Filtering of Markov Renewal Queues, II: Birth-Death Queues

Filtering of Markov Renewal Queues, II: Birth-Death QueuesAuthor(s): Jeffrey J. HunterSource: Advances in Applied Probability, Vol. 15, No. 2 (Jun., 1983), pp. 376-391Published by: Applied Probability TrustStable URL: http://www.jstor.org/stable/1426441 .

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Adv. Appl. Prob. 15, 376-391 (1983) Printed in N. Ireland

@ Applied Probability Trust 1983

FILTERING OF MARKOV RENEWAL QUEUES, II: BIRTH-DEATH QUEUES

JEFFREY J. HUNTER,* University of Auckland

Abstract

In this part we extend and particularise results developed by the author in Part I (pp. 349-375) for a class of queueing systems which can be formulated as Markov renewal processes. We examine those models where the basic transition consists of only two types: 'arrivals' and 'departures'. The 'arrival lobby' and 'departure lobby' queue-length processes are shown, using the results of Part I to be Markov renewal. Whereas the initial study focused attention on the behaviour of the embedded discrete-time Markov chains, in this paper we examine, in detail, the embedded continuous-time semi-Markov processes. The limiting distributions of the queue-length processes in both continuous and discrete time are derived and interrelationships between them are ex- amined in the case of continuous-time birth-death queues including the M/M/1/M and M/M/1 variants. Results for discrete-time birth-death queues are also derived.

MARKOV RENEWAL PROCESSES; MARKOV CHAINS; SEMI-MARKOV PROCESSES;

QUEUE LENGTHS; LIMITING DISTRIBUTIONS; STATIONARY DISTRIBUTIONS

1. Introduction

A queueing system can be regarded as consisting of three entirely separate components: an arrival lobby through which all arrivals pass unheeded instan-

taneously at the time of their arrival, the queue facility consisting of the actual

waiting line and the service mechanism, and a departure lobby through which all customers pass immediately following their service completion. The arrival

doorman, the queue supervisor and the departure doorman supervise and observe their respective components. A doorman is permitted to view the

queue facility only when a customer moves through the lobby under his control. Each of these supervisors has an entirely different view of the

queueing process. In this paper we assume that the queue-length process, as seen by the queue

supervisor, is a Markov renewal process (MRP) with arrivals and departures

Received 13 November 1981; revision received 14 April 1982. * Postal address: Department of Mathematics and Statistics, University of Auckland, Private

Bag, Auckland, New Zealand.

376



Filtering of Markov renewal queues, II 377

occurring singly. The only realistic practical situation where this holds is the class of birth-death queueing models. For continuous-time birth-death queues arrivals and departures cannot occur simultaneously (with probability 1) and

consequently this system is a special case of the model considered in [6] with no feedback. For discrete-time birth-death queues, however, arrivals and

departures may occur at the same time and consequently we need to utilise the full generality of the feedback model presented in [6] to examine the embedded Markov chains present in the system.

From Part I [6] we deduce that that queue-length processes as seen by the arrival and departure doormen are also MRP. Furthermore the results of this earlier paper enable us to examine the stationary and limiting properties of the embedded Markov chains. The main thrust of this paper is, however, centred around the properties of the continuous-time semi-Markov processes (SMP) associated with each embedded MRP. We carry out this study under the restriction that the queue-length process, observed by the queue supervisor, changes by either +1 or -1 at any transition epoch, ruling out feedback-type queues. This enables us to find explicit expressions for the elements of the semi-Markov kernels of each MRP. The relevant general theory is covered in Section 3 and its subsections.

Section 4 summarises the results as they pertain to birth-death queues including the well-known M/M/1/N and MI/M/1 queues. This paper gives a unified presentation for such models. The approach taken enables us to derive rigorously some of the 'folklore' results as well as to obtain some new results.

2. Description of the model

Let X(t) be the number of customers in the queue facility at time t. Let

{T,, n O0}, with To 0, be the epochs at which X(t) changes value and X, X(T, +), the number of customers in the system following the nth transition. We assume that {(X,, T,); n >

-0} is an MRP with semi-Markov kernel

(or matrix) Qx(t) [Q1i(t)] and state space f= {0, 1, ... , N}, (Noo). Furthermore, we assume that the only non-zero elements of the kernel are

OQ,+1(t), (0?--i--5-N-1),

and Oi,i-1(t), (1<-i?---N). The following example gives a practical example that falls naturally into this

model.

Example 1. Birth-death queue in continuous time. If {XA} and {1_1}

are the usual birth and death rates with Ao= 0 (AN =0 when N < oo) then, (cf. [3], p. 316),

( ,i,+l(t)= [A,/(A,+ C)][1-exp (-(A, + gL)t)],

(0- i

-N-1), (1)

Q.,_l(t) =

[/_q/(A, +

pz,)][l -exp (-(A, + pz)t)], (1?_

i ? N).



378 JEFFREY J. HUNTER

In the next example transitions which leave X, unchanged are permitted. Although the general theory of Section 3 excludes such a model we examine its embedded discrete-time processes, using the results of [6], in Section 4.3.

Example 2. Birth-death queue in discrete time. Suppose QO1(t)=pjFj(t) where

a•Ji/(ai + dJi), j= i+1,

(0_5i_-N-1) (2) pij =aipi/(ai + iiP,), j=i, (1 5 i _N),

jl&,g3/(a, +

&iP,, j 1, (1

:_5-i :_-N) and

F,(t)= O U(t- kA)(&,)-k-i(1

- , 3), (05 i 5N), k=1

under the convention that ci- 1- ai,#- 1-i (0 5 i5 N) with aNO-0, 3o - 0 and U(x)= 1 for x -20 and 0 for x <0.

This specifies the semi-Markov kernel for a discrete-time queueing model with {ai} and {Jo3} being the probabilities, respectively, of an arrival and

departure at the end of an interval of length A if, at the end of the previous interval, there were i customers in the queue facility.

Now {T,}={Ta)} U {T•,} where TV) (or Td)) are times at which the nth arrival (or departure, respectively) occur within the facility. (T~ a) , T0od -= 0). Let X~1 = X(Ta)) and X~ =- X(T1d)) so that X(a) (or X d)) is the number of customers in the queue facility and arrival (or departure) lobby immediately following the nth arrival (or departure, respectively).

In developing the results of this paper we assume, in general, that N is finite. The arguments can, in most cases, be extended to cover the case of unlimited

waiting rooms, when N=oo, but care must be taken. A discussion of the

appropriate generalisation is presented at the conclusion of each section

throughout the paper.

3. General theory

3.1. The embedded Markov renewal processes. First observe that Ox has the

representation Qx = QA + QD where

QA(t) =[QAi(t)]

with QA-ij (t), ii+1, tO, j i + 1;

OD (t) - [Di (t)] with ODi,(t) i-1(t) j = i-i.




Since Ox has the same form as Q, the kernel of the MRP discussed in Section 3 of Part I, [6], but with the feedback component OFQ 0, we can utilise the

'filtering' results of that section.

Using the standard notation for convolutions we define the following functions:

for i<j, Kii(t) QOi,i+1 * Qi+l,i+2 * * * Oi-l,(t); for j < i, L,(t) -- i,i-1 * Oi-1,i-2 *

' * *j+1j(t)?

Theorem 1. If {(X,, T,)} is an MRP with state space f = {0, 1, ... , N} and semi-Markov kernel Ox then

(a) {(X a), T(,))} is an MRP with state space 9 and semi-Markov kernel (a) = (= oO ) * QA = [f ] where

Lii-1 *

Q-j,, j1 j

i, 1

_5 i

- N;

(3) Of = ] -1ji, j=i+l1, 0_-i_5N-1;

10, otherwise.

(b) {(X ), T~d))} is an MRP with state space 9 and semi-Markov kernel

Q"d) = (TOO 0 ) * A = [Q4)] where

Kj+I *

Qj+i.j, OiN-, i N-1; (4)

Q = ] O +

1,j, 1 - i - N, j = i - 1;

,0, otherwise.

Proof. The structure of the kernels follows from Theorem 2(a) and (b) of [6]. Expressions (3) and (4) follow upon extraction of the elements or, directly, by a

sample path argument.

Corollary 1A. Let {(X,, T,)} be an irreducible persistent MRP with state space S= {0,

1,- - - , N}.

(a) If X(- I- X~- 1 then, provided X0o 0 or equivalently XoA)

a 0, {(Xa-),

Tn ))} is an irreducible persistent MRP with state space (a-) = {0, 1, - , N- 1} and semi-Markov kernel

O(a• = [O~~- where

(5) Of-)= ="a)

0 :iN-1,

0:-j:N-1.

(b) If Xo - N or equivalently

X0'o # N, {(X n, Tfd))} is an irreducible persis-

tent MRP with state space f(d) = {0, 1, NN-1} and semi-Markov kernel Q(d)= [Q(d, (i, j)E p(d)

Proof. (a) Q l (t) = Qol(t) is a proper distribution function so that if for any n >O X(<a ) O, X

(•1) #0. Further, since Q (t) =0 for all ie $, state 0 can never be entered and thus, for the {(X ", T ")} process, O must be a transient state and the states {1, 2, ... , N} form an irreducible and hence a persistent




(by the finiteness condition) set. The conclusion follows by a relabelling of the states.

(b) Similarly, 0(d) W

=, (b) Similarly, N,N

_(t) =

OQNN-(t) is a proper distribution and OQ(t)= 0

for all i e cf from which we deduce state N is transient.

In the case of unlimited waiting rooms Theorem 1 holds but with N taken as

+oo. The required extension of Corollary 1A is not so straightforward. State 0 is still transient in {(Xa, TaT )} but there are now no transient states in the

{(X T, T )} process. The irreducibility results still hold but the classification of the states is intimately connected with the existence of solutions to the

stationary equations of the associated embedded Markov chain (cf. [3], p. 135).

3.2. The embedded Markov chains. In this section we consider the existence of limiting and stationary distributions for the discrete-time Markov chains

{X,}, {Xn)} and {Xn } whose transition matrices are given by Px = E[Qi(+00)] [Pi], p(a) = [p,2)] and p(d)= [d)], where if

aii - Kii(+oo) = Pi,i+1Pi+,i+2 ' '

Pi-1,i, (i <),

- / Li(+?0) =

Pij, -1Pi-1,i-2 Pi+1,i ( < i),

and aii - P - 1, (i -

0); then

a) ii,i-ijPi-i ,ij 05i-N, 1 5j

_ 5min (i + 1, N),

o0, otherwise;

?) i.+Pi+1.i, 05iN, max (0,

i-1)-j-N-1, I 0, otherwise.

Firstly, Px is the transition matrix of an irreducible, persistent, periodic with

period 2, Markov chain. Thus a limiting distribution {lim.oo P{X, = k}} does

not exist but the Markov chain does have a stationary distribution, which follows as a special case of Theorem 5 of [6].

Theorem 2. The stationary distribution {rr,},

(0 _5 i 5 N), of the Markov chain

{X,} is given by

Wi = (aoil3,/o) I (aok/1ko) , O 5 i 5 N.

Since {X } and {X (} are not irreducible Markov chains we need to restrict attention to their irreducible sub-chains with state spaces '(")= {1, 2, ... , N} and '(d) =

{0, , , ... , N- 1} and transition matrices P(") and P(d, as given,

respectively, in Theorems 4(a) and 4(d) of [6], (with PF 0). By using Theorem 8 of [6] or by solving the appropriate stationary equations

we obtain, after simplification, the following results.




Theorem 3. (a) The stationary distribution {:~ a)} of the irreducible (sub)-Markov chain

{Xn"} with state space ?(')

is given by

(a)" = (ola0i/1i_,0) (aok/0k-1,o) , 15 i 5 N.

k=1

(b) The stationary distribution {7? d)} of the irreducible (sub)-Markov chain

{Xn,} with state space

S,(d) is given by

(rd) =

(ao,i+/lo) [Y

(ao,k+1I)ko)]=05

O i5 N- 1. k=0

Since {X(a)}

and {X(d)} are aperiodic Markov chains ia)j} and {.r(d)} are in fact limiting distributions.

Observe that {1r}, {("a)} and { d)} are always different probability distribution since they are defined on different state spaces i.e. Sf, f(9) and S(d) respectively. However, {X("-)} and {X(d) have the same state space.

Corollary 3A. The limiting distributions of the sub-Markov chains {Xn~-} and {X•)},

{r••"-} and

{r)}, are equivalent with

(-N-1(d)

(

[ 1kO

-1

ia-) =trd)_=

(ao,i+l/io)EL=

(ao,k+1I/ko) , 05 iO N- 1.

Proof. Since ,ra-) = (a)" the result follows directly from Theorem 3.

Corollary 3A is also a special case of Theorem 12 of [6]. Besides Cooper [4], p. 155, Krakowski [8] gives an intuitive argument to establish the same result for a wide class of queues.

Corollary 3B. If {}, { }a)j and {i d) are as given by Theorems 2 and 3,

Ti= (Wa)+I d)/2, 05 iN,

(with r(a) =0 and Tr(d) 0).

Corollary 3B is easily verified. It is a special case of Theorems 9 and 10(b) of [6] although it is intuitively obvious since, in the long run, there must be an

equal number of arrivals and departures. The results of this section hold when N = oo if appropriate care is taken with

the limiting operations. For example Corollary 3A will hold provided kZ=o (ao,k+lflko) < 00

3.4. The embedded semi-Markov processes. Suppose sup, T, = +oo, as is the case for Example 1. Then (cf. [3], p. 316) X(t) = X,,

T,• =t< Ta+ implying

that {X(t), t > 0} is the minimal SMP associated with the MRP {(X,, T,)}. Similarly




if X(a)(t) X "(a), T(a) t <

Ta)1 and

X(d)(t) Xd), T() - t < Tnd),

then {X(a)(t), t _ 0} and {X(d)(t), t O} are the minimal SMP associated with the MRP {(X(a), T())} and {(X d), Td))}.

X(a)(t) (X(d)(t)), the number of customers in the system immediately after the most recent arrival (departure) prior to time t, is the continuous-time

queue-length process as observed by the arrival (departure) doorman. Since it is customary to observe the arrival lobby queue-length process just prior to an arrival we shall examine the X("-)(t) process where X("-)(t) = X(a)(t) - 1 which is the minimal SMP associated with {(Xn-), T ))}.

If the {(X,, T,)} process is irreducible with state space 9S and the Qii(t) are not step functions, (as is the case for Example 1), then (cf. [3], p. 342, [10], p. 104) X(t) has a stationary distribution {p1}, (i e sf), which is also the limiting distribution. These conditions also imply that X("-)(t) (or X(d)(t)) has a limiting distribution {p?a)}, (i = J(a-)), (or { pd)}, (i E (-(d)), respectively). Prior to deriv-

ing these distributions we introduce some notation. For those pairs of states

(i, j) such that pi > 0 we define

Fi(t) Qi (t)/pij = P{T,++I-

Tn t I X, = i, Xn+1=

}1, the conditional d.f. of T,,+1 - T. given X, = i and X,,+ = j; and

mii = t dFii(t) = p1/iPii where i = Jt dQij(t).

Theorem 4. Under the conditions stated above:

(a) pA =

•?1Jk/_ E 1rk/x, 0

<- i

<_-N, k=0

where the {}ri} are given by Theorem 2 and the {i_}

are given by

mo01, i = 0,

ti = Pi i-1Mii-1+Pi i+1mii+1, 15 i ?N- 1,

mN,N-1, i = N;

(b) p •a- •-)a(a-) (a- -, i N- 1,(a /k=0

where the {?ra-)} are given by Corollary 3A and the {g("a-)} are given by

Z +1,(Pt+1m,+

+ m+,) +pi+1,i+2i+0,i+2,

i 1 N- 2,

(a-) _ =

N-i

1. l




N-1 (c)1 IAld) =~(d) (d)/ •" (d) (d)

(c) P1 7 i k k L

where the {7r?d)} are given by Corollary 3A and the {iLd')}

are given by N-1

f ao,1+1(Pl+1l,lml+l,

+

m1,+1l),

i = 0, (d) 1=0

N-1

t ai,l+l(P1+1,1m1+11

+ M,' 1+1) + ' ,ll' Pi, -lr 'i-

1 l i:--N- 1

Proof. These results all follow by the key renewal theorem as applied to MRP (cf. [10], p. 104).

In (a) the {tN}

are the mean holding times in state i, i.e.

i ,= IE[T+- T I X, = i]= IE[Tn+x- T, I X, = i, Xn+1 = j]Pij

S,) (<

A little more care is needed in the derivations of the {La-)} ({,(d)"}) the mean holding times in i for the MRP {(X"-), T•"))} ({(Xn), Tj(d)}).

For (b), first observe that for 0-i5N-1,

N-1 i0

("a-) = (a-) where (a-) = t dU -(t). j=0

Now, from (3) and (5), for 0 - j 5 i _

N- 1,

S= td(oi+,,,,i O,-1 *.. * + *

j+1(t))

=( -) td(E+,i*Fi,i* * _ Fi+,i * F i,+(t)), = (a

where pi-' = QON-(+o) = +3j+,lpj,+1.

In this above integral we have a convolu- tion of proper distribution functions so that the integral is in fact the mean of a sum of random variables giving

1i =

Oi+,iPi,i+l(r +l,i - ,i_1-+

4 ? ? + mi+l,i +

miij+1).

Similarly, for 15 j = i+ 1 _N- 1,

(a-) i,i+1 =Pi+l,i+2W'i+l,i+2*

Hence, for 0 -

i -

N- 2, after simplification, i+1 i+1 k-1 i+1

S a-)

mk,k -

Oi+l,jPj,j+l i+,P,+,+ j=0

k=-1 j=0 =0




The required expression follows by utilising the observation that

Cik 0Ai+1,jP,j+1 = Ii+1,k-X1

The expression for i = N follows analogously. For (c), with similar notation, for 0 - i 5 j - N-1, from (4),

i(d) =

tdQd)(t)= td(Qi,i+* O i+1,i+2 *

*

,+x* Oj+j(t)),

It d(,Fi,+ * Fi+l,i+2

**... * Fjl * Fj+lj(t)),

= aij+XPj+lj+(i,i+1+

mi+l,i+2+ + miji+1+ mi+l,),

while, for 1 i j + 1 5N- 1, gid- =

P,-mi- Now, for 1? i 1 N- 1, gd'= irLj11 iLd), with the sum extending from 0 to

N- 1 when i = 0. Substitution and utilisation of the result that

=kN•l ati,i+XPi+X,i = ai,k+1 leads to the stated results.

For continuous-time birth-death queues mi,i-1= =-i,+1 = mi. This leads to a

further simplification of the results of Theorem 4.

Corollary 4A. If the mi = n (j = i - 1, i + 1) then Theorem 4 holds but with

(a) A = mi, O 5 i _ N;

i+1

(b) A!a-) fim, 0<5i<sN_-

1 1=0

N

I=i

Corollary 4B. If the mij = mi (j = i- 1, i + 1) then the mean holding times

{(ta-)} and {tld)} can be found by the following algorithms:

(a) ga-)= a-)l Pi+,i+ +1

1i 5i;N-1,

with Io-) = Pxomo

+ m1.

(b) ild'= 1 d1Pi' + M+ , O5 i 5 N-2,

with fL1 = PN-1,NmN + mN-1.

The results of Theorem 4 and its corollaries extend naturally to models with unlimited waiting rooms by taking the limit as N tends to oo.

Corollary 4C. Under the conditions for Theorem 4: (a) If CP is finite thenr {r(a-)} # {pi }. (b) For any $ then {"(d)} {p(d)}

Proof. (a) Equality of the two limiting distributions will follow if and only if g"-' is constant for all i,

(0?- i N- 1). However from Theorem 4(b), since




13N, = 13N-1,, it is easily verified that - = • + m, NN_1. Since mNN/-1

f 0

equality of the i•a)

is clearly impossible. (b) From Theorem 4(c), since ao,l+1 = a1,+1 it is seen that ~4o = 1Id)+mo

and thus for any such queueing system (N<oo or N=oo) Ca)'• ~# AI and equality of the mean holding times can never be attained. Consequently

4. Applications

4.1. Birth-death queues in continuous time. For such queues the semi- Markov kernel is given by (1) and the transition probabilities are Pi,i+l =

A/(A•, + p) and p1,,il = =t1/(A. + p,), (LO = 0, AN =0). In the terminology of Sec- tion 3.4, mj,+1 = mj,

(_O i _-N-1),

and mji_1=mi, (1? i 5N), where mi= 1/(Ah +p~). Let us define Ap Ad/j, (1 i <N-1), with

po-1 and pi?-0. Then,

substitution in Theorems 2, 3 and 4 yield the following expressions for the limiting and/or stationary distributions. (See also Corollary 13B of [6].)

Theorem 5. For the birth-death queue with state space 'P= {0, 1, - - N}:

(a) Tr = Po "

' Pi-1(1 +p~)io, 1 i <N,

with N oIT = 2 Po ..." -1

(b) {(--

= Td)

where (d) =

- wri Po " "1 Pirod,

15i N-1,

with N W(d) = Po i

(C) Pi =[(Ao - 0

/ Ai--1)/(gl1 ... i)]Po, 1i5N,

with

Po= 1+ Y ;(Ao 0 ...k-JAA1 ... Ak)

k=l 1

/N-1 (P -

i 0

N-E: i N

-1- with

i+l

I"-' = i+l, /(-l

+ fLi), 0

-i-SN-1; 1=0

(e) p = k= o05i<N-1, /N,-1 - _




with N

L•d) = a 1,1+(hX + 0L5), O

__i 5N-1.

Some of these distributions have appeared in the literature previously with different derivations.

{p,}, commonly known as the 'outside observer's distribution', is usually derived via the infinitesimal generator of the embedded Markov chain in continuous time by solving the associated stationary equations (e.g. [3], p. 273).

{(?,} can be obtained via the {pj} distribution (cf. [10], p. 714) where it is shown that -rn = c(X, + p,)pi).

The {(a-•-' distribution and its equality to the {rT (d)} distribution also appears in the literature [9] with an alternative derivation. Krakowski [8], using intuitive 'plausibility' arguments gives the result of Theorem 5(b) concerning

{(a-}(. Furthermore, Cooper ([4], p. 194) presents an exercise, attributed to P. J. Burke (1968, unpublished), concerning the derivation of the 'arriving customer's distribution' {ra~-j} where it is shown that Xi, (a-) = (a-). Such a result follows from Corollary 3A from which Theorem 5(b) follows.

The expressions for the limiting distributions of an arriving (departing) customer's distribution in continuous time {p?-)} ({p}(d)) are new.

Corollary 5A. For the birth-death queue with state space 5f = {0, 1, ? ? , N}:

/N-1 N

i' = , Akpk = /i+1Pi+1 Lkpk, iN- 1. k=O /k=1

(Assuming S' = {0, 1,... .}, Krakowski [8] derived the first equality of Corol-

lary 5A. See also [4], p. 85.)

Corollary 5B. Let {raia-)}

be the limiting arriving customer's distribution for a birth-death queue with birth rates Ni = (N- i)A, (0 5 i 5 N). Let {-lp,} be the limiting outside observer's distribution for a birth-death queue with birth rates NlX = (N - 1 - i)A, (0 5 i

_ N-1). Then

{(•a--)}= {_XPi

Cooper ([4], pp. 82-6) gives an intuitive argument to establish Corollary 5B for such finite-source models with 'quasi-random' inputs. The conclusion of

Corollary 5B is still valid if N•i

= N-li--1, (15 i-

N- 1). Corollary 4C implies that, for finite systems, equality between {T?a-)} and

{p, },

or between {d)a} and {pd)}, is not possible. If we simplify either the arrival or service processes can we find simple relationships between such limiting distributions? Theorem 4 and Corollary 4B aid in deriving the following results which assist us in such an examination.




Theorem 6. For the birth-death queue with state space = {0, 1, ... , N} (a) if the arrival rates A, = A, (0 -

i -

N- 1), then jkla- = 1/A, (0 - i -

N- 2), and 4-14 = (1/A) + (1/1N) implying

p_

= t

-/(1) +

/

-'O-i--(a-)N-2, P I

1 +----(a-) - 1+kN(a-)

N 1;

(b) if the service rates p, = , (1- i- N), then t -)= (1/A0o)+(1/pt) and

gCid)= 1/I, (1 - i 5 N) implying

( +1A (d)1 + o 1r(d) i = 0, (d)

0 IAi

0

pi Lo 1. (d)

/k + ir (d)

1<i-<5N.

In passing we note that the expressions for {j"-')} and {( d)} as derived in the theorem also follow by intuitive arguments. For example, under the condition of Theorem 6(b), p d) is the mean time between departures with i in the system. When i >0 the services take an exponential length of time with mean 1/~ while when i = 0 we have to wait for an arrival (mean time 1/A0) and then wait for a service completion (mean time 1/v) before the next transition occurs in the {(X~'), Tnd))} process. Similar observations hold for the {(ta-)} under the condition of Theorem 6(a) with care taken when i = N- 1, since in this case we have to wait for a departure (with mean time 1/1tN) and then wait for the next arrival (with mean time 1/A) before the next transition occurs in the {(X•a-), T~))} process.

Corollary 6A. For birth-death queues with state space $1 = {0, 1, - -... , N}: (a) if Ai = A, (0 5 i - N- 1), then

(i) 1a-) = p1/(l-PN), Oi5N -1;

(ii) p =-)-

Pi 0

i-5-i -N

-2;

PN--1 + PN, i = N- 1;

(b) if Itk = t, (1 - i -N),

then

(i) = Pi+1/(1 -Po), 05i 5 N - 1;

( d) Po+ P13, i = 0,

pIIM, 1 --_ i <5N.

These results are all new except for Corollary 6(a)(i) which appears in [8]. (Corollary 6(a)(i) also holds for M/G/1/N queues as shown in [5], p. 252.)




Let us now examine the birth-death queue with a countably infinite state

space. For such models considerable simplification results when we assume either a Poisson input, A, = A for all i t 0 or a negative exponential service-time

distribution, fk = p for all it 1. (Note that no finite-state-space model can have a Poisson input since AN- 0.) The results which follow are deduced by taking appropriate limits.

Theorem 7. For the birth-death queue with state space 1' = {0, 1, 2, -- (a) if A. = A, (i _ 0), then

-{ld) }= {la-} = {pia-)}= {p};

(b) if m

= , (i=t

1), then

{l"a-)} = { l)} = {Pi+lI/(- PO)- ,

and

(d Po + Pi, i = 0,

p p+, i= 1.

The equality of the outside observer's distribution {pi} with the arriving customer's distribution {?ia"-)} (or equivalently with the departing customer's distribution {1rd)}) for single-server queueing system with Poisson input and

general service times is well known, (e.g. [3], p. 352, [4], p. 65, [5], p. 236, [7], p. 118). Theorem 7(a) provides us with a rigorous proof in the birth-death context substantiating the claim made in [8]. The remaining results in Theorem 7 are new.

4.2. The M/M/1/N and MI/M/1 queueing models. The M/M/1/N queue with traffic intensity p is a birth-death queue with state space = {0, 1,- , N}, XA= A (0 - i

_ N-1), 1,= t (1_5 i

_ N) and p= AIX.

Theorem 8. For the M/M/1/N queue with traffic intensity p: if p# 1 then

(1--p)/2(1 - p), i = 0,

(a) (i) 'i = (1-p2)pi-1/2(1 _ N), 1i--<N-1 (1- p)pN-1/2(1- pN), i = N;

(b) (i) ir" = !a)i = (1 - p)p'(1- pN), O0 i<5N-1;

(c) (i) Pi = (1- p)p'/(1- pN+1), O i -- N;

I ((1 -p)p(1- pN+l), 05i --N-2, (d) (i)

Pi-= (1 -

p2)pN-1/(1--PN+I), i=N-l; i = 0,

1(e) (i) p 1p)pi+/(1 pN+X), iN-1;




while if p = 1 then

(a) (ii) = 1/2N, i = 0, N,

i11/N, 1

-5 i -5 N- 1;

(b) (ii) a-) = Id)= 1/N, 5 i N- 1

(c) (ii) Pi = 1/(N + 1), O -5 i - N;

(d) (ii) P(a-)= 2/(N+1), i=5N-2;

2d) 2/(N + 1), i = 0,

(e) (ii) pa =- 1.I1/(N+ 1), 1-5i N-1.

Apart from the results (c) (i) and (c) (ii), which are well known, and the

results (b) (i) and (b) (ii), which follow from results for M/G/1/N and

GI/M/1/N queues, the remaining results have not specifically appeared in the literature.

Corollary 8A. For the M/M/1 queue with traffic intensity p <1:

{1drw)} = {Ira-) = (a-=}a- j= {pi} = {(1 - p)p'};

(1- p)/2, i = 0,

(1- p2)pi-1/2, i- -1; and

(d) 1- p2, i = 0,

(1 Pp)pi+1

The {pi}, { rId)} and {I.(a-)} distributions are well known; {,ri} is discussed in

[10], p. 105, whilst {pId)} and {pIa-)} follow from results deduced for M/G/1 and

GI/M/1 queues. In particular, for M/G/1 queues, even though {(X,,

T.)} is not an MRP, {X )}

is a Markov chain and {(Xn), T n)} is an MRP. For such models, if p is the usual

traffic intensity then, provided p <1,

(d) p1- 2, i = 0,

P (d) i>2'

Similarly, for GI/M/1 queues, {(X,, T,)} is not an MRP but {X(a-)} is a Markov chain and {(Xa-'), T'(a)} is an MRP with {pIa-)}= a-) ([5], p. 36).

4.3. Birth-death queues in discrete time. When such queues were introduced

in Example 2 we saw that it was possible for an arrival and a departure both to

occur in the same interval implying a transition from a state to itself. Conse-




quently this type of queue is not covered by the theory of this paper. However, we can use the results of [6] to examine the embedded discrete-time Markov chains.

If we define, for 0 5 i 5 N, Aa, - aii, i• p, i - ai, q - with AN = 0,

g•o= 0, PN = O0, qN = 1 then it is easily verified that the transition probabilities of

the {X,j Markov chain are identical to those of the feedback model as discussed in Example 1 of [6].

Let us use an asterisk to denote reference to the relevant feedback model. Then {T,} = {*T,}, {Ta)}={*TV)} and {Td)} ={*T)"'}. Consequently {}ri}= {*i}, {.I a)}= {*(i)} and {)} = {*Io)

If we define p,=aii/dljiP, (1?i N-1), with Po-1, PN-O, then from

Theorem 13 and Corollary 13A of [6] we obtain, after simplification for case

(c),

( 1 (a) 1ri = Po " "

+ pi- i --1+'ro,

1 - i -_N,

with

ITo = Po""

P 1

(b) a) =P ?"-

1_N (b) =Po ''' Pi-1 o

with (1) 1-T=

- o Po " " " Pi-1

/1 \ J

Li =1

The { Ia-)} distribution is not readily available. We use an argument similar to that used in establishing the {w?'-)} distribution in Corollary 15A of [6]. The

requisite generalisation implies that

( ) =(() , ( - -) 1 <

i--"

- i

i ,

with r i)= TN-1N-1-

An inductive procedure (starting from i = N- 1, N- 2, - -, M and showing the result true for i = M- 1) leads to the conclusion that {(a-)}= {d)}, akin to the result established in Corollary 3A above.

Acknowledgement

A substantial part of the research reported in this paper was carried out whilst the author was the recipient of a College of Engineering Visiting




Professorship within the Department of Industrial Engineering and Operations Research, Virginia Polytechnic Institute and State University. The author wishes to acknowledge the many helpful discussions he held with Professor

Ralph L. Disney concerning the results contained herein.

References

[1] TINLAR, E. (1969) Markov renewal theory. Adv. Appl. Prob. 1, 123-187. [2] CINLAR, E. (1975) Markov renewal theory: a survey. Management Sci. 21, 727-752. [3] CINLAR, E. (1975) Introduction to Stochastic Processes. Prentice-Hall, Englewood Cliffs, N.J. [4] COOPER, R. B. (1972) Introduction to Queueing Theory. MacMillan, New York. [5] GROSS, D. AND HARRIS, C. M. (1974) Fundamentals of Queueing Theory. Wiley, New York. [6] HUNTER, J. J. (1983) Filtering of Markov renewal queues, I: Feedback queues. Adv. Appl.

Prob. 15, 349-375. [7] KLEINROCK, L. (1975) Queueing Systems, Volume 1: Theory. Wiley, New York. [8] KRAKOWSKI, M. (1974) Arrival and departure processes in queues. Rev. Franc. Automat.

Informat. Recherche Opirat. 8 V-I, 45-56. [9] NATVIG, B. (1975) On the input and output processes for a general birth-and-death

queueing model. Adv. Appl. Prob. 7, 576-592. [10] Ross, S. M. (1970) Applied Probability Models with Optimization Applications. Holden

Day, San Francisco.



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Filtering of Markov Renewal Queues, II: Birth-Death Queues