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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

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

Such models were proposed by Zheng and Zipkin [

In [

The present problem corresponds to a random walk in the quarter plane (as

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 [

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 [

The remainder of this paper is organized as follows. In Section

We let

In Figure

A sketch of the transition rates for the random walk.

For

where the integral is over a small loop about

We next evaluate

We will show that the asymptotics are quite different whether

A sketch of the parameter domain and the transition curve

For

where

and

and

Note that, in view of (

In Theorem

Next we take

For

and

For

Thus when

The results in (

Next we study the transition range

Since we will now have the state variables

For

and

and

where

If

Next we consider some different asymptotic limits, those of “light” and “heavy” traffic. Light traffic corresponds to infrequent arrivals, where

The expression in (

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

We solve the difference equation(s) in (

From (

If we define

The coefficient of

We derive Theorems

We need to understand the singularities of the integrands in (

Consider first (

Next we let

If

As

Now consider

Now we consider ranges with

Next we take

The expansion of

Next we consider

We thus reexamine (

Now consider

Now consider the case

We can easily compute higher order terms in the expansion, and our analysis shows that the asymptotic series will now involve powers of

Next we analyze how the results in Theorem

The asymptotic results for

We begin by considering the diagonal probabilities

Now consider

For

When

If we let

This completes the analysis of the transition range where

We establish Theorem

For

For the first heavy traffic limit (HTL1) we let

Hence the limiting form of (

We can also derive the HTL1 limits directly from the asymptotic formulas in (

In the second heavy traffic limit, HTL2, we again let

In HTL2 we also obtain

If

For

Here we study the roots of

Now we consider the possibility of having complex roots of

If

We have shown that any root of

Chalres Knessl work was partly supported by NSA Grant H 98230-11-1-0184. Haishen Yao was supported by PSC-CUNY research award no. 64349-0042.