On Spectral Properties of Finite PopulationProcessor Shared Queues

On Spectral Properties of Finite Population Processor Shared Queues

Abstract

We consider sojourn or response times in processor-shared queues that have a finite population of potential users. Computing the response time of a tagged customer involves solving a finite system of linear ODEs. Writing the system in matrix form, we study the eigenvectors and eigenvalues in the limit as the size of the matrix becomes large. This corresponds to finite population models where the total population is . Using asymptotic methods we reduce the eigenvalue problem to that of a standard differential equation, such as the Hermite equation. The dominant eigenvalue leads to the tail of a customer’s sojourn time distribution.

Keywords: Finite population, processor sharing, eigenvalue, eigenvector, asymptotics.

1 Introduction

The study of processor shared queues has received much attention over the past 45 or so years. The processor sharing (PS) discipline has the advantage over, say, first-in first-out (FIFO), in that shorter jobs tend to get through the system more rapidly. PS models were introduced during the 1960’s by Kleinrock (see [1], [2]). In recent years there has been renewed attention paid to such models, due to their applicability to the flow-level performance of bandwidth-sharing protocols in packet-switched communication networks (see [3]-[5]).

Perhaps the simplest example of such a model is the -PS queue. Here customers arrive according to a Poisson process with rate parameter , the server works at rate , there is no queue, and if there are customers in the system each gets an equal fraction of the server. PS and FIFO models differ significantly if we consider the “sojourn time”. This is defined as the time it takes for a given customer, called a “tagged customer”, to get through the system (after having obtained the required amount of service). The sojourn time is a random variable that we denote by . For the simplest model, the distribution of depends on the total service time that the customer requests and also on the number of other customers present when the tagged customer enters the system.

One natural variant of the -PS model is the finite population model, which puts an upper bound on the number of customers that can be served by the processor. The model assumes that there are a total of customers, and each customer will enter service in the next time units with probability . At any time there are customers being served and the remaining customers are in the general population. Hence the total arrival rate is and we may view the model as a PS queue with a state-dependent arrival rate that decreases linearly to zero. Once a customer finishes service that customer re-enters the general population. The service times are exponentially distributed with mean and we define the traffic intensity by . This model may describe, for example, a network of terminals in series with a processor-shared CPU. This may be viewed as a closed two node queueing network.

The finite population model does not seem amenable to an exact solution. However, various asymptotic studies have been done in the limit , so that the total population, or the number of terminals, is large. If is large it is reasonable to assume either that , the arrival rate of each individual customer, is small, of the order , or that the service rate is large, of the order . Then will remain as . Previous studies of the finite population model were carried out by Morrison and Mitra (see [6]-[11]), in each case for . For example, the moments of the sojourn time conditioned on the service time are obtained in [8], where it was found that the asymptotics are very different according as (called “normal usage”), (called “heavy usage”), or (called “very heavy usage”). In [9] the unconditional sojourn time distribution is investigated for and the three cases of , in [10] the author obtains asymptotic results for the conditional sojourn time distribution, conditioned on the service time , in the very heavy usage case , and in [11] the results of [10] are generalized to multiple customer classes (here the population is divided into several classes, with each class having different arrival and service times). In [6] the authors analyze the multiple class model and obtain the unconditional sojourn time moments for in the normal usage case, while in [7] heavy usage results are obtained.

In this paper we study the spectral structure of the finite population model as . We denote the sojourn time by and its conditional density we call with

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Here denotes the number of other customers present in the system immediately before the tagged customer arrives, and thus . Then we define the column vector . satisfies a system of ODEs in the form where is an tridiagonal matrix, whose entries depend on and . The eigenvalues of are all negative and we denote them by with the corresponding eigenvectors being . We shall study this eigenvalue problem for and three cases of : , and . In each case we obtain expansions of the and then the , for various ranges of . Often the eigenvectors can be expressed in terms of Hermite polynomials for . Since is a finite matrix the spectrum is purely discrete, but as the size of the matrix becomes large we sometimes see the eigenvalues coalescing about a certain value. Ordering the eigenvalues as , the tail behavior of and for is determined by the smallest eigenvalue , where is the unconditional sojourn time density with

 p(t)=N−1∑n=0pn(t)Pr[N(0−)=n]. (1.1)

It is interesting to note that while previous studies (see [7]-[9]) of the finite population model lead to the scaling , the spectrum involves the transition scale .

Our basic approach is to use singular perturbation methods to analyze the system of ODEs when becomes large. The problem can then be reduced to solving simpler, single differential equations whose solutions are known, such as Hermite equations. Our analysis does make some assumptions about the forms of various asymptotic series, and about the asymptotic matching of expansions on different scales. We also comment that we assume that the eigenvalue index is ; thus we are not computing the large eigenvalues here.

This paper is organized as follows. In section 2 we state the mathematical problem and obtain the basic equations. In section 3 we summarize our final asymptotic results for the eigenvalues and the (unnormalized) eigenvectors. The derivations are relegated to section 4. Some numerical studies appear in section 5; these assess the accuracy of the asymptotics.

2 Statement of the problem

Throughout the paper we assume that time has been scaled so to make the mean service time .

The conditional sojourn time density satisfies the following linear system of ordinary differential equations, or, equivalently, differential-difference equation:

 p′n(t)=ρ(1−n+1N)pn+1(t)+nn+1pn−1(t)−[ρ(1−n+1N)+1]pn(t),0≤n≤N−1. (2.1)

The above holds also at if we require to be finite, and when (2.1) becomes

 p′N−1(t)=N−1NpN−2(t)−pN−1(t). (2.2)

The initial condition at is

 pn(0)=1n+1, (2.3)

and we note that (2.3) follows from integrating (2.1) from to .

Here we focus on obtaining the eigenvalues (and corresponding eigenvectors) for the matrix , which corresponds to the difference operator in the right-hand side of (2.1), specifically

 A=⎡⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣−1−ρ(N−1N)ρ(N−1N)000⋯001/2−1−ρ(N−2N)ρ(N−2N)00⋯0002/3−1−ρ(N−3N)ρ(N−3N)0⋯00⋮⋮⋮⋮⋮⋱⋮⋮00000⋯N−1N−1⎤⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦

which has size . It follows that the solution of (2.1)-(2.3) has the spectral representation

 pn(t)=N−1∑j=0e−νj(N,ρ)tcjϕj(n;N,ρ). (2.4)

Here are the eigenvalues of , indexed by and ordered as , is the eigenvector corresponding to eigenvalue , and the spectral coefficients in (2.4) can be calculated from (2.3), hence

 1n+1=N−1∑j=0cjϕj(n),0≤n≤N−1.

From (2.1) we can easily establish orthogonality relations between the , and these lead to an explicit expression for the :

 cj=∑N−1n=0(ρN)nN!(N−n−1)!ϕj(n)∑N−1n=0(ρN)n(n+1)N!(N−n−1)!ϕ2j(n).

We note that the tail of the sojourn time, in view of (2.4), is given by

 pn(t)∼c0ϕ0(n)e−ν0t,t→∞. (2.5)

This asymptotic relation holds for and large times. Our analysis will assume that but for sufficiently large (2.5) must still hold. We shall study the behavior of for and for various ranges of the parameter .

We shall show that the behavior of the eigenvalues is very different for the cases , , and (more precisely ). Furthermore, within each range of the form of the expansions of the eigenfunctions is different in several ranges of . Our analysis will cover each of these ranges, but we restrict ourselves to the eigenvalue index being . Note that the matrix has eigenvalues so that could be scaled as large as . We are thus calculating (asymptotically for ) only the first few eigenvalues and their eigenfunctions. Obtaining, say, the large eigenvalues, would likely need a very different asymptotic analysis. When we shall see that the eigenfunctions and their zeros are concentrated in a narrow range of , which represents the fraction of the customer population that is using the processor. Since are functions of the discrete variable , by “zeros” we refer to sign changes of the eigenvectors. If the eigenvalue index were scaled to be also large with , we would expect that these sign changes would be more frequent, and would occur throughout the entire interval .

We comment that to understand fully the asymptotic structure of for requires a much more complete analysis than simply knowing the eigenvalues/eigenfunctions, as the spectral expansion may not be useful in certain space/time ranges. However, (2.5) will always apply for sufficiently large times, no matter how large is, as long as is finite.

We also comment that in (1.1), by results in [12], we have

 Pr[N(0−)=n]=(ρN)n(N−1)!(N−1−n)!/N−1∑l=0(ρN)l(N−1)!(N−1−l)!, (2.6)

which says that the distribution of coincides with the steady state distribution of in a finite population queue with population . The tail of the unconditional sojourn time is then

 p(t)∼c′0[N−1∑n=0ϕ0(n)(ρN)n(N−1)!(N−1−n)!]e−ν0t,t→∞, (2.7)

where .

3 Summary of results

We give results for the three cases: , and . Within each case of the eigenvectors have different behaviors in different ranges of .

3.1 The case ρ<1

For the eigenvalues are given by

 νj=(1−√ρ)2+c1√N+c2(j)N3/4+c4(j)N+O(N−5/4),j≥0 (3.1)

where

 c1=2√ρ√1−√ρ,c2(j)=(2j+1)√ρ(1−√ρ)3/4 (3.2)

and

 c4(j)=√ρ(22√ρ−3ρ−15)16(1−√ρ)−38√ρ(1−√ρ)j(j+1). (3.3)

We observe that the leading term in (3.1) () is independent of the eigenvalue index and corresponds to the relaxation rate in the standard queue. For the standard -PS model (with an infinite customer population), it is well known (see [13], [14]) that the tail of the sojourn time density is

 p(t)∼ke−(1−√ρ)2texp[−3(π2)2/3ρ1/6t1/3]t−5/6,t→∞ (3.4)

where is a constant. This problem corresponds to solving an infinite system of ODEs, which may be obtained by letting in the matrix (with a fixed ). The spectrum of the resulting infinite matrix is purely continuous, which leads to the sub-exponential and algebraic factors in (3.4).

The result in (3.1) shows that the (necessarily discrete) spectrum of has, for and , all of the eigenvalues approaching , with the deviation from this limit appearing only in the third () and fourth () terms in the expansion in (3.1). Note that in (3.1) is independent of . Comparing (3.4) to (2.7) with (3.1) and , we see that the factors in (3.4) are replaced by . Note that (3.4) corresponds to letting and then in the finite population model, while (2.7) has with a finite large . The expansion in (3.1) breaks down when becomes very large, and we note that when , the three terms , and become comparable in magnitude. The expansion in (3.1) suggests that can be allowed to be slightly large with , but certainly not as large as , which would be needed to calculate all of the eigenvalues of . As we stated before, here we do not attempt to get the eigenvalues of large index .

Now consider the eigenvectors , for and . These have the expansion

 ϕj(n)=k0ρ−n/2[Φ(0)j(y)+N−1/8Φ(1)j(y)+O(N−1/4)] (3.5)

where and are related by

 n=√N√1−√ρ+N3/8y (3.6)

and

 Φ(0)j(y)=e−z2/4Hej(z),z=√2(1−√ρ)3/8y, (3.7)
 Φ(1)j(y)=[23βz2+23β(8j+4)]ddz[e−z2/4Hej(z)]+(α2−23β)ze−z2/4Hej(z), (3.8)

with

 α=√ρ2(1−√ρ)−7/8,β=−(1−√ρ)1/84√2. (3.9)

Here is the Hermite polynomial, so that and . The constant is a normalization constant which depends upon , and , but not . Apart from the factor in (3.5) we see that the eigenvectors are concentrated in the range and this corresponds to . The zeros of the Hermite polynomials correspond to sign changes (with ) of the eigenvectors, and these are thus spaced apart.

We next give expansions of on other spatial scales, such as , and , where (3.5) ceases to be valid. These results will involve normalization constants that we denote by , but each of these will be related to , so that our results are unique up to a single multiplicative constant. Note that the eigenvalues of are all simple, which can be shown by a standard Sturm-Liouville type argument.

For (hence ) we find that

 ϕj(n)∼k1ρ−n/2gj(x)eN1/4f(x),N→∞,0

where

 f(x)=2√x−23√1−√ρx3/2,
 gj(x)=(1−√1−√ρx)jex2/4x1/4[1+(1−√ρ)1/4√x]2j+1exp[(2j+1)(1−√ρ)1/4√x]. (3.11)

The normalization constants and are related by

 k0=k1N−j/8(−1)j2−5j/2−1(1−√ρ)(j+1)/8exp[14(1−√ρ)+2j+1+43(1−√ρ)−1/4N1/4], (3.12)

by asymptotic matching between (3.5) and (3.10).

Next we consider and scale . The leading term becomes

 ϕj(n)∼k2ρ−n/2G(ξ,j)exp[NF(ξ)+N1/2F1(ξ)+N1/4F2(ξ,j)] (3.13)

with

 F(ξ) = 12ξlogρ+ξlog2+12ξ−12√ρ√ρξ2+4ξ(1−√ρ) (3.14) −ξlog[2√ρ−ρξ+√ρ2ξ2+4ρξ(1−√ρ)] +ρ−4√ρ+22ρlog[ρξ+2−2√ρ+√ρ2ξ2+4ρξ(1−√ρ)2(1−√ρ)] +12log[(ρ−2√ρ+2)ξ+2(1−√ρ)−(√ρ−2)√ρξ2+4ξ(1−√ρ)2(1−√ρ)],
 F1(ξ)=c1ρlog[2(1−√ρ)+ρξ+√ρ2ξ2+4ρξ(1−√ρ)2(1−√ρ)], (3.15)
 F2(ξ,j)=c2(j)ρlog[2(1−√ρ)+ρξ+√ρ2ξ2+4ρξ(1−√ρ)2(1−√ρ)] (3.16)

and can be written as the integral

 G(ξ,j)=ξ−3/4exp[∫ξ0(34v−H(v,j))dv] (3.17)

where

 H(ξ,j)=1(1−ξ)eF′−e−F′{c4(j)+ρ√ρ−eF′−1ξe−F′+(1−√ρ2ξ)[F′′+(F′1)2]}

and and are, respectively, the derivatives of (3.14) and (3.15). Note that and are independent of the eigenvalue index , while and do depend on it. The constants and are related by , by asymptotic matching between (3.10) and (3.13).

Finally we consider the scale . The expansion in (3.10) with (3.11) develops a singularity as and ceases to be valid for small . For we obtain

 ϕj(n)∼k3ρ−n/212πi∮1zn+111−zexp(11−z)dz, (3.18)

where the contour integral is a small loop about , and by asymptotic matching of (3.10) as with (3.18) as ,

3.2 The case ρ>1

We next consider with . The eigenvalues are now small, of the order , with

 νj=1N(ρρ−1+ρj)+o(N−1). (3.19)

Note that now we do not see the coalescence of eigenvalues, as was the case when , and the eigenvalue index appears in the leading term in (3.19). The form in (3.19) also suggests that the tail behavior in (2.5) is achieved when . Now the zeros of the eigenvectors will be concentrated in the range where , and introducing the new spatial variable , with

 n=N(1−1ρ)+√NX, (3.20)

we find that

 ϕj(n)∼k0Hej(√ρX) (3.21)

where is again the Hermite polynomial, and is again a normalization constant, possibly different from (3.5). On the -scale with we obtain

 ϕj(n)∼k1ξ1ρ−1(ξ−1+1ρ)j,ξ≠1−1ρ (3.22)

and and are related by

 k0=ρ−j/2(1−1ρ)1ρ−1N−j/2k1, (3.23)

by asymptotic matching between (3.21) and (3.22). Note that and for the first two eigenvectors (), (3.21) is a special case of (3.22). The expression in (3.22) holds for and for , but not for or . For the latter we must use (3.21) and for we shall show that

 ϕj(n)∼k2eiπρ/(ρ−1)2πi∫C(1−z)−ρρ−1(z−1ρ)1ρ−1zndz, (3.24)

where is a closed loop that encircles the branch cut, where and , in the -plane, with the integrand being analytic exterior to this cut. By expanding (3.24) for and matching to (3.22) as we obtain

 k2=k1N11−ρ(−1)j(1−1ρ)j−1ρ−1Γ(ρρ−1). (3.25)

In contrast to when , the eigenvectors now vary smoothly throughout the interval (cf. (3.22)) but their sign changes are all concentrated where and the spacings of these changes are of the order , and approximately the same as the spacings of the zeros of the Hermite polynomials (cf. (3.20) and (3.21)).

3.3 The case ρ≈1

Finally we consider the case and introduce the parameter by

 ρ=1+γN1/3,−∞<γ<∞.

This case will asymptotically match, as , to the results and, as , to the results. Now a two-term asymptotic approximation to the eigenvalues is

 νj=f(γ)N−2/3+g(j,γ)N−1+o(N−1) (3.26)

where

 γ−A+2A2=0, (3.27)
 f(γ)=1A+1A4, (3.28)
 g(j,γ)=1A6−3A3+12+(j+12)√1+4A3. (3.29)

Thus given we must solve (3.27) to get and then compute and . We can explicitly invert (3.27) to obtain

 A(γ)=γ2+bγ+b23bwithb=[γ3+27+3√81+6γ3]1/3. (3.30)

We also note that if we have and then , , and . The expression in (3.26) shows that the eigenvalues are small, of the order , and to leading order coalesce at . The second term, however, depends linearly on the eigenvalue index .

The expansions of the eigenvectors will now be different on the four scales , , and . It is on the third scale that the zeros of become apparent, and if we introduce by

 n=N2/3A(γ)+√NU,−∞

we obtain the following leading order approximation to the eigenvectors

 Missing or unrecognized delimiter for \Big (3.32)

Thus again the Hermite polynomials arise, but now on the scale , which corresponds to . The spacing of the zeros (or sign changes) of for the case is thus which is comparable to the case , and unlike the case where (cf. (3.6)) the spacing was .

On the spatial scale , (3.32) ceases to be valid and then we obtain

 ϕj(n)∼k1G(V,j)eN1/3F(V),V=nN2/3≠A(γ) (3.33)

where this applies for all except , where (3.32) holds, and is given by

 F(V)=⎧⎨⎩∫V0F′+(w)dw,0≤V≤A(γ)∫A0F′+(w)dw+∫VAF′−(w)dw,V>A(γ), (3.34)
 F′±(V)=12[V−γ±√(V−γ)2−4(f(γ)−1V)], (3.35)

where is as in (3.28). Note that is independent of the eigenvalue index , and if then , which follows from (3.27) and (3.28). Thus the discriminant in (3.35) vanishes when , and in fact it has a double zero at this point. Thus the sign switch in as crosses is needed to smoothly continue this function from to . The function is given by

 G(V,j)=V−1/4[V−A(γ)]jexp{∫V0[14w−jw−A−H(1)+(w,j)]dw},0≤V

and

 G(V,j) = A(γ)−1/4[V−A(γ)]jexp{∫A0[14w−jw−A−H(1)+(w,j)]dw} (3.37) ×exp{−∫VA[jw−A+H(1)−(w,j)]dw},V>A.

We can show that as and as , so that the integrals in (3.36) and (3.37) are convergent at and .

By expanding (3.33) for we can establish the asymptotic matching of the results on the -scale (cf. (3.33)) and the -scale (cf. (3.32)) and then relate the constants and , leading to

 k0=k1A−1/4(1+4A−3)−j/4N−j/6exp{N1/3∫A0F′+(w)dw+∫A0[14w−jw−A−H(1)+(w,j)]dw}.

Next we consider the scale with , and now the eigenvectors have the expansion

 ϕj(n)∼k2G(ξ,j)eN1/3F(ξ) (3.38)

where

 F(ξ)=f(γ)logξ=(1A+1A4)logξ (3.39)

and

 G(ξ,j)=ξg(j,γ)−γf(γ)exp[1−γf(γ)ξ] (3.40)

where and are as in (3.28) and (3.29). The constants and are related by

 k2 = k1N[g(j,γ)−γf(γ)]/3A−g(j,γ)+γf(γ)+j−1/4exp{∫A0[14w−jw−A−H(1)+(w,j)]dw} ×exp{N1/3[f(γ)3logN−f(γ)logA−1−γf(γ)A+∫A0F′+(w)dw]}.

For , (3.18) holds with , but now .

To summarize, we have shown that the eigenvalues have very different behaviors for (cf. (3.1)), (cf. (3.19)) and (cf. (3.26)). In the first case, as , the eigenvalues all coalesce about which is the relaxation rate of the standard model. However, higher order terms in the expansion show the splitting of the eigenvalues (cf. and in (3.1)), which occurs at the term in the expansion of the . For the eigenvalues are small, of order , but even the leading term depends upon (cf. (3.19)). When there is again a coalescence of the eigenvalues, now about where is given by (3.30). The splitting now occurs at the first correction term, which is of order . We also note that the leading order dependence of the on occurs always in a simple linear fashion. Our analysis will also indicate how to compute higher order terms in the expansions of the and , for all three cases of and all ranges of . Ultimately, obtaining the leading terms for the reduces in all 3 cases of to the classic eigenvalue problem for the quantum harmonic oscillator, which can be solved in terms of Hermite polynomials.

4 Brief derivations

We proceed to compute the eigenvalues and eigenvectors of the matrix above (2.4), treating respectively the cases , and , in subsections 4.1-4.3. We always begin by considering the scaling of , with , where the oscillations or sign changes of the eigenvectors occur, and this scale also determines the asymptotic eigenvalues. Then, other spatial ranges of will be treated, which correspond to the “tails” of the eigenvectors.

4.1 The case ρ<1

We recall that so that means that the service rate exceeds the maximum total arrival rate . When and the distribution of in (2.6) behaves as for , which is the same as the result for the infinite population -PS queue.

We introduce

 y=(n−√N√1−√ρ)N−3/8=nN−3/8−N1/8√1−√ρ

as in (3.6), and set and . Then setting