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We analyze a discrete-time Geo/Geo/1 queueing system with preferred customers and partial buffer sharing. In this model, customers arrive according to geometrical arrival processes with probability

The queuing system in which the interarrivals and service times are positive integer values of random variables is called the discrete-time queuing system. Compared with the continuous time, the discrete-time queue is more suitable for the modeling and performance analysis of computer systems, telecommunication network systems, manufacturing and production systems, traffic systems, and health-care systems. Moreover, the discrete-time modeling can be used to approximate the continuous system in practice, but the reverse is generally not true [

In the last 20 years a number of the discrete-time queueing models have been studied, details of which may be found in papers [

The intention of the present paper is to study the Geo/Geo/1 queueing system with preferred customers and partial buffer sharing. As everyone knows, the analysis of a queue system with partial buffer sharing is much more difficult than that with infinite capacity. In the partial buffer sharing mechanism, the normal arrivals can only access the buffer if the buffer occupancy is less than a predetermined value

The remaining sections of this paper are organized as follows: in the next section, the model description and assumptions are stated. In Section

We consider a Geo/Geo/1 queueing system with preferred customers and partial buffer sharing. In this system, the server may start his service only at discrete-time dots. Thus the service time axis can be decomposed into an integer number of time slots. Without loss of generality, we assume that it is the unit service time slots. It is supposed that all queueing activities of arriving and departing happen at the slot boundaries. That is, they may take place at the same time. But, for convenience, we assume, like the disposed method by many papers concerning the discrete-time queue, that a potential arrival occurs in the interval

We assume that the arrival process is Bernoulli process and the intertime intervals of the arrivals follow the same geometric distribution with rate

If an arriving customer finds the server idle, he commences immediately their services. Otherwise, if the server is busy at the arrival epoch, the arrival either interrupts the customer being served to commence his own service with probability

The queueing system has a threshold for buffer control. If the total number of customers in the system is equal to or more than threshold

The system can be described by the process

In this section we will analyze the discrete-time Geo/Geo/1 queueing system with preferred customers and partial buffer sharing.

Using a recursive method, we can calculate the steady-state probability distribution of the system. From (

That is,

Also by (

We finally obtain

It is clear that if

Then we obtain that the sufficient and necessary condition for the stability of the system is

Thus we get the following theorem.

The sufficient and necessary condition for the stability of the system is

(1) When

(2) When

In each moment the queue length of the system is either less than critical value

Now we compute the expected values of the length of the queue

We will discuss a virtual busy period in the system which is different from the original system. The intention of the present paragraph is to facilitate discussion of the sojourn time of a customer. Only in this case, we suppose that the probability of an arrival is equal to

We will use the generating function

The probability generating function of the virtual busy period is

It is not difficult to prove that expression

In this paragraph, we will discuss the actual busy period of the system. An actual busy period

Note that, in the condition that there is no arriving of the preferred customers in the first

The first term in the right-hand side of (

We define

Here, we only list the computing process of the fourth term in (

Multiplying by

Now we show that the solution of (

Define

Thus, (

For any

Hence, within the unit circle by Rouché’s theorem, the number of roots of

Further, for

Thus, the following result is proven.

The solution of (

The mean of busy period is

(1) For

(2) For

(3) For

The intention of this section is to study the distribution of the time that a customer spends in the server in the steady state. If a new preferred customer arrives to the system, the service will be interrupted. Hence the sojourn time of a customer in the server is the elapsed service times plus a whole service time again after interruptions. We let

We define

The distribution of the time that a customer spends in the server in the system follows a geometric distribution with parameter

The mean time of a customer in the server is

In this section we attempt to present the distribution of the virtual sojourn time that a customer spends in the system. A virtual sojourn time of a customer in the system is defined as the period of time that a customer spends in the system from the beginning of its service till the moment of its departure. About this system, we will reckon the possible interruptions time on the sojourn time. We denote by

We also use the generating function

The mean time of a customer in the system is given by

Let

We also use the generating function

The mean sojourn time of a customer in the system is given by

Let

In this section, we provide some numerical examples to show the effect of the main parameters

Case

The free probability versus threshold

In Case

The free probability versus

The loss probability versus threshold

The loss probability versus

The mean system size versus threshold

The mean system size versus

The mean of busy period versus threshold

The mean of busy period versus

The mean sojourn time in the system versus

The mean sojourn time in the system versus

The mean waiting time versus

The mean waiting time versus

As everyone knows, the analysis of a queue system with partial buffer sharing is much more difficult than that with infinite capacity, especially under the priority mechanism. In this paper, we study a discrete-time Geo/Geo/1 queue with preferred customers and partial buffer sharing and carry out an extensive investigation. A sufficient and necessary condition for the stability of the system (Theorem

The authors declare that there is no conflict of interests regarding the publication of this paper.