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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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