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arXiv:math/0611526v1 [math.PR] 17 Nov 2006

A note on insensitivity in stochastic networks

Stan Zachary

Maxwell Institute for Mathematical Sciences, Heriot-Watt University

Edinburgh

We give a simple and direct treatment of insensitivity in stochastic networks which

is quite general and which provides probabilistic insight into the phenomenon. In the

case of multi-class networks, the results generalise those of Bonald and Prouti` ere (2002,

2003).

1Introduction

It is well-known that many stochastic networks—notably queueing and loss networks—

have stationary distributions of their level of occupancy which depend on certain input

distributions only through the means of the latter. This phenomenon of insensitivity has

been studied by various authors over an extended period of time, in varying degrees of

generality and abstraction, and using a variety of techniques.

In the present paper we revisit this topic to develop an insight of Pechinkin (1983, 1987)

to give a very simple and direct treatment of insensitivity. In particular the approach

avoids those based on brute-force calculations, the consideration of phase-type distribu-

tions (Schassberger, 1978, Whittle, 1985, Bonald and Prouti` ere, 2002, 2003), or the use of

quite complex machinery for handling generalised semi-Markov processes (Burman, 1981,

Schassberger, 1986)—although such processes are implicit in the current approach. It fur-

ther avoids assumptions about, for example, continuity of distributions, necessary for some

of the above approaches, and also explicitly identifies the entire stationary distributions

of the networks concerned, showing that, where insensitivity obtains, these stationary dis-

tributions have a particularly simple and natural form. Pechinkin used his insight, which

involves what is in effect a coupling argument together with induction, to give probabilistic

proofs of the insensitivity of a number of single-class loss systems with state-dependent ar-

rival rates—results originally proved analytically by Sevastyanov (1957). He also indicated

the wider applicability of the approach in the single-class case. In the present paper we

give a substantial reformulation of the underlying idea, under more general conditions and

showing that its most natural expression is in terms of balance equations. This consider-

ably simplifies its application to single-class systems—notably the quite complex coupling

constructions are no longer needed. It further makes possible the extension of the idea to

the multi-class networks considered in Section 3. The main aim is to provide probabilistic

insight, notably for multi-class networks. Indeed it is shown that insensitivity is simply a

byproduct, under appropriate conditions, of probabilistic independence.

We study networks in which individuals arrive at various classes at rates which may

depend on the state of the entire system, bringing workloads which are independent and

identically distributed within classes and which have finite means. Within each class

0American Mathematical Society 1991 subject classifications. Primary 60K20

Key words and phrases. insensitivity, stochastic network, partial balance

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workloads are reduced at rates which may again be state-dependent (when the rate is

constant workloads may be identified with lifetimes in classes), and on completion of its

workload an individual moves to a different class or leaves the system, with probabilities

which may yet again be state-dependent.

In order to obtain insensitivity we typically require that an individual joining a class is

immediately served, i.e. has its workload reduced, at a rate which is the same as that

of an individual immediately prior to leaving the class (where in each of these cases the

number of individuals in each class of the system is the same)—more generally that the

service discipline should define a network which is symmetric in the sense of Kelly (1979).

The most common example is that of processor-sharing networks, but other possibilities

are well-known, for example, “last-in-first-out preemptive resume” networks. We shall

concentrate on a very broad class of processor-sharing networks, introduced by Bonald

and Prouti` ere (2002) and including, for example, traditional loss networks and processor-

sharing Whittle and Jackson networks, as special cases). We shall also indicate the simple

modifications required to deal with other possibilities.

For the above class of processor-sharing networks, Bonald and Prouti` ere used phase-type

arguments to show that, under conditions which correspond to the satisfaction of the

appropriate partial balance equations, the stationary distribution of the number of indi-

viduals in each class is insensitive to the workload distributions, subject to the means of

the latter being fixed and to the distributions themselves being drawn from the broad class

of Cox-type distributions (dense in the class of all distributions on R+). In the present pa-

per we formally consider all workload distributions on R+with finite means, and identify

also the stationary residual workload distributions. However, as stated above our main

aim is to give a direct and probabilistically natural treatment. It turns out (and is in

many cases well-known) that, when the appropriate partial balance equations are satis-

fied for such a network, then the stationary distribution of the entire system, including

the specification of residual workloads, is such that departures from each class are exactly

balanced by arrivals to that class—in a sense again to be made precise below. Indeed, for

single-class systems, this is the essence of Pechinkin’s insight. What is of interest is that

same idea extends to establish insensitivity for the very much more general networks con-

sidered here, and indeed appears also to establish insensitivity in more abstract settings

such as that considered by Whittle (1985), though we do not formally consider this more

abstract environment here.

In order to fix ideas, it is convenient to consider first, in Section 2, single-class networks.

Here the extension of previous ideas is not too difficult. Nevertheless it is desirable to give

a careful treatment of this case, avoiding notational complexity while preserving rigour, so

as both to establish the underlying principle and also to set the scene for the multi-class

networks which we consider in Section 3.

2Single-class networks

Consider an open system with a single class of individual (customer, call, or job). Indi-

viduals arrive as a Poisson process with state-dependent rate α(n), where n is the number

of individuals currently in the system. Arriving individuals have workloads which are

independent of each other and of the arrivals process with a common distribution µ on

R+which we assume to have a finite mean m(µ). While there are n individuals in the

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system, their total workload is reduced at a rate β(n) ≥ 0, where we assume β(n) > 0 if

and only if n > 0; an individual departs the system when its workload is reduced to zero.

By suitably redefining the rates β(n) if necessary, we may, and do, assume without loss of

generality that the mean workload m(µ) = 1.

We consider first the processor-sharing case. Here when there are n > 0 individuals in the

system, the workload of each is simultaneously reduced at a rate β(n)/n, and the set-up

described above becomes a fairly general description of a single-class processor sharing

system. A special case is the simple Erlang loss system, in which, for some α,β > 0, we

have α(n) = αI(n < C) for some capacity C ≤ ∞ (where I is the indicator function) and

β(n) = nβ for all n ≥ 0. Here individuals are typically referred to as calls, and workloads

correspond to call durations (since β(n)/n is independent of n). A further special case

is the M/GI/m/∞ processor-sharing queue, in which, again for some α,β > 0, we have

α(n) = α for all n and β(n) = min(n,m)β for all n and some fixed m.

We represent the system as a Markov process (X(t))t≥0by defining its state at any time t

to be the number n of individuals then in the system together with their residual workloads

at that time. (An alternative is to record, for each individual, the workload completed at

time t.) For given n > 0 these workloads form an (unordered) set, and may be regarded as

taking values in the quotient space Snobtained from Rn

be obtained from each other under permutation of their coordinates. The σ-algebra B(Sn)

on Snis similarly formed in the obvious manner from the Borel σ-algebra on Rn

state space S for the process (X(t))t≥0is then the union of the Sn, n ≥ 0, where the set S0

is taken to consist of a single point, and its associated σ-algebra B(S) consists of those sets

which are countable unions of sets in the σ-algebras B(Sn). The process (X(t))t≥0is thus

an instance of a piecewise-deterministic Markov process (Davis, 1984, 1993). However, we

avoid the need for most of the general machinery for handling such processes.

+by identifying points which may

+. The

We define the probability distribution ¯ µ on R+to be the stationary residual life distribu-

tion of the renewal process with inter-event distribution µ, that is, if µ has distribution

function F then ¯ µ has distribution function G given by

G(x) = 1 −

?∞

x

(1 − F(y))dy

(recall that m(µ) = 1). Note that the “residual life” here should be thought of as a residual

workload rather than a time. For each n ≥ 1, define also the probability distribution ¯ µnon

Snto be the product of n copies of the distribution ¯ µ, again with the above identification of

points in Rn

n copies of the distribution ¯ µ and θ is the projection from Rn

Thus ¯ µnrepresents the joint distribution of the residual lives at any time in a set of n

independent stationary renewal processes each with inter-event distribution µ; we define

also ¯ µ0to be the probability distribution concentrated on the single-point set S0. For

each n, we also regard ¯ µnas a distribution on S, assigning its total mass one to the

set Sn. Finally, for any distribution π on Z+, define the distribution ¯ µπon S by ¯ µπ=

?

workloads.

+(more formally, ¯ µn(A) = ¯ µn(θ−1(A)), A ∈ B(Sn), where ¯ µnis the product of

+into the quotient space Sn).

n∈Z+π(n)¯ µn. Thus ¯ µπassigns probability π(n) to the event that there are n individuals

in the system, and, conditional on this event, assigns the distribution ¯ µnto their residual

Theorem 1. Suppose that the distribution π on Z+is the solution of the balance equations

π(n + 1)β(n + 1) = π(n)α(n),n ≥ 0, (1)

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and that

?

n≥0

π(n)α(n) < ∞. (2)

Then the distribution ¯ µπon S is stationary for the process (X(t))t≥0, and in particular

the distribution π is stationary for the associated number of individuals in the system.

Remark 1. The condition (2) ensures that, under stationarity, individuals arrive at the

system at a finite rate.

Proof of Theorem 1. In order to exclude pathological behaviour in the argument below,

we make the one additional assumption that the distribution µ has no atom of probability

at zero. This is without loss of generality: in the case that µ does have such an atom, the

evolution of the system may clearly be equivalently described by redefining α, β and µ so

as to remove it, and the result of the theorem is easily obtained via this reparametrisation.

Analogously to the definition of ¯ µn, for each n ≥ 1, define the probability distribution ˆ µn

on Snto be the product of n − 1 copies of the distribution ¯ µ and a single copy of the

distribution µ, yet again with the above identification of points in Rn

ˆ µn(A) = ˆ µ(n)(θ−1(A)), A ∈ B(Sn), where ˆ µ(n)= ¯ µn−1× µ and θ is again the projection

from Rn

+into the quotient space Sn.) We again regard ˆ µnas a distribution on S, assigning

mass one to the set Sn.

Consider now the modified process (ˆ X(t))t≥0 on S describing the system in which the

workload distribution is again µ and in which, when there are n ≥ 1 individuals in the

system, individual workloads are again reduced at rate β(n)/n; however, for the modified

system, (a) an individual departing on completion of its workload is immediately replaced

by another bringing an independent workload with distribution µ, (b) external arrivals

to the system are not accepted. Thus, for the modified system, the number of individu-

als remains constant, and conditional on this being n, the system behaves as a set of n

independent renewal processes, each of which has stationary residual workload distribu-

tion ¯ µ. Hence, for any distribution π′on Z+, the distribution ¯ µπ′ on S is stationary for

the process (ˆ X(t))t≥0.

Let (Pt)t≥0and (ˆPt)t≥0be the semigroups of transition kernels associated respectively with

the processes (X(t))t≥0and (ˆ X(t))t≥0. For any a > 0, let Dabe the class of functions f

on S taking values in [0,1] and satisfying the continuity condition

+. (More formally,

|(Ptf(x) − f(x))| ≤ atfor all x ∈ S and t > 0, (3)

where Ptf(x) =?

expectation of f(X(t)) when (X(t))t≥0 is given initial distribution ν); similarly define

νˆPtf.

Now compare the behaviour of the processes (X(t))t≥0and (ˆ X(t))t≥0, each started with

the distribution ¯ µπ; so as to simplify the description below we couple these two processes

so that they agree until the time of the first arrival or workload completion. We then have

(see the further explanation below) that, with this common initial distribution, for any

SPt(x,dy)f(y). For any such f and for any distribution ν on S, define

Sf(x)ν(dx) and, for any t > 0, define νPtf = ν(Ptf) (so that νPtf is thealso νf =

?

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a > 0, f ∈ Da, and h > 0,

¯ µπPhf − ¯ µπˆPhf = E?f(X(h)) − f(ˆ X(h))?

= h

?

= h

?

= o(h)

n≥0

π(n)?α(n)(ˆ µn+1f − ¯ µnf) + β(n)(¯ µn−1f − ˆ µnf)?+ o(h)

[π(n)α(n) − π(n + 1)β(n + 1)](ˆ µn+1f − ¯ µnf) + o(h)

(4)

n≥0

(5)

as h → 0 (recall in (4) that β(0) = 0); further the above convergence as h → 0 is uniform

over f ∈ Dain the sense that (5) may be written as

sup

f∈Da

|¯ µπPhf − ¯ µπˆPhf| = o(h)as h → 0(6)

(again see below). To show (4) note first that, from the above coupling and for any h > 0,

we have f(X(h)) = f(ˆ X(h)) except where there is either at least one external arrival or at

least one workload completion in [0,h]. It follows from the definition of ¯ µπthat, conditional

on the number of individuals initially being n the probability of an external arrival in [0,h]

is α(n)h + o(h) as h → 0, and that an arriving individual finding the distribution of the

system to be ¯ µnchanges this to ˆ µn+1in the case of the process (X(t))t≥0 and leaves

it unchanged in the case of the process (ˆ X(t))t≥0. Similarly, again conditional on the

number of individuals initially being n (and recalling that m(µ) = 1), the probability of

a workload completion in a time interval [0,h] is β(n)h + o(h) as h → 0, and that under

the distribution ¯ µn, conditional on such a completion taking place, the residual workload

distribution becomes ¯ µn−1in the case of the process (X(t))t≥0and ˆ µnin the in the case of

the process (ˆ X(t))t≥0. Further it follows from the conditions (1) and (2) that, under the

initial distribution ¯ µπ, the probability of two or more arrivals or workload completions in

[0,h] is o(h) as h → 0. That the relation (4) now holds as h → 0 with the uniformity over

f ∈ Darequired for (6) follows easily from these results and from the definition of Da. To

see this note that, since f ∈ Daimplies that f takes values in [0,1], the contribution to

the error term in (4) resulting from the neglect of the possibility of two or more arrivals

or workload completions in [0,h] is uniformly o(h) as h → 0 as required. Similarly the

terms ˆ µn+1f − ¯ µnf and ¯ µn−1f − ˆ µnf in (4) are obtained by treating the precise time of

the first arrival or workload completion within [0,h] as if it were time h; (recalling that

ˆ µn+1, etc, are probability measures) it follows from (3) that the consequent error in each

of the above two terms is bounded by 2ah, so that the further contribution to the error

term in (4) is O(h2) as h → 0, again with uniformity over f ∈ Da. The relations (5), and

hence (6), are now immediate from the balance equations (1). Since the distribution ¯ µπ

is stationary for the process (ˆ X(t))t≥0, it now follows from (6) that, again for any a > 0

and h > 0,

sup

f∈Da

|¯ µπPhf − ¯ µπf| = o(h)as h → 0.

Further, it is straightforward that if f ∈ Da, then also Ptf ∈ Dafor any t > 0. Standard

manipulations using the semigroup structure of (Pt)t≥0, e.g. the consideration of increas-

ingly refined partitions of the interval [0,t], now give that, for all a > 0, f ∈ Da, and

t ≥ 0,

¯ µπPtf = ¯ µπf.(7)

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Finally, we show that it follows from (7) that ¯ µπPt = ¯ µπfor all t > 0, so that ¯ µπis

stationary for (X(t))t≥0as required. It is sufficient to show that, for any n ≥ 1 and any

set A ∈ B(Sn) whose inverse image in Rn

+under the mapping θ defined above is a product

of intervals in R+, we have

¯ µπPtIA= ¯ µπIA, (8)

where IAis the indicator function of the set A. It follows from the piecewise deterministic

form of the process (X(t))t≥0that we may choose a sequence of functions (fk, k ≥ 1) such

that, for each k, (i) fk∈ Dafor some a > 0 and (ii) fkand IA agree except on a set

whose Lebesgue measure (under θ−1) in Rn

+tends to zero as k → ∞. Since ¯ µπ, and so also

¯ µπPt, are non-atomic distributions, the result (8) now follows by using (7) with f = fk

and letting k → ∞.

Remark 2. Suppose that the equations (1) above are multiplied by the signed measure

(ˆ µn+1− ¯ µn) to give

π(n)α(n)(ˆ µn+1− ¯ µn) = π(n + 1)β(n + 1)(ˆ µn+1− ¯ µn),n ≥ 0.(9)

These equations have an obvious interpretation as representing, under the distribution ¯ µπ

and for each n ≥ 0, a detailed balance of flux between Snand Sn+1, not just with regard

to the total probability assigned to each of these spaces, but also with regard to the

distribution of the residual workload sizes: the intuition underlying the derivation of (5)

above—which is also that of Pechinkin’s coupling approach—is that, under ¯ µπ, an arrival

finding n individuals in the system transforms the residual workload distribution from ¯ µn

to ˆ µn+1, while a departure from the system when it contains n+1 individuals transforms

the residual workload distribution from what would have been ˆ µn+1, if the individual had

remained in the system with a renewed workload, to the distribution ¯ µn.

In the case where we do not have processor-sharing, i.e. in which it is no longer the case

that at any time all workloads are being reduced at the same rate, it is necessary at any

time to distinguish the individuals in the system. Thus each Snabove is replaced by Rn

and the state space S is replaced by S∗=?

at any time all service effort is devoted to the last individual to arrive at the system. If at

any time there are n individuals in the system, we may index these by i = 1,...,n in the

order of their arrival, and no individual changes index during its time in the system; as

usual arrivals occur as a Poisson process with rate α(n), and the workload of individual n

is now being reduced at rate β(n), while that of the remaining individuals is being reduced

at rate 0. As previously, define the probability distribution ¯ µ on R+to be the stationary

residual life distribution of the renewal process with inter-event distribution µ, and, for

each n ≥ 0, let the distribution ¯ µnon Rn

For each n ≥ 1, let the distribution ˆ µnon Rn

the distribution ¯ µ and a single copy of the distribution µ, with the latter assigned to the

nth coordinate of Rn

+. Finally, for any distribution π on Z+, define the distribution ¯ µπon

S by ¯ µπ=?

above: under the distribution ¯ µπ, and relative to the modified process considered in the

proof of Theorem 1, an arrival finding n individuals in the system transforms the residual

workload distribution from ¯ µnto ˆ µn+1, while a departure from the system when it contains

+

n≥0Rn

+. We consider as an example the case

of the single-server queue with “last-in-first-out preemptive resume” discipline, in which

+be now the (ordered) product of n copies of ¯ µ.

+be the (ordered) product of n − 1 copies of

n∈Z+π(n)¯ µnas before. With these (re)definitions, both Theorem 1 and its

proof remain unchanged as stated. Again the underlying reason is as given in Remark 2

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n+1 individuals transforms the residual workload distribution from ˆ µn+1to ¯ µn. Note that

this balance does not obtain in the case of, for example, a “first-in-first-out” discipline,

and here, as is again well known, we do not have the above insensitivity.

3 Multi-class networks

Consider now a multi-class network.

adaptations to other disciplines may be made as in the single-class case.

{1,...,N} denote the set of classes, and let n = (ni, i ∈ I) where ni is the number

of individuals in each class i. An individual entering class i acquires a workload which has

distribution µiwith nonzero finite mean m(µi); we again assume without loss of generality

that m(µi) = 1; the workload of each individual in class i is reduced at a state-dependent

rate φi(n)/ni, where φi(n) > 0 if and only if ni> 0. Individuals arrive at each class i

from outside the network as a Poisson process with state-dependent rate φ0i(n); on com-

pletion of its workload in any class i an individual moves to class j with state-dependent

probability φij(n)/φi(n) or leaves the network with probability φi0(n)/φi(n), where

We concentrate on the processor-sharing case—

Let I =

?

j∈I

φij(n) + φi0(n) = φi(n) (10)

(there are no problems in allowing the possibility φii(n) > 0). The workloads, arrivals

processes and routing decisions are all independent.

As in the single-class case, we represent the system as a Markov process (X(t))t≥0 by

defining its state at any time to be the vector n introduced above together with the

residual workloads at that time of the set of individuals in each class. For given n, these

workloads take values in the space Snwhich is the ordered product of the spaces Sni, i ∈ I,

where, as previously, each Sniis formed from Rni

space which may be obtained from each other under permutation of their coordinates

(and where again the set S0contains a single point). The state space S for the system is

the union of all the possible Sn, and the spaces Snand S are endowed with the obvious

σ-algebras B(Sn) and B(S).

Analogously to the single-class case, for each i ∈ I, define the probability distribution ¯ µi

on R+to be the stationary residual life distribution of the renewal process with inter-event

distribution µi(as previously the residual life should be interpreted as a residual workload).

For each i ∈ I and for each ni≥ 1, define as previously the distribution ¯ µi

the product of nicopies of the distribution ¯ µi(again with the above identification of points

in Rni

+)—representing the joint distribution of the residual lives in a set of niindependent

stationary renewal processes each with inter-event distribution µi; define also ¯ µi

probability distribution concentrated on the single-point set S0. For each n ∈ ZN

the distribution ¯ µnon Snto be the (ordered) product distribution which, for each i ∈ I,

assigns the distribution ¯ µi

nito Sni. We again regard the distribution ¯ µnas a distribution

on S, assigning its total mass one to the set Sn. For any positive distribution π on ZN

we define the distribution ¯ µπon S by ¯ µπ=?

It is notationally convenient to expand the set I to I′= {0} ∪ I, treating 0 as an extra

class feeding external arrivals to, and receiving departures from, the network. (However,

the components of the state n of the network remain indexed in the original set I.) For

completeness we define φ00(n) = 0 for all n, and also φij(n) = 0 for all i ∈ I, j ∈ I′, and

+by identifying points within the latter

nion Snito be

0to be the

+, define

+,

n∈ZN

+π(n)¯ µn.

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n such that ni= 0 (so that (10) above remains valid for such n also). For each i ∈ I, let

eibe the N-dimensional vector whose ith component is 1 and whose other components

are 0, and let e0be the N-dimensional vector all of whose components are 0. For each n

and each i,j ∈ I′define the vector Tj

and Tjn = Tj

0n = n + ej.

in = n − ei+ ej; define also Tin = T0

in = n − ei

Theorem 2. Suppose that the distribution π on ZN

+satisfies the partial balance equations

π(n)

?

j∈I′

φij(n) =

?

j∈I′

π(Tj

in)φji(Tj

in),

n ∈ ZN

+,i ∈ I′, (11)

where, for n and i ∈ I such that ni= 0 we interpret the right side of (11) as zero (recall

that when ni= 0 we have φij(n) = 0 for all j ∈ I′so that (11) is automatically satisfied

in this case). Suppose also that

?

n∈ZN

+

π(n)

?

i∈I

φ0i(n) < ∞. (12)

Then ¯ µπis stationary for the process (X(t))t≥0, and in particular π is stationary for the

associated numbers of individuals in the system. Conversely, if a distribution π on ZN

stationary for the numbers of individuals in the system for all µ = (µi, i ∈ I) such that

m(µi) = 1 for all i ∈ I, then π satisfies the equations (11).

+is

Proof. Suppose first that π satisfies the equations (11). As in the proof of Theorem 1, we

again assume without loss of generality that each distribution µihas no atom of probability

at zero.

For each n and for each i such that ni≥ 1, define also the residual workload distribution ˆ µi

on Snby ˆ µi

ni×?

workload distribution ¯ µj, except only that a single individual in the class i is given the

workload distribution µi. For each n, define also ˆ µ0

Again as in the proof of Theorem 1, define the process (ˆ X(t))t≥0on S to be that appropri-

ate to the modified system in which there are no arrivals, departures, or transfers between

classes; rather each individual in each class i, on completion of its workload, acquires a

new independent workload with distribution µi. Thus the occupancy of the system re-

mains constant; conditional on this being n, individual workloads in any class i such that

ni> 1 are again reduced at rate φi(n)/niand the system behaves as a set of independent

renewal processes. Further, for any distribution π′on ZN

stationary for (ˆ X(t))t≥0.

Again let (Pt)t≥0and (ˆPt)t≥0be the semigroups of transition kernels associated respectively

with the processes (X(t))t≥0 and (ˆ X(t))t≥0, and, for any a > 0, let Dabe the class of

functions f on S taking values in [0,1] and satisfying the earlier continuity condition (3).

Comparison of the behaviour of the processes (X(t))t≥0and (ˆ X(t))t≥0, each started with

the distribution ¯ µπand coupled as in the earlier proof until the time of the first external

n

n= ˆ µi

j?=i¯ µj

njwhere ˆ µi

niis defined as in the proof of Theorem 1. Thus ˆ µi

n

corresponds to each individual in each class j having independently the stationary residual

n= ¯ µn.

+, the distribution ¯ µπ′ on S is

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arrival or workload completion, now gives that, for any a > 0, f ∈ Da, and h > 0,

¯ µπPhf − ¯ µπˆPhf

= E?f(X(h)) − f(ˆ X(h))?

= h

?

+

??

??

n∈ZN

π(n)

?

i∈I′

?

j∈I′

φij(n)

?

ˆ µj

Tj

inf − ˆ µi

nf

?

+ o(h) (13)

= h

?

n∈ZN

+

i∈I′

?

j∈I′

π(n)φji(n)ˆ µi

Ti

jnf −

?

i∈I′

?

j∈I′

π(n)φij(n)ˆ µi

nf

?

+ o(h)

= h

?

n∈ZN

+

i∈I′

?

j∈I′

π(Tj

in)φji(Tj

in)ˆ µi

nf −

?

i∈I′

?

j∈I′

π(n)φij(n)ˆ µi

nf

?

+ o(h)

= h

?

n∈ZN

+

?

i∈I′

??

j∈I′

π(Tj

in)φji(Tj

in) −

?

j∈I′

π(n)φij(n)

?

ˆ µi

nf + o(h)

= o(h) (14)

as h → 0, with uniformity of convergence over all f ∈ Da, so that we may write

sup

f∈Da

|¯ µπPhf − ¯ µπˆPhf| = o(h) as h → 0(15)

(recall again that φij(n) = 0 whenever ni = 0, so that there is no difficulty with the

lack of a formal definition of ˆ µi

nin this case). The identity (15) is simply the multi-class

version of the identity (6) in the proof of Theorem 1, and is similarly obtained, albeit

with a slightly more compact notation; in particular, conditional on the common initial

distribution of the two processes being given by ¯ µn, in the time interval [0,h] where h is

small, a transition from i to j in the original system—where either i or j may be 0—occurs

with probability π(n)φij(n)h + o(h) as h → 0, and in this case the distribution of the

process (X(t))t≥0 becomes ˆ µj

Tj

(13) now holds with the required uniformity of convergence follows, using also (12), as in

the earlier proof; finally the results (14), and so also (15), follow from the partial balance

equations (11). Since the distribution ¯ µπis stationary for the process (ˆ X(t))t≥0, it now

follows from (15) that, again for any a > 0 and h > 0,

inwhile that of the process (ˆ X(t))t≥0 becomes ˆ µi

n; that

sup

f∈Da

|¯ µπPhf − ¯ µπf| = o(h) as h → 0.

That ¯ µπis now stationary for (X(t))t≥0follows as in the proof of Theorem 1.

Now suppose that a distribution π on ZN

in the system for all µ = (µi, i ∈ I) with m(µi) = 1 for all i ∈ I.

π then necessarily satisfies the partial balance equations (11) is given by Bonald and

Prouti` ere (2002, 2003). In summary, consider the case in which in every class the workload

distribution is exponential with mean 1, and, for any fixed class i, compare this with the

case in which, for some 0 < λ < 1, the workload distribution in class i is replaced by

a mixture of two distributions, obtained by choosing with probability λ an exponential

distribution with mean λ−1, and with probability 1 − λ the distribution concentrated on

0. Both these models may be (re)formulated as simple Markov jump processes—in the

latter case the transition rates into and out of the class i are reduced by a factor λ. Since

+is stationary for the numbers of individuals

A proof that

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π is stationary in both cases, comparison of the (full) balance equations for stationarity

yields the partial balance equations (11).

Example 1. Processor-sharing Whittle networks. Suppose that for some ν ≥ 0, some

strictly positive function Φ on ZN

such that p00= 0, we have, for each n,

+, and some stochastic matrix P = (pij, i ∈ I′, j ∈ I′)

φ0j(n) = νp0j,

φij(n) =Φ(Tin)

j ∈ I′,

Φ(n)

pij,i ∈ I,j ∈ I′,

where we again make the convention that Φ(Tin) = 0 whenever ni= 0. Then it is readily

checked that the partial balance equations (11) are satisfied by

π(n) = aΦ(n)

?

i∈I

ρni

i,(16)

for any a > 0 and positive solution ρ = (ρi, i ∈ I) of the equations

ν =

?

?

j∈I

ρjpj0,

ρi=

j∈I

ρjpji+ νp0i,i ∈ I.

Thus in particular the stationary distribution π given by (16) for the number of individuals

of each type in the system is insensitive to the µi(recall our assumption m(µi) = 1 for all

i). For the case where P is irreducible and ν > 0, the above equations for ρ have a unique

solution. Again when P is irreducible and when ν = 0 (corresponding to a closed network)

π remains uniquely determined, up to a multiplicative constant, by (16). The case where

Φ(n) =?

processor-sharing networks, by Bonald and Prouti` ere (2002, 2003).

i∈Iλni

ifor positive constants (λi, i ∈ I) characterises processor-sharing Jackson

networks. Further discussion of Whittle networks is given by Serfozo (1999) and, for

Example 2. Networks with no internal transitions. Suppose that φij(n) = 0 for all i,j ∈ I

and for all n ∈ ZN

+, so that no transitions are possible between the classes in I. The

partial balance equations (11) then reduce to the detailed balance equations

π(n)φi0(n) = π(Tin)φ0i(Tin),

n ∈ ZN

+,ni≥ 1,i ∈ I. (17)

(In the case of a single class, these equations further reduce to the equations (1).) An ex-

ample is given by a traditional (uncontrolled) loss network—see, for example, Kelly (1986).

This is naturally processor-sharing. Here workloads are identified with call durations and,

for some set A ⊂ ZN

+such that n ∈ A implies Tin ∈ A for all n and i such that ni≥ 1

(A is typically defined by capacity constraints), we have

φ0i(n) = νiI(Tin ∈ A)

φi0(n) = σini,

for some vectors (νi, i ∈ I) and (σi, i ∈ I) of strictly positive parameters. The equa-

tions (17) are then satisfied by

π(n) = a

?

i∈I

κni

i

ni!, (18)

10

Page 11

where κi= νi/σifor each i, and where a is naturally chosen to be a normalising constant.

As was originally shown by Burman et al (1984), we therefore again have insensitivity of

the occupancy distribution π of the network. The stationary distribution of the residual

call durations is as identified by Theorem 2.

Other examples of processor-sharing networks with no internal transitions are given by

those used to model connections in communications networks with simultaneous resource

requirements and variable bandwidth requirements—see, for example, Bonald and Mas-

souli´ e (2001) and de Veciana et al (2001). Here it is far from automatic that the detailed

balance equations (17) are satisfied.

Acknowledgements

The author is most grateful to Serguei Foss, Takis Konstantopoulos and Ilze Ziedins for

some very helpful discussions, and to the referee for a careful reading of the manuscript

and some most helpful suggestions.

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References

[1] Bonald, T. and Massouli´ e, L. (2001). Impact of fairness on Internet performance. In

Proceedings of ACM SIGMETRICS 2001. Cambridge, Massachusetts.

[2] Bonald, T. and Prouti` ere, A. (2002). Insensitivity in processor-sharing networks. Per-

formance Evaluation, 49, 193–209.

[3] Bonald, T. and Prouti` ere, A. (2003). Insensitive bandwidth sharing in data networks.

Queueing Systems, 44, 69–100.

[4] Burman, D. Y. (1981). Insensitivity in queueing systems. Adv. Appl. Prob., 13, 846–

859.

[5] Burman, D. Y., Lehoczky, J. P. and Lim, Y. (1984). Insensitivity of blocking proba-

bilities in a circuit switching network. J. Appl. Prob. 21, 850–859.

[6] Davis, M. H. A. (1984). Piecewise deterministic Markov processes: a general class of

non-diffusion stochastic models. J Royal Stat Soc, Ser B 46, 353–388.

[7] Davis, M. H. A. (1993). Markov Models and Optimization. Chapman and Hall, Lon-

don.

[8] de Veciana, G., Lee, T.-J. and Konstantopoulos, T. (2001). Stability and performance

analysis of networks supporting elastic services. IEEE/ACM Trans. on Networking,

9, 2–14.

[9] Kelly, F. P. (1979). Reversibility and Stochastic Networks. Wiley, Chichester.

[10] Kelly, F. P. (1986). Blocking probabilities in large circuit-switched networks. Adv.

Appl. Prob. 18, 473–505.

[11] Pechinkin, A. V. (1983). An invariant queueing system. Math. Operationsforsch. und

Statist., ser. optimization, 14, No. 3, 433–444.

[12] Pechinkin, A. V. (1987). A new proof of Erlang’s formula for a lossy multichannel

queueing system. Soviet J. Comput. System Sci. 25, 165–168. Translated from: Izv.

Akad. Nauk. SSSR. Techn. Kibernet, (1986), No. 6, 172–175 (in Russian).

[13] Schassberger, R. (1978). Insensitivity of steady-state distributions of generalized semi-

Markov processes with speeds. Adv. Appl. Prob. 10, 836–851.

[14] Schassberger, R. (1986). Two remarks on insensitive stochastic models. Adv. Appl.

Prob. 18, 791–814.

[15] Serfozo, R. F. (1999). Introduction to Stochastic Networks. Springer, Berlin.

[16] Sevastyanov, B. A. (1957). An ergodic theory for Markov processes and its application

to a telephone system with refusals. Theory Probability. Appl., 2, 104–112.

[17] Whittle, P. (1985). Partial balance and insensitivity. J. Appl. Prob., 22, 168–176.

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