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We consider a storage allocation model with a finite number of storage spaces. There are

We consider the following storage allocation model. There are

We can consider the storage spaces as parking spaces of a restaurant. The primary spaces are in a lot right next to the restaurant, and the secondary spaces are located somewhere further away from the restaurant. Lower ranked spaces will be closer to the restaurant so it is natural for a customer to use the first-fit policy. Since spaces are occupied and emptied at random times, this model is called a dynamic storage allocation model. Design and analysis of algorithms for dynamic storage allocation are a fundamental part of computer science [

In the language of queueing theory, the model with finite secondary storage spaces can be called the

We let

We focus here on only the steady state distribution but comment that the transient behavior of the standard Erlang loss model can be analyzed by singular perturbation methods of the type employed here (see [

There has been much past work on the model with an infinite (secondary) storage capacity (

The solution of the finite capacity model with

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

We consider a system with

The joint process

Steady state transition rates.

The process

The total number,

We recently obtained in [

We next introduce the parameters

The asymptotic structure of the joint distribution will be very different for four main regions in the

Four regions of the

The presence of the different regions can be explained intuitively. If

In the analysis it proves sometimes useful to use the variables

We begin by giving asymptotic results for

For

Here and throughout the paper, we use the convention that

It will prove convenient to express some of our results in terms of the three constants

Define the constants

We note that the relation

Now we consider the joint distribution;

For

which can be recast as the limit

and this applies for

which holds for

When

We next study the transitions between the three limiting results in Proposition

For

where

As

The complicated distribution in (

The asymptotic expansion of

First consider region

Region

For

where

where

where

where

where

with

where

We can view (

For

We next consider regions

Region

Region

We thus define

For

where

with

The expression in (

The expression in (

In contrast to regions

The expansion in (

In Figure

For

For

Next we consider region

Region

For

The expansion in (

The expansion in (

Again, different expansions must be given near

We conclude by noting that, for all regions of parameter space

This completes our summary of the asymptotic expansions in the three main regions,

We next analyze the four boundary segments of the state space rectangle, namely, the line segments

After treating the four boundary segments we will give asymptotic results that are valid near the four corner points,

For

where

where

so that

The expression in (

where

that satisfies

so that

For

where

We note that the regions of validity of the expansions in Proposition

For region

Next we consider points

For

where

where

Now the expressions in (

where

where

Note that the asymptotics are most complicated in (

For

where

that satisfies

The expansion of

where

and the expansion of

where

We see that for each of the three cases the dependence of

For

Thus (

Next we examine the four corners of the state space, where

For

This gives the expansion when there are but a few primary spaces occupied and a few secondary spaces empty. Next we consider the corner point

For

where

where

We note that (

For

where

which when

and

For region

We can show also that the expansions for

Now consider (

The results in (

Next we examine the corner

For

where

where

and

where

and

When

Finally we give results that apply near the transition curves that separate

For

For

For

For

For

Thus the transition from

As we approach

For

With the scaling