A DIFFUSION MODEL FOR TWO PARALLEL QUEUES WITH PROCESSOR SHARING: TRANSIENT BEHAVIOR AND ASYMPTOTICS

We consider two identical, parallel M/M/1 queues. Both queues are fed by a Poisson arrival stream of rate I and have service rates equal to #. When both queues are non-empty, the two systems behave independently of each other. However, when one of the queues becomes empty, the corresponding server helps in the other queue. This is called head-@the-line processor sharing. We study this model in the heavy traffic limit, where p 1/#1. We formulate the heavy traffic diffusion approximation and explicitly compute the time-dependent probability of the diffusion approximation to the joint queue length process. We then evaluate the solution asymptotically for large values of space and/or time. This leads to simple expressions that show how the process achieves its stead state and other transient aspects.


Introduction
Queueing systems are used in a wide variety of applications, such as computer and communications networks and manufacturing systems.In analyzing such models, one typically wishes to compute the probability distribution of some stochastic process.Obtaining the full time-dependent distribution is a difficult task for all but simple models.
Here we consider the following model, which is sometimes referred to as head-@ the-line processor sharing of parallel queues.There are two parallel M/M/1 queues, each fed by independent Poisson arrival streams with rate I.Each of the two servers works at rate #.When both queues are non-empty, each server tends to its own queue.However, if the first (resp., second) queue becomes empty, the first (resp., here we give very precise results and also indicate how to obtain full asymptotic series.We obtain the results by the saddle point and related methods for evaluating integrals [3].
These authors consider single queues with the processor sharing service discipline.
The main focus in these papers is the calculation of the sojourn time distribution of a tagged customer, as well as its moments.The queue length distribution in the M/M/1-PS model is the same for the PS and FIFO service disciplines.It is the so- journ time distribution that is very different for PS and FIFO service.If we condi- tion the sojourn time on the total service that the tagged customer must receive, then the PS discipline is much more efficient at servicing shorter jobs.In [4], the authors analyzed the M/M/1-PS model and computed the Laplace transform of the sojourn time distribution, conditioned on the job size.In [16], the GI/M/1-PS model is analyzed.In particular, simple expressions are given for the first two (unconditional) sojourn time moments.The M/G/1-PS model was analyzed by Yashkov [20-22] and Ott [15].The response time distribution is computed in [20], the transient distribu- tion of the number of customers present is analyzed in [21] and a good survey of work on processor-shared queues appears in [22].In [15], the author extended the results in [20] to calculate the joint distribution of the sojourn time and of the number of cus- tomers present upon the departure of the tagged customer.Some approximations for the more difficult GI/G/1-PS model are given in [17].
We believe that the structure of P(x, y, T) revealed here (both exact and asympto- tic) will also arise in other diffusion (and also discrete) models corresponding to two or more coupled queues.Other explicit solutions to diffusion models arising in queue- ing networks are given in Newell [14], Foschini [8], Knessl and Tan [19] and Avram  [1].In [1], the author classifies the steady-state densities of a large class of two-dimen- sional diffusion models according to their tail behaviors as x and/or y---,oe.We believe that such a classification should also be possible for the transient behavior, and this work may be viewed as a first step in that direction.We also mention that the diffusion approximation analysis presented here should be extendible to problems with general arrivals and/or service.However, these generalizations are likely to lead to somewhat more complicated PDEs and boundary conditions (BCs) then those in (2.10)-(2.12).
In Section 2, we formulate the model.In Section 3, we summarize and briefly discuss our main results.The detailed calculations are presented in Sections 4 and 5.
Note that this means that the initial queue lengths are assumed to be large, of the order O(e-1).
If the queue is stable (i.e., p < 1), we will set a + 1, and then (2.7) defines in terms ofp.In the unstable case (p > l), we will set a -1.Ifp=l, wetakea=0 and then (2.8) corresponds to viewing p(m,n,t) on large space/time scales, with m, n, t--c and m/X/, n/V/ fixed.
We also have the normalization condition (2.13) i i P(x,y,T)dx dy-lfor all O. 0 0 (2.14) We do not consider the corner condition (2.5) in formulating the heavy traffic diffusion model.We will show that P(x,y,T) becomes infinite near the origin x y 0, and hence (2.9) cannot be the correct asymptotic approximation to the discrete probabilities p(m,n,t) for small values of x and y (more precisely for (x,y)= O(c), which is the same as (m, n) O(1)).A proper analysis of the corner region would in- volve analyzing the discrete model (2.2)-(2.5),with # + #ca.However, we will show that such a detailed treatment is not necessary to determine P(x, y, T), which is the heavy traffic diffusion approximation valid away from the corner.The total pro- bability mass in the corner region is asymptotically smaller than that on the (x,y) scale.However, calculating the higher order terms in the series (2.9) (e.g., the func- tion p(1)) would necessitate a careful treatment of the corner region.
We shall obtain an explicit solution for the leading order diffusion approximation P(x,y,T).Then we shall obtain detailed asymptotic results for this limiting density, that apply for x and/or y and/or T large.The final results are summarized in Section 3 and the details of the calculations are given in Sections 4 and 5.

Summary of Main Results
In Section 4 we solve (2.10)-(2.14)and obtain the following integral representations for P(x, y, T).
We note that Pll is the Laplace transform of PII" When a > 0 (i.e., p < 1), PI! has simple pole at 0 0, which determines the limiting behavior of Pll as T+oo" PII(X, y, c) lomo[OPii(x y; O)] e (x+u)2+(y+u 0 (3.7) As Tcz, we have P I--O and thus (3.7) gives the steady-state density of the diffusion approximation, and this agrees with the result previously obtained by Knessl [9] and Morrison [12].When a <_ 0 (i.e., p >_ 1), the poles is absent and we now have P I, P II ---c as T---oc.
Now observe that the total number of customers Xl(t + N2(t in the two queue network belmves as the standard M/M/1 model with an arrival rate -2, and service rate 2#.In appendix A, we show after a lengthy calculation that P(, z-; O)d -"(z-0)/ z-0l 0 where z 0 x 0 + Y0 and P is the Laplace transform of P(x, y, T) over time.
We denote by (X(T), Y(T)) the diffusion process that approximates our discrete queue.Then the density of X(T)+ Y(T) should be precisely that of the heavy traffic diffusion approximation to the standard M/M/1 model.Denoting this density by (z,T), it satisfies Ozz + az; z,T > 0 z + aP-0; z-0, T>0 (3.9) lr o e(z-Zo).
1 in the time-derivative of (3.9) arises due to the fact that the discrete The factor of g model has a total arrival rate -2,, rather than ,.We let (z;0)- f -rV(z,T)dT.Then, solving (3.9) for the Laplace transform , we obtain 0 z.
precisely the right side of (3.8).This shows that (z;0)-f P(w,z-w;O)dw, and this must be the case.0 Since the solution in Theorem 1 is quite complicated, we evaluate it asymptotically for large values of space and/or time.This yields simpler formulas that show more clearly the basic qualitative features of the joint density.First we take initial conditions x 0 Y0 0. In view of (2.8), this does not mean that the discrete process (Nl(t),N2(t)) starts at the origin (0,0), but rather that m 0 and n o are of order o(-1).When x 0-y0=0, we have PI-O and hence P-PII.The following re- sults are established in Section 5.
To discuss the asymptotic results, we refer to the case where x, y, T are all large and of comparable magnitudes as "interior" asymptotics, as this corresponds to the interior of the (x, y, T) space.When two of (x, y, T) are large and the third is not, we call the resulting asymptotic expansions "face" asymptotics; when only one of (x,y,T) is large, we call it "edge" asymptotics.Thus, (a)-(c) refers to interior asymptotics, (d)-(g) to face asymptotics, and (h) and (i) refer to edge asymptotics along the T-axis or edge.Along the face where x,y+ with T fixed, we can easily show that (a) remains valid.Along edges where x+ with y,T fixed (resp.y with x, T fixed) we can show that (d)(resp. (g)) remains valid.Hence, it is necessary to give different expressions along only two of the three faces and one of the three edges.
Parts (a)-(c) show that for each fixed, large x and y, there is a critical time T-T. at which the process has reached its steady-state.For times T < T., transient effects are important and the leading term in the expansion of P depends upon time, while for T > T., the leading term depends only upon x and y.We refer to the cylindrical surface T T.(x,y) as a "front"; as time increases, this surface moves outward and eventually covers the entire space.Such fronts were previously found in other, one-dimensional queueing models (see [18,23]).
Part (h) gives the standard "relaxation rate" approximation.Such asymptotics are discussed for single server queues in Cohen [5] and for two tandem M/M/1 queues in Blanc [2].Our analysis shows that for large times T, the behavior of P is different according as x, y O(1) (cf.(h)); x, y O() (cf.(i)); or x, y O(T) (cf. (a)-(c)).We also note that along the face y 0, the transition in the behavior of P occurs at T x/, which is where the cylindrical front intersects the plane y 0.
Theorem 4: For x o yo O and p l, we set a O; the asymptotic expansions are as follows: We observe that if p >_ 1, P0 as Toc for any fixed x,y.The expression in Theorem 4, part_(e) is in fact the exact result for P-PII when a-0; for with x, y O(v/T), this cannot be simplified any further.This is similar to the diffusion approximation for the standard M/M/1 queue for p > 1 and shows that the two queues decouple in this limit.However, it is important to note that Theorem 3 (a) is more general than (3.10) as the former assumes that x,y,T are large, but x/T and y/T are not necessarily close to one.The coupling of the two queues is evident by the form of the function L(x,y, T).The results in part (d) again give the standard relaxation rate asymptotics.
Next we give analogous results for non-zero initial conditions (x0, Y0)" Theorem 2": For fixed conditions x o and Yo and p < 1, we set a + 1.
Theorem 3": For fixed initial conditions x 0 and Yo and p > 1, we set a--1.Asymptotic expansions of P(x,y,T) are as follows:

4T
We note that when Theorem 3* (a)is specialized to the scaling x-T-O(V/) and y-T-O(V@), we again obtain the decoupled Gaussian form (3.10), a.nd the dependence on the initial conditions disappears in this limit.It is also possible to show that the double integral f f [expression in Theorem 4* (e)] dxdy-1, which 0 0 shows that when p 1 (a 0), the density is concentrated in the range x,y O(V/), as we might expect.For p > 1, Theorem 3* (a) shows that the density is concentrated where x-T, y-T-O(v@).For p < 1, it is concentrated for x,y- O(1), and we can easily show that the double integral of the right-hand side of Theorem 2* (h) vanishes.
All our asymptotic results apply for some or all of x,y,T large but with fixed initial conditions xo, y o.Other interesting insights may be seen by evaluating P(x,y, T;xo, Yo) for large initial conditions (i.e.x 0 and Y0 large), but we do not con- sider these here.Experience with other models (see [18,23]) shows that asymptotics for large initial conditions better reveal how the two-dimensional process (X(T), Y(T))interacts with the barriers (reflecting boundaries) at x 0 and y 0.

Derivation of the Integral Representations
We obtain the representations for P(x,y,T)in Theorem 1 by solving (2.10)-(2.14).
We note that PI-+O as T-oc and thus (4.1) cannot represent the full density of the diffusion model, especially when p < 1.The solution (4.4) to (4.3) is a particular solution to the Green's function problem.However, we now show that there also exists a non-zero solution to the corresponding homogeneous problem (obtained by dropping the delta function in (4.3)) that plays a role in ultimately determining P(x,y,T).
This completes the analysis.

Asymptotic Expansions
We establish Theorems 2-4 (and also 2"-4") by evaluating Theorem 1 in various asymptotic limits.We do not give all the technical details, but simply sketch the main ideas.

4T2
Thus the saddle lies on the positive real axis and we can easily show that the directions of steepest descent at (0 are arg(-0) /2.From (5.3), we also note that the integrand has a simple pole at (= 1/2 and a branch point at 0. For (a) x,y,T P(x,y,T) L(x,y,T)exp[ap(x,y,T)] If we specialize the result in Theorem 3 (a) to x-T-O(x/) and y-T- 1); y,Tc use (d)-(f) and the symmetry P(x, y, T; Xo, Yo) P(Y, x, T; Yo, Xo) use (b) and the symmetry P(x, y, T; Xo, Yo) P(Y, x, T; Yo, Xo) (d) x,y O(1); Tcx P(x, y, T) e (x + y)-ff)+ y sinh \-ff-)cosh \-- and the symmetry P(x, y, T; Xo, Yo) P(Y, x, T; Yo, Xo) + u)2)l dt du.