On the Nonsymmetric Longer Queue Model : Joint Distribution , Asymptotic Properties , and Heavy Traffic Limits

We consider two parallel queues, each with independent Poisson arrival rates, that are tended by a single server. The exponential server devotes all of its capacity to the longer of the queues. If both queues are of equal length, the server devotes ] of its capacity to the first queue and the remaining 1 − ] to the second. We obtain exact integral representations for the joint probability distribution of the number of customers in this two-node network.Then we evaluate this distribution in various asymptotic limits, such as large numbers of customers in either/both of the queues, light traffic where arrivals are infrequent, and heavy traffic where the system is nearly unstable.


Introduction
We consider a nonsymmetric version of the longer queue model.Here there are two parallel queues, each fed by a Poisson arrival stream.There is but a single server who tends to the longer of the two queues.If the number of customers in each queue is the same, then the server devotes ] of its capacity to the first queue and 1 − ] to the second queue, with 0 < ] < 1.We let  1 ( 2 ) denote the number of customers in the first (second) queue, the two arrival rates are  1 and  2 , and the server works at rate .Note that the total number of customers,  1 +  2 , in the two-node network behaves as the standard //1 model, so in the steady state we have the geometric distribution Prob[ 1 +  2 = ] = (1 −  1 −  2 )( 1 +  2 )  , where  1 =  1 /,  2 =  2 /, assuming the stability condition  1 +  2 < 1.The "symmetric case" corresponds to  1 =  2 (thus  1 =  2 ) and ] = 1/2, and this was analyzed in detail by Flatto [1].
Such models were proposed by Zheng and Zipkin [2] to study problems in inventory control.In [2] finite capacities were assumed in the two queues, and the authors studied numerically the steady state probabilities (, ) = Prob[ 1 = ,  2 = ], in terms of the capacity size and also for different service disciplines, such as the longer queue (LQ) discipline here, and also the first-come-firstserved discipline.
In [1] the author used two-dimensional generating functions and analyticity arguments and obtained explicit expressions for (, ), in the symmetric case, as contour integrals.Then asymptotic results were derived for the joint distribution (, ), as  and/or  becomes large, and also the marginal tails and various conditional limit laws were obtained.In this paper we generalize some of the results of [1] to the nonsymmetric model, and we will show that now many of the asymptotic results become quite different.As in [1] we assume that the model is preemptive, so if  1 =  2 (and thus the server works on both queues) and a new arrival occurs to the first queue, then the server switches immediately all its capacity to the first queue.The more difficult nonpreemptive version of the LQ model was studied by Cohen [3], in the case of Poisson arrivals and general service times.The problem is reduced in [3], using generating functions, to a functional equation which is furthermore converted to a Riemann-Hilbert boundary value problem.
For the present model the analyticity arguments are fairly simple, and we focus mostly on the asymptotic properties of the solution.We will show that these asymptotics are quite different from those of the symmetric model in [1].After obtaining exact integral representations for (, ), and also the marginal probabilities () = Prob[ 1 = ] and P() = Prob[ 2 = ], we asymptotically evaluate these integrals for  and/or  large.We use standard techniques, such as the Laplace method, saddle point method, singular analysis, and the Euler-MacLaurin formula.Good general references on the asymptotic evaluation of integrals and sums are the books [17][18][19][20][21].
In addition to this model being interesting on its own, many variants of shortest queue problems, such as ones with multiple servers and finite capacities, can be asymptotically reduced to LQ models of the type considered here (see [22,23]).For example, in [23] we showed that the finite capacity version of the standard symmetric SQ model (analyzed in [4,5]), where  1 ,  2 ⩽  and  is the capacity, asymptotically reduces to the symmetric LQ model in [1], if we consider the process ( −  1 ,  −  2 ), which measures the number of spots available in the two waiting rooms.Then having a thorough understanding of the nonsymmetric LQ model and its asymptotics will allow us to analyze, at least in some asymptotic limits, nonsymmetric variants of SQ models.
The remainder of this paper is organized as follows.In Section 2 we summarize all of the main results, both exact and asymptotic.They are listed in Theorems 1-5, and some discussion/interpretation appears following each theorem.In Section 3 we briefly derive the exact expressions for (, ); in Section 4 we derive asymptotic properties of (, ) for ,  large.In Section 5 we derive light traffic (where  1 ,  2 → 0) and heavy traffic (where  1 +  2 ↑ 1) results.

Problem Statement and Summary of Results
We let ( 1 ,  2 ) be the numbers of customers in the two parallel queues, and let (, ) = Prob[ 1 = ,  2 = ] be the joint queue length distribution in the steady state.The two arrival rates are  1 and  2 , the exponential server works at rate , and  1 =  1 /,  2 =  2 /.If  1 >  2 ( 1 <  2 ) the server works on the first (second) queue, but if  1 =  2 the server works at rate ] on the first queue and rate (1 − ]) on the second, with 0 < ] < 1.The symmetric case corresponds to  1 =  2 and ] = 1/2.We henceforth assume the stability condition  1 +  2 < 1.
Note that the elementary difference equations ( 6) and ( 7) may be solved immediately to obtain (, 0) and (0, ), up to multiplicative constants.In Section 3 we analyze ( 1)-( 9) to obtain (, ) in the forms of contour integrals, which we summarize below.

Theorem 1.
For  1 +  2 < 1, the steady state distribution is as follows: (i)  > : where the integral is over a small loop about  = 0, den () (ii)  < : (iii)  = : We next evaluate (, ) in various asymptotic limits, to gain more insight into the structure of the joint distribution.Writing (, ) = (, ;  1 ,  2 , ]) to emphasize the dependence on the model parameters, we clearly have the symmetry relation Thus it is sufficient for the asymptotics to assume that Advances in Operations Research and we also note that the expressions in Theorem 1 are consistent with (22).We will show that the asymptotics are quite different whether In Figure 2 we sketch the curve ( 1 +  2 ) 2 =  1 −  2 in the ( 1 ,  2 ) parameter plane, for  1 +  2 < 1.Note that the curve may also be written as and passes through the points (0, 0) and (1, 0).We first give results for (, ) when ( 1 +  2 ) 2 >  1 −  2 , and note that this includes the symmetric case of Flatto [1].
Note that, in view of ( 12) and ( 16), Advances in Operations Research In Theorem 2 we listed the expansions of (, ) in order of decreasing , from  = ∞ in (25) to  = 0 in (46).In the symmetric case  1 =  2 and ] = 1/2, and then (, ) = (, ) and Theorem 2 is consistent with the results of Flatto in [1] (there the cases  = (1),  → ∞ and  = (1),  → ∞ were not considered, and the limits where  ≈   ,  *  were not treated in as much detail).Note that (34) and (38) correspond to "product form" approximations to (, ).The expressions in (25) and ( 46) are actually exact when  = 0 and  = 0, respectively, in view of our comments below (9).Our analysis in Section 4 will also indicate how to compute higher order correction terms in the various asymptotic series.
Next we study the transition range ( 1 +  2 ) 2 ≈  1 −  2 in parameter space.This will lead to a new set of asymptotic results which will show, for example, how the formula in (52) for (, ) changes to the purely geometric approximation in (35).To quantify the closeness to the curve ( 1 + 2 ) 2 =  1 − 2 in Figure 2 we write where  → 0 + and  can have either sign.If  = 0 we are exactly on the transition curve.For small , in certain ranges of  = /, the results in Theorems 2 and 3 still apply.For example, if  = (1) and  → ∞ or  = (1) and  → ∞ Theorems 2 and 3 agree, and then no transition range result is needed.As 26), and then (27) will hold for all  ∈ (1, ∞) in the transition case.Thus for  = / > 1 the transition case will require a new asymptotic result only if  ≈ 1.For  < 1 the asymptotic result in (42) will apply for  <  *  , where now, since 37) and ( 54)).Then if  < 1, we will need different asymptotic results only when  ∈ [ 0  , 1], including  ≈  0  and  ≈ 1, where Note that  0  is the limit of both   and  *  , if we replace  1 − 2 by ( 1 +  2 ) 2 =  2 , as (37) leads to (setting since  → 0 + corresponds to  → .Similarly, (54) leads to   →  0  in this limit.Since we will now have the state variables  and  large, and also  small, it is necessary to relate these.In Theorem 4 we summarize the transition case results, scaling  and  in terms of .
and L is defined by the contour integral where Br + is to the right of all singularities of the integrand, including the pole at  =  2 if  > 0.
Next we consider some different asymptotic limits, those of "light" and "heavy" traffic.Light traffic corresponds to infrequent arrivals, where  1 ,  2 → 0. Heavy traffic corresponds to nearly unstable systems, where  1 +  2 ↑ 1.It turns out that the present model has two possible heavy traffic limits.In the first, which we call HTL1, we have  1 +  2 ↑ 1 with a fixed 0 <  2 < 1.Then most of the probability mass will occur in the range where  and  are large, but with  −  = (1).More precisely, if  = 1 −  1 −  2 → 0 + then  and  must be scaled to be ( −1 ), but with the difference  −  fixed.In the second heavy traffic limit (HTL2) we again set  = 1− 1 − 2 → 0 + but now let  2 → 0 + , with  2 = ().Now the probability mass will become more spread out, with appreciable mass anywhere in the range  >  > 0, where  = / and  = /.The light and heavy traffic results are summarized below as Theorem 5.

Theorem 5. (i) Light traffic: For
Here Br + is a vertical contour in the -plane, which lies to the right of all singularities.
The expression in (78) applies for  > ,  < , and  = , and in the light traffic limit the discontinuity of (, ) along the diagonal will appear only in the higher order terms.In HTL1, (79)-(81) show a piecewise geometric distribution in the ℓ variable, and an exponential density in .For HTL2, writing (82) as (, ) ∼  2 F(, ) we can easily show that ∫ ∞ 0 ∫ ∞  F(, )  = 1 so that to leading order the probability mass concentrates where ,  = ( −1 ) with  > .From (84) we have (, ) = ( 2 ) but the total mass along the main diagonal is (), which is smaller than the mass in (82).Then also ( − 1, ) = ( 2 ) with total mass ∑  ( − 1, ) = (), which is comparable to that along the main diagonal.The diagonals with  −  ⩾ 2 have mass ( − ), which is smaller still.The integrands in (82), (84), and (86) have branch points at  = −(2 + 1) 2 /(4), are analytic at  = 0, and may have poles at This completes our summarization of the exact and asymptotic results.Despite the seeming complexity and the many separate cases, all the results follow from fairly standard asymptotic evaluations of the integrals in Theorems 1, as we will show in Sections 4 and 5.

Asymptotics of the Joint Distribution
We derive Theorems 2-4 by expanding asymptotically the integrals in Theorem 1.We will use a combination of the saddle point method and singularity analysis.Good general references on techniques for asymptotically evaluating integrals can be found in [17][18][19][20][21].
We need to understand the singularities of the integrands in (10), (15), and (19).There are clearly branch points where  =   and  =  *  , with Since  1 ⩾  2 we have   ⩽  *  and for  1 >  2 the branch point at  *  is farther from the origin than the one at   .In fact,  *  will never play a role in the asymptotics.The integrands are also singular at  = 0, where (10) has a pole of order  + 1, and (15) has a pole of order  + 1.The only possible other singular points are at the zeros of den().We can easily verify that  = 1 is a simple zero of all four functions num  (), num  (), num(), and den(), so all the integrands are analytic at  = 1.In the appendix we study in detail the algebraic equation den() = 0, and show that the only possible zero is at  = then the two branch points are the only singularities of the functions (), (), and ℎ() in ( 12), ( 16), and (19).If ( 1 +  2 ) 2 >  1 −  2 , which is clearly true in the symmetric case, then  =   is a simple pole of these functions (since den  (  ) ̸ = 0).In view of ( 105) and (106), we have 1 <   <   <  *  , if the pole is present and the stability condition  1 +  2 < 1 holds.
If 1 <  <   ,   >   and in deforming the loop in (10) to the saddle point contour || =   () we must take into account the contribution from the residue at the pole   .But we have (  ) ⩾ (  ) with equality only if  =   , when   =   .Thus the pole contribution dominates the saddle contribution and we have − num  (  ) den  (  ) . (115) From ( 13) we find that and from ( 11) As  → ∞ we have   → 0 and we must then reconsider the asymptotics of (10).From (28)   = ( −1 ) as  → ∞ so we scale  = / in the integral in (10) and consider the limit  → ∞ with  =  (1).Then and ( 10) becomes asymptotically Since   + (0) = − 2 / √ ( 1 +  2 + 1) 2 − 4 1 we see that (119) is the same as (25).Now consider  = / ≈   , where the saddle and pole are close to each other.This is a standard problem that is discussed, for example, in [17,18].We now expand the integrand in (10) about  =   , ultimately scaling  −   = ( −1/2 ), and then the integrand will approach a limiting value as  → ∞.We have, by Taylor/Laurent series, 3 ) . (121) From ( 11) and ( 122) we find that since, at  =   ,   =   .It follows that Advances in Operations Research 13 where  was defined above (31).Also, from (11) we find that Then setting and scaling  as in ( 32), (121) becomes Then we use ( 120) and ( 126) to get where Br − is a vertical contour with Re(V) < 0. To obtain (127) we shifted the original contour in (10) into the circle || =   −  1 ,  1 > 0, and note that || <   implies to leading order that Re(V) < 0. The integral in (127) can be expressed in terms of a parabolic cylinder function of order  = −1 (see (61)), which can be expressed in terms of the standard error function, using the identity With ( 128) and ( 117), (127) becomes the same as (31), so we have derived the leading term for the range −  = ( −1/2 ).

Advances in Operations Research
The expansion of ℎ() will be in powers of √  − , but only the odd powers will contribute to the asymptotics.Now, where we used the binomial expansion of √ 1 −  and Stirling's formula.It follows that the leading term for (, ) is, in view of ( 138) and (139), with a correction that is ( −1 ) relative to the leading term, which may be computed from the ((  − ) 3/2 ) term in (138), and a refined Stirling approximation of the factorials in (139).Some of the algebra in our calculations is simplified by introducing  =  1 +  2 and  = √ 1 −  2 .Then  in (136) factors as and After some calculation we find that so that  −  < 0 and then ( − )/(2 2 ) is the same as the constant  in (53).We have thus established (52).
But (  ) =   / by (157), and then (165) with (166) give precisely the approximation in (57)-(60).Note that  is the same as (   −   )/ 2 .We can easily compute higher order terms in the expansion, and our analysis shows that the asymptotic series will now involve powers of  −1/4 .Actually, the leading term in (57), which has a Gaussian dependence on  1 (hence on ), can be obtained by simply expanding the saddle point approximation in (42), for  ↑   .However, the ( −1/4 ) correction term is necessary to see the transition to the range  >   , where (55) applies.In view of (62) the correction term becomes comparable to the leading term when .This completes the derivation of Theorem 3, where Next we analyze how the results in Theorem 2 transition to those in Theorem 3, as ( 1 +  2 ) 2 decreases through  1 −  2 (> 0).We could simply assume that ( 1 +  2 ) 2 =  1 −  2 and then obtain the necessary asymptotic results.But to see the transition it is necessary to also analyze cases where ( 1 +  2 ) 2 ≈  1 −  2 .To make this more precise we write as in (63), and assume that  → 0 + .Then  ≷ 0 according as Since only the product  is important, we can set  = +1, 0, −1, according to the cases Then we must relate the small parameter  to the large parameters , , and we show below that a natural scaling is to take ,  = ( −2 ) as  → 0 + .The asymptotic results for  → ∞ and  = (1) are the same in Theorems 2 and 3, and thus no transition is needed here.We can write these results in terms of, say, ( 1 , ) rather than ( 1 ,  2 ) and expand for small  to somewhat simplify the expression in ( 25), but we will not do so here.If ( 1 +  2 ) 2 <  1 −  2 the saddle point approximation in (27) applies for all  > 1, while if ( 1 +  2 ) 2 >  1 −  2 it applies only for  >   .But   → 1 (cf.(26)) as ( 1 +  2 ) 2 →  1 −  2 so it will apply for any fixed  > 1.But  ≳ 1 will require a separate analysis.We also note that the sector 1 <  <   , where the product form solution in (34) applies, shrinks to zero.Thus if such an approximation will play a role here, it must be contained near  = 1.
For  < 1 ( > ) we need only consider the ranges  ∈ ( 0  , 1) and  ≈  0  , where  0  is in (64), with  0  being the limit of both  *  and   , as  →  or  → 0 + .For  >  0  the saddle at  *  () exceeds the branch point   and hence the latter determines the asymptotic behavior of (, ).For a fixed , we scale  =   + / and use We again expand () in the form in (169), with  now replaced by   , where Using (161) with   replaced by  0  , along with (180) and (181), the integral in (7) becomes Here we let  = − and used (171) and (172) to approximate  and .Scaling  as in (178) and evaluating the integral in (182) similarly to (173) leads to (70).If  −  = (1), the same analysis applies, as then we can simply replace  by 1 and  2 by  1 in (70) and ( 71), but must maintain the factor [ * + (  )] − .
If we let  → 0 + but consider even larger values of , with  =  1 / 4 = ( −4 ), a slightly different expansion applies.Now both  and √  −  in (186) become ().Setting  −   = −/√, we use in (15) to ultimately obtain the expression in (74), which involves the contour integral in (76).The function L(; ) can be expressed as an infinite sum of parabolic cylinder functions, as This completes the analysis of the transition range where ( 1 +  2 ) 2 ≈  1 −  2 , and we have thus established Theorem 4.