condition is a trivial bonus of the proof.

The minimality assumption is essential for the validity of the

sufficiency part of the theorem. For a counter-example without it take

a birth and death process with

(2.17) 0 = 0 , pn< + ,< + -

(2.17) E~0 O PjInwO n=O nn

where pn = X 0l X, . n-1/4 -"n' n = 1,2,...,p = 1. Such a birth

and death process is explosive (see [5]). However,

(2.18) z0 =0,n X0 n i v

Zn+l = z PU ,n

satisfies the equations q ciiz• = -1, i = 1,2,..., for any zI.

(1 qOjzj < + holds trivially.) For z1 sufficiently large znJ1-0

will be positive for all n since the negative series in (2.18) is

convergent.

9

!Kingman (7] proved this theorem for bounded qi by a different

"J method. Since boundedness of the q, guarantees the minimality

assumption but is not necessary for it to hold, Theorem 2 would con-

stitute an extension of Kingman's result. An application of the suf-

ficiency condition to parallel queues can also be found in (7].

1

10

%

BIBLIOGRAPHY

[1] K. L. Chung, Markov Chains with Stationary Transition Probabilities,

Springer-Verlag, Berlin, 1960.

[2] C. Derman, "A solution to a set of fundamental equations in Markov

chains," Proc. Amer. Math. Soc., Vol. 5 (1954), pp. 332-334.

"(3] W. Feller, An Introduction to Probability Theory and Its Applications,

Wiley, New York, 1950.

[4] F. G. Foster, "On the stochastic matrices associated with certain

queueing processes," Ann. Math. Stat., Vol. 24 (1953), pP. 355-360.

[51 S. Karlin and J. McGregor, "The differential equations of birth-

and-death processes, and the Stieltjes moment problem," Trans. Amer.

Math. Soc., Vol. 85 (1957), pp. 489-546.

[6] S. Karlin and J. McGregor, "The classification of birth and death

processes," Trans. Amer. Math. Soc., Vol. 86 (1957), pP. 366-400.

[7] J. F. C. Kingman, "Two similar queues in parallel," Ann. Math.

Stat., Vol. 32 (1961), pp. 1314-1323.

[8] D. G. Kendall and G. E. H. Reuter, "The calculation of the ergodic

projection for Markov chains and processes with a countable

infinity of states," Acta Math., Vol. 97 (1957), PP. 103-144.

[9] R. G. Miller, Jr., "Stationarity equations in continuous time

Markov chains," Technical Report No. 80 (1962), Stanford University,

Contract Nonr-225 (52), (NR-342-022), accepted for publication

in Trans. Amer. Math. Soc.

[10] G. E. H. Reuter, "Denumerable Markov processes and the associated

contraction semi-groups on I," Acta Math., Vol. 97 (1957), pp.

1-46.

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