Performance parameters of queueing systems with preemptive priority in transient regime: Model M/M/1

First Author\[1\]<sup>\*</sup>, Second Author<sup>2</sup>  
(Double Blind Review: Please do not type or edit anything here until final-camera ready submissions)

*<sup>1</sup>First affiliation, City and Country (Please do not type or edit anything here, until final camera-ready paper submission)*

*<sup>2</sup>Second affiliation, City and Country (Please do not type or edit anything here, until final camera-ready paper submission)*

<table>
<tbody>
<tr class="odd">
<td>ARTICLE INFO</td>
<td></td>
<td>ABSTRACT</td>
</tr>
<tr class="even">
<td><p><em>Article history:</em></p>
<p>Received</p>
<p>Revised</p>
<p>Accepted</p>
<p>Online first</p>
<p>Published 1 March 2024</p></td>
<td></td>
<td><p>In Markov chain theory, performance parameters are indicator of the proper management of a queue. In this field, an abundant literature exist, particularly in the steady state, which is not the case in the transient state.</p>
<p>It is in this context that we can question whether it is possible to establish the equations of the performance measures in the transient regime with absolute priority given the complexity of the study of Markov chains in a transient regime.</p>
<p>To achieve this, we think that, if we base ourselves on the theories of state probabilities, the Kolmogorov equations, the Laplace transforms, the matrices of transitions and the convergent series, then it is possible to establish the expressions of the numbers of non-priority customers and their time in the queue and in the system.</p>
<p>The purpose of this paper is to analyze the M/M/1 transient performance measures with absolute priority. Its originality lies in the fact that we have determined the expressions of the performance measures of non-priority customers in a transient regime. Indeed, very few publications are made in this area at this time.</p>
<p>A numerical application was treated to illustrate the theory evoked above. This reflection could soon be carried out in a fuzzy environment.</p></td>
</tr>
<tr class="odd">
<td><p><em>Keywords:</em></p>
<p>Performance measurement</p>
<p>Queuing system</p>
<p>Transient regime</p>
<p>Absolute priority</p>
<p>Non-priority customers</p>
<p><em>DOI:</em></p>
<p>10.24191/jcrinn.v9i1</p></td>
<td></td>
<td></td>
</tr>
</tbody>
</table>

# IntroductioN

The notion of rational queue management is essential today in several areas, among others: public transport, service at a counter, reception in a medical office, etc. In all these cases, managers and clients need efficient services, because waiting has a cost. Thus, to size a service center, as (Agnès Lagnoux et Claudie Chabriac, 2012) points out, a necessary condition is that it is capable of absorbing the average arrival and service flow rates of all categories of customers.

Initiated since the beginning of the 20th century, with the work of the Danish engineer AK Erlang on the management of Copenhague’s telephone networks, queuing phenomena were only generalized and developed in other areas after the second world war with the remarkable contributions of Palm, Chapman, Kolmogorov, Pollaczek ,… Currently, much work has been carried out on the study of performance parameters with or without priority in steady state, among others (Deepak Gupta,Aarti Saini, et al., 2022), (Baudouin Adia et all, 2022); but very rare in transient regime where we can cite (Alonge w’Omatete et al., 2022). In this existing literature, nothing is done on the performance parameters with absolute priority in the transient state of the waiting system.

The fundamental question we want to address is whether it is possible to establish the performance parameters of the M/M/1 model with absolute priority in transient conditions. The question being worrying, we believe that there would be scientific methods based in particular on state probabilities, Kolmogorov equations, Laplace transforms, transition matrices, convergent series, etc. which would facilitate the evaluation of measurements performance of the M/M/1 queuing system with absolute priority in transient conditions. Particular emphasis is placed on non-priority customers.

Given that this reflection is limited to the calculation of performance parameters in priority transient regime, our approach is structured in two sections: the first relates to generalities on state probabilities, Kolmogorov equations, Laplace transforms and transition matrices. The second, analyze the system performance measurements and the numerical example.

In this reflection, we carry out a mathematical analysis of the performance parameters which are the indicators of the good management of a queue, the case under examination is the determination of performance measures in transient mode, with absolute priority. The discipline retained is FIFO, with the consequence that the service of a non-priority customer can only completely result in the absence of a priority customer in the system. The method used is based on state probabilities, Kolmogorov equations, Laplace transforms and transition matrices.

**State probability**

Consider a stochastic process with time space T and state space \(\ E = \left\{ X\left( \Omega \right),\ \ t \in T \right\}\). According to (Mabela M., et al., 2021), the conditional probability for the system to be in the state \(e_{k}\)at time \(\ t_{k}\), knowing that it was in the state \(e_{j}\)at time \(t_{j}\)and in the state \(e_{i}\)at time \(t_{i}\)is written:

\(p_{e_{i}e_{j}e_{k}}\left( t_{i},t_{j},t_{k} \right)\mathbb{= P}\left\lbrack X\left( t_{k} \right) = e_{k}\mid X\left( t_{j} \right) = e_{j},\ X\left( t_{i} \right) = e_{i} \right\rbrack\) (1)

With \(e_{i},e_{j},e_{k} \in E\), or simply

\(p_{i,j,k}\left( t_{i},t_{j},t_{k} \right)\mathbb{= P}\left\lbrack X\left( t_{k} \right) = k\mid X\left( t_{j} \right) = j,\ X\left( t_{i} \right) = i \right\rbrack\)

Let us note by

\(P\left( t \right) = \left( p_{i,j}\left( t \right) \right)\ où\ \ p_{i,j}\left( t \right)\mathbb{= P}\left( \left\lbrack X_{t^{'} + t} = j\mid X_{t^{'}} = i \right\rbrack \right)\ \),

so

\(P\left( t \right) = e^{\text{Lt}} = \sum_{k = 0}^{+ \infty}{\frac{t^{k}}{k!}L^{k}}\) (2)

**Kolmogorov equations**

> Or \(\overrightarrow{\pi}\left( t \right) = \overrightarrow{\pi}(0)P(t)\), its derivative (Dassa Meriyam, 2019) :

\[\overrightarrow{\pi'}\left( t \right) = \overrightarrow{\pi}(0)P'(t) = \overrightarrow{\pi}\left( t \right)L\]

Where \(\text{L\ }\)is the infinitesimal generator of the stochastic process defined by

\(L\left( i,j \right) = \left\{ \begin{matrix}
p_{i,i} = - \lambda \\
p_{i,i + 1} = \lambda \\
p_{i,j}\ \ \  = 0 \\
\end{matrix} \right.\ \) (3)

The life and death processes can be used to model the evolution of a population over time, the number of customers in a queue, in the system, etc. They are continuous Markov processes \(\left( \ T = \mathbb{R}_{+} \right)\)with values in \(E\mathbb{= N}\)such that the only possible non-negligible transitions from \(N\)are to \(N + 1\ \ ou\ N - 1\).

We will ask \(a_{i,i + 1} = \lambda_{i}\text{\ \ et\ \ }a_{i,i - 1} = \mu_{i}\)where \(i \geq 1\ \ et\ \ a_{0,0} = \lambda_{0}\)represents \(\lambda_{i}\)the birth rate from the state \(i\)and \(\mu_{i}\)the death rate from the state \(i\).

The following system is called “Kolmogorov equations”

\(\left\{ \begin{matrix}
{p^{'}}_{0}\left( t \right) = - \lambda p_{0}\left( t \right) + \mu_{1}p_{1}\left( t \right)\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ } \\
{p^{'}}_{n}\left( t \right) = \lambda_{n - 1}p_{n - 1}\left( t \right) - \left( \lambda_{n} + \mu_{n} \right)p_{n}\left( t \right) + \mu_{n + 1}p_{n + 1}(t) \\
\end{matrix} \right.\ \) (4)

**Laplace transforms**

We define a causal function as any function defined on \(\mathbb{R}\), zero on \(\rbrack - \infty,0\lbrack\)and continuous piecewise on \(\lbrack 0, + \infty\lbrack\). If \(f\)is a causal function, the Laplace transform of \(f\)is defined by (Norbert Verdier et al., 2022):

\(F\left( p \right)\mathcal{= L}\left( f \right)\left( p \right) = \int_{0}^{+ \infty}{e^{- pt}f\left( t \right)\text{dt}} = \lim_{x \rightarrow + \infty}{\int_{0}^{x}{e^{- pt}f\left( t \right)dt,}}\) (5)

for the values of \(p\)for which this integral converges.

**Properties**

  - *The Laplace transform is linear* :

\(\mathcal{L}\left( af + bg \right) = a\mathcal{L}\left( f \right) + b\mathcal{L(}g)\). (6)

  - *Effect of a translation* :

either \(a > 0\) and \(g\left( t \right) = f\left( t - a \right)\text{\ \ \ alors}\mathcal{\text{\ \ L}}\left( g \right)\left( p \right) = \ e^{- ap}\mathcal{L}\left( f \right)\left( p \right).\) (7)

  - *Effect of multiplication by an exponential* :

If \(g\left( t \right) = e^{\text{at}}f\left( t \right),\ \ \ avec\ a\mathbb{\in R\ \ }\text{alors}\mathcal{\text{\ \ \ \ L}}\left( g \right)\left( p \right)\mathcal{= L(}f)(p - a)\). (8)

  - *Derivation* .

Let be \(f\)a causal function differentiable on \(\rbrack 0, + \infty\lbrack\). So, for everything \(p\)for which both members have meaning,

\(\mathcal{L}\left( f^{'} \right)\left( p \right) = p\mathcal{L}\left( f \right)\left( p \right) - f(0^{+})\). (9)

  - *The reciprocal of the Laplace transform*

If \(F\left( p \right)\mathcal{= \ L}\left( f\left( t \right) \right)\text{\ \ \ \ \ so\ \ f}\left( t \right) = \mathcal{L}^{- 1}\left( F\left( p \right) \right)\ \) (10)

***Consequences***.

\(\text{if\ f}\left( t \right) = 1\ \ \ so\ \ F\left( p \right) = \frac{1}{p}\). (11)

\(\text{if\ f}\left( t \right) = \sin{(at)}\text{\ so\ \ \ F}\left( p \right) = \frac{a}{p^{2} + a^{2}}\). (12)

\(for\ ft) = \cos{\left( \text{at} \right)\ }\text{we\ have\ \ \ \ F}\left( p \right) = \frac{p}{p^{2} + a^{2}}\). (13)

\(\text{for\ f}\left( t \right) = e^{- \alpha t}\text{\ \ we\ have\ \ \ \ F}\left( p \right) = \frac{1}{p + \alpha}.\) (14)

\(\text{si\ f}\left( t \right) = t^{n}\text{\ \ \ \ \ so\ \ \ \ \ F}\left( p \right) = \frac{n!}{p^{n + 1}}\). (15)

**Transition matrix and graph of a CMTD**

The advantage of this approach is that we are interested in the state of the system only at particular moments \(t_{n}\)in the evolution of the process, by means of state and transition probabilities.

Let us denote by \(P\), the one-step matrix \(P^{(1)}\)and by\(P_{x,y} = P_{x,y}^{(1)}.\)

For everything \(t,\ t\mathbb{' \in N}\), we have:

\(P\left( t + t^{'} \right) = P\left( t \right).P(t^{'})\)

Especially,

\(P^{(n + 1)} = P^{(n)}.P^{(1)} = P^{(n)}\text{P\ \ and\ \ \ }P^{(n)} = P^{n}\).

The one-step transition probabilities are given by:

\(p_{i,j}\mathbb{= P}\left\lbrack X_{t} = j\mid X_{t - 1} = i \right\rbrack\ \forall i,j \in E\ \ such\ as\ \sum_{j \in E}^{}{p_{i,j} = 1}\). (16)

The transition probabilities with \(n\)steps (steps) denoted by \(P_{i,j}^{(n)}\)are given by:

\(p_{i,j}^{(n)}\mathbb{= P}\left\lbrack X_{t + t'} = j\mid X_{t} = i \right\rbrack\). (17)

The transition matrix \(P\)characterizing the chain is often useful for creating a graph of the chain, where the states are represented by points and the transition probability \(P_{x,y} > 0\)by an arc oriented above which the value of is noted \(p_{x,y}\).

**Results**

**Performance parameters of priority clients in system M/M/1 transient state**

Note that this category of customers is served as in a standard queue (Baudouin Adia et all, 2022).

Starting from the state probabilities described by the Kolmogorov equations, according to (Mabela, M., 2023) and (Ritha. W and Rajeswari, 2021).

For \(\lambda_{n} = \lambda > 0,\ \ \mu_{n} = \mu > 0\ \), system (4) is written:

\(\left\{ \begin{matrix}
{p^{'}}_{x}\left( t \right) = \lambda p_{x - 1}\left( t \right) - \left( \lambda + \mu \right)p_{x}\left( t \right) + \mu p_{x + 1}\left( t \right)\ ;\ \ \ x \geq 1 \\
{p^{'}}_{0}\left( t \right) = - \lambda p_{0}\left( t \right) + \mu p_{1}\left( t \right)\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ } \\
\end{matrix} \right.\ \) (18)

The Laplace transforms (5) and (9) applied to \({p^{'}}_{x}\left( t \right)\text{\ and\ }p_{x}\left( t \right)\)are written:

\(\left\{ \begin{matrix}
\mathcal{L}\left( {p^{'}}_{x}\left( t \right) \right) = sp_{x}^{*}\left( s \right) - p_{x}(0) \\
\mathcal{L}\left( p_{x}\left( t \right) \right) = p_{s}^{*}\left( s \right)\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ } \\
\end{matrix} \right.\ \) (19)

Substituting ( 19) into (18) we obtain:

\(\left\{ \begin{matrix}
sp_{x}^{*}\left( s \right) - p_{x}\left( s \right) = \lambda p_{x - 1}^{*}\left( s \right) - \left( \lambda + \mu \right)p_{x}^{*}(s) + \mu p_{x + 1}^{*}(s) \\
sp_{x}^{*}\left( s \right) - p_{0}\left( 0 \right) = - \lambda p_{0}^{*}\left( s \right) + \mu p_{1}^{*}\left( s \right)\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ } \\
\end{matrix} \right.\ \) (20)

Returning to initial conditions where \(p_{x}\left( 0 \right) = 0,\ if\ x \neq 0\ and\ p_{0}\left( x \right) = 1,\ if\ x = 0\), (20) becomes:

\[\left\{ \begin{matrix}
sp_{x}^{*}\left( s \right) = \lambda p_{x - 1}^{*}\left( s \right) - \left( \lambda + \mu \right)p_{x}^{*}\left( s \right) + \mu p_{x + 1}^{*}\left( s \right)\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\ (21)\ \ \ \ \ \ \ \ \ \ \  \\
sp_{x}^{*}\left( s \right) - 1 = - \lambda p_{0}^{*}\left( s \right) + \mu p_{1}^{*}\left( s \right)\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\ \ \ \ \ (22)\ \ \ \ \ \ \ \ \  \\
\end{matrix} \right.\ \]

According to (Y cart B, 2014), the \(p_{x}^{*}(s)\)are solutions of equation (21). By setting \(k\left( s \right) = \frac{p_{x + i}^{*}\left( s \right)}{p_{x - (i + 1)}^{*}\left( s \right)};\ \ \ (i = 0,1)\), we obtain the characteristic equation associated with (21),

\(\mu k^{2}\left( s \right) - \left( s + \lambda + \mu \right)k\left( s \right) + \lambda = 0\) (23)

Whose roots are given by:

\(\alpha\left( s \right) = \frac{1}{2\mu}\left\lbrack s + \lambda + \mu + \left( \left( s + \lambda + \mu \right)^{2} - 4\mu\lambda \right)^{\frac{1}{2}} \right\rbrack\) (24)

\(\beta\left( s \right) = \frac{1}{2\mu}\left\lbrack s + \lambda + \mu - \left( \left( s + \lambda + \mu \right)^{2} - 4\mu\lambda \right)^{\frac{1}{2}} \right\rbrack\) (25)

The solution to (21) is written according to (Ycart B., 2014):

\(p_{x}^{*}\left( s \right) = A\left( s \right)\left( \alpha\left( s \right) \right)^{x} + B(s)\left( \beta\left( s \right) \right)^{x}\) (26)

With \(\ \ 0 < \beta\left( s \right) < 1 < \alpha\left( s \right)\ \ \ \ for\ \ s > 0\),

The root \(\alpha(s)\)being rejected because \(\left( s \right) \notin \lbrack 0,1\rbrack\), the linearity of the Laplace transform leads to the following:

\(\sum_{x = 1}^{n}{p_{x}^{*}\left( s \right)} = \frac{1}{s},\ \ (s > 0)\) (27)

The series \(\sum_{x = 1}^{n}{p_{x}^{*}\left( s \right)}\)being convergent implies that

\(\sum_{x = 1}^{\infty}\left\lbrack B\left( s \right)\left( \beta\left( s \right) \right)^{x} \right\rbrack = \frac{B(s)}{1 - \beta(s)}\) (28)

The relations (27) and (28) give:

\(\frac{1}{s} = \frac{B\left( s \right)}{1 - \beta(s)}\ \ \  \Longrightarrow \ \ \ \ \ \ \ \ B\left( s \right) = \frac{1}{s}\left( 1 - \beta\left( s \right) \right)\) (29)

And (26) becomes:

\(p_{x}^{*}\left( s \right) = \frac{1}{s}\left( 1 - \beta\left( s \right) \right)\left( \beta\left( s \right) \right)^{x} = \frac{1}{s}\left( \left( \beta\left( s \right) \right)^{x} - \left( \beta\left( s \right) \right)^{x + 1} \right)\) (30)

By deriving the relation (25) we obtain:

\(\beta^{'}\left( s \right) = \frac{1}{2\mu}\left( 1 - \frac{s + \lambda + \mu}{\left( \left( s + \lambda + \mu \right)^{2} - 4\mu\lambda \right)^{\frac{1}{2}}} \right)\) (31)

The relations (25) and (31) at the point \(s = 0\)give:

\(\beta\left( 0 \right) = \left\{ \begin{matrix}
1\ \ \ \ \ \ \ \ \ \ \ \ if\ \lambda \geq \mu \\
\frac{\lambda}{\mu}\ \ \ \ \ \ \ \ \ \ if\ \ \lambda < \mu \\
\end{matrix} \right.\ \) (32)

\(\beta^{'}\left( 0 \right) = \left\{ \begin{matrix}
 - \frac{1}{\lambda - \mu}\ \ \ \ \ \ \ if\ \ \lambda \geq \mu \\
\frac{\lambda}{\mu}\left( \frac{1}{\lambda - \mu} \right)\ \ \ if\ \ \lambda < \mu \\
\end{matrix} \right.\ \) (33)

The case \(\lambda < \mu\) interests us more compared to the other, because the opposite leads to explosion.

Developments in entire series of \(\left( \beta\left( s \right) \right)^{x}\text{\ \ if\ \ }\left( \beta\left( s \right) \right)^{x + 1}\ \); according to Mac-Laurin. By putting \(f\left( x \right) = \left( \beta\left( s \right) \right)^{x}\text{\ \ \ et\ \ \ g}\left( x \right) = \left( \beta\left( s \right) \right)^{x + 1}\)we obtain:

\(f\left( x \right) = \sum_{n = 0}^{\infty}{x!\left( \frac{\lambda}{\mu} \right)^{x}\left( \frac{1}{\lambda - \mu} \right)^{x}\frac{s^{n}}{n!} = \left( \frac{\lambda}{\mu} \right)^{x}\left( \frac{\lambda - \mu}{\left( \lambda - \mu \right) - s} \right)\ \ ;\ \ \ \ \ \left| \frac{\lambda}{\lambda - \mu} \right| < 1}\) (34)

\(g\left( x \right) = \sum_{n = 0}^{\infty}{(x + 1)!\left( \frac{\lambda}{\mu} \right)^{x + 1}\left( \frac{1}{\lambda - \mu} \right)^{x + 1}\frac{s^{n + 1}}{(n + 1)!} = \left( \frac{\lambda}{\mu} \right)^{x + 1}\left( \frac{\lambda - \mu}{\left( \lambda - \mu \right) - s} \right)\ \ ;\ \ \ \left| \frac{s}{\lambda - \mu} \right| < 1}\)

(35)

Substituting (34) and (35) into (30) we obtain:

\(p_{x}^{*}\left( s \right) = \frac{1}{s}\left\lbrack \left( \frac{\lambda}{\mu} \right)^{x}\left( \frac{\mu - \lambda}{s - \left( \lambda - \mu \right)} \right) + \left( \frac{\lambda}{\mu} \right)^{x + 1}\left( \frac{\mu - \lambda}{s - \left( \lambda - \mu \right)} \right) \right\rbrack\) (36)

(8) allows us to write (with ci .\(p_{x}\left( 0 \right) = 0,\ x \neq 0)\ :\)

\(p_{x}^{'}\left( t \right) = \ \mathcal{L}^{- 1}\left( sp_{x}^{*}\left( s \right) \right) = \mathcal{L}^{- 1}\left\lbrack \left( \frac{\lambda}{\mu} \right)^{x}\left( \frac{\mu - \lambda}{s - \left( \lambda - \mu \right)} \right) + \left( \frac{\lambda}{\mu} \right)^{x + 1}\left( \frac{\mu - \lambda}{s - \left( \lambda - \mu \right)} \right) \right\rbrack\) (37)

(6) and (14) applied to (37) gives the following:

\(p_{x}^{'}\left( t \right) = - \left( \frac{\lambda}{\mu} \right)^{x}\left( \lambda - \mu \right)\ e^{- \left( \mu - \lambda \right)t} + \left( \frac{\lambda}{\mu} \right)^{x + 1}(\lambda - \mu)e^{- (\lambda - \mu)t}\) (38)

The integration of (38) in \(\lbrack 0,t\rbrack\)gives:

\(p_{x}\left( t \right) = \left( \frac{\lambda}{\mu} \right)^{x}e^{- \left( \mu - \lambda \right)t} - \left( \frac{\lambda}{\mu} \right)^{x + 1}e^{- \left( \mu - \lambda \right)t} + c\) (39)

\(p_{x}\left( 0 \right) = 0,\ \ \ C = \left( \frac{\lambda}{\mu} \right)^{x}\left( 1 - \frac{\lambda}{\mu} \right)\)

Hence the general solution is

\(p_{x}\left( t \right) = \left( \frac{\lambda}{\mu} \right)^{x}\left( 1 - \frac{\lambda}{\mu} \right)\left( 1 - e^{- \left( \mu - \lambda \right)t} \right)\) (40)

For \(x = 0\),

\(p_{0}\left( t \right)\mathbb{= P}\left\lbrack x\left( t \right) = 0 \right\rbrack = \left( 1 - \frac{\lambda}{\mu} \right)\left( 1 - e^{- \left( \mu - \lambda \right)t} \right)\) (41)

Knowing that \(\rho_{1} = \frac{\lambda}{\mu}\), and posing: \(\alpha = \mu - \lambda\ \); (40) and (41) become:

\(p_{x}\left( t \right) = {\rho_{1}}^{x}\left( 1 - \rho_{1} \right)\left( 1 - e^{- \alpha t} \right)\)(43)

\(p_{0}\left( t \right)\mathbb{= P}\left\lbrack x\left( t \right) = 0 \right\rbrack = \left( 1 - \rho_{1} \right)\left( 1 - e^{- \alpha t} \right)\) (44)

Therefore, the performance parameters are such as:

  - *Server utilization rate at a time* \(\mathbf{t}\).

\(\mathfrak{u}\left( t \right) = 1 - p_{0}\left( t \right) = 1 - \left( 1 - \rho_{1} \right)\left( 1 - e^{- \alpha t} \right) = \rho_{1} + (1 - \rho_{1})e^{- \alpha t}\). (45)

  - *System throughput on a date* \(\mathbf{\text{\ t}}\ \):

\(d\left( t \right) = \left( 1 - p_{0}\left( t \right) \right)\mu = \mu\left( 1 - \left( 1 - \rho_{1} \right)\left( 1 - e^{- \alpha t} \right) \right) = \lambda + \alpha e^{- \alpha t}\). (46)

  - *Average number of clients in system at the moment* \(\mathbf{\text{t~}}\):

\({\overline{N}}_{s}\left( t \right) = \sum_{x = 0}^{\infty}xp_{x}\left( t \right) = \sum_{x = 0}^{\infty}{x\rho^{x}\left( 1 - \rho_{1} \right)\left( 1 - e^{- \alpha t} \right)} = \frac{\rho_{1}(1 - e^{- \alpha t})}{1 - \rho_{1}}\), (47)

  - *Average number of customers in queue on a date* \(\mathbf{t}\):

\({\overline{N}}_{f}\left( t \right) = \sum_{x = 0}^{\infty}{(x - 1)}p_{x}\left( t \right) = \sum_{x = 0}^{\infty}xp_{x}\left( t \right) - \sum_{x = 1}^{\infty}{p_{x}\left( t \right)} - p_{0}(t) = \frac{{\rho_{1}}^{2} - ({\rho_{1}}^{2} - \rho_{1} + 1)e^{- \alpha t}}{1 - \rho_{1}}\)

(48)

  - *Average stay time of a customer in the system during a period*\(\mathbf{\text{t~}}\)

\({\overline{T}}_{s}\left( t \right) = \frac{{\overline{N}}_{s}\left( t \right)}{d(t)} = \frac{\rho_{1}(1 - e^{- \alpha t})}{(1 - \rho_{1})(\lambda + \alpha e^{- \alpha t})}\). (49)

  - *Average time a customer waits in line for an instant* \(\mathbf{t}\)**:**

\({\overline{T}}_{f}\left( t \right) = \frac{{\overline{N}}_{f}\left( t \right)}{d(t)} = \frac{{\rho_{1}}^{2} - ({\rho_{1}}^{2} - \rho_{1} + 1)e^{- \alpha t}}{(1 - \rho_{1})(\lambda + \alpha e^{- \alpha t})}\). (50)

**Non-Priority customers Performance Settings**

Note that for this category of customers, the task is not so easy because their services are strongly disrupted by the arrivals and services of priority customers in the system. Service to a non-priority customer can only result in the total absence of a priority customer in the system.

Let \(X\left( t \right)\text{\ et\ Y}\left( t \right)\), be the numbers of respectively priority and non-priority customers at the moment \(\mathbf{t}\). As \(\left( X\left( t \right),Y\left( t \right) \right)\)it is a discrete state, the continuous-time Markov chain and its techniques will allow us to describe the limit distribution, when it exists, from a diagram.

Consider \(\lambda_{1}\left( \theta \right)\text{\ et\ }\lambda_{2}(\theta)\), the arrival rates of priority and non-priority customers respectively in the system during the period \(\lbrack 0,\theta\rbrack\)and by \(\gamma\left( \theta \right)\ et\ \delta(\theta)\), the rates of priority and non-priority customers respectively served during the period \(\lbrack 0,\theta\rbrack\).

  - The arrival rate in the system is the sum \(\lambda = \lambda_{1} + \lambda_{2}\ \);

  - Or \(p = \frac{\lambda_{1}}{\lambda}\ \ et\ q = \frac{\lambda_{2}}{\lambda}\ \)the respective proportions of priority and non-priority customers in the system;

  - The average times of services in the system are given by their weighted averages \(\frac{1}{\gamma}\text{\ \ et\ }\frac{1}{\delta}\ \)respectively for priority and non-priority customers;

  - The average service rate in the system is given by:

\(\frac{1}{\mu} = p\left( \frac{1}{\gamma} \right) + q\left( \frac{1}{\delta} \right) = \frac{1}{\lambda}\left( \frac{\lambda_{1}}{\gamma} + \frac{\lambda_{2}}{\delta} \right)\) (51)

  - The traffic intensity in the system is given by:

\(\rho = \frac{\lambda}{\mu} = \frac{\lambda_{1}}{\gamma} + \frac{\lambda_{2}}{\delta} = \rho_{1} + \rho_{2}\). (52)

Consider the following transition diagram:

![](65a293d863222_media/media/image1.png)

Fig1. Transition diagram

***Description of the transition diagram***:

  - Arrival of a priority client, the system goes from state \(\left( m,n \right)\)to state \((m + 1,n)\)with a transition rate \(\lambda_{1}\).

  - The arrival of a non-priority customer, the system goes from state \(\left( m,n \right)\)) to the state \((m,n + 1)\)with a transition rate \(\lambda_{2}\).

  - The service of a non-priority customer is over; the system transitions from state \(\left( 0,n \right)\)to state \(\ \left( 0,n - 1 \right),\ \ (n \geq 1)\), with a transition rate \(\delta\).

  - The service of a priority customer is completed; the system transitions from state \((m,n)\)to state \((m - 1,n)\)with a transition rate \(\text{\ γ}\).

From this description follow the following balance equations:

\(\left( \lambda_{1} + \lambda_{2} \right)p_{\left( 0,0 \right)}\left( t \right) = \gamma p_{1,0}\left( t \right) + \delta p_{0,1}(t)(1 - e^{- \alpha t})\) (53)

\(\left( \lambda_{1} + \lambda_{2} + \gamma \right)p_{m,0}\left( t \right) = \gamma p_{m + 1,0}\left( t \right) + \lambda_{1}p_{m - 1,0}(t)(1 - e^{- \alpha t})\) (54)

\(\left( \lambda_{1} + \lambda_{2} + \delta \right)p_{0,n}\left( t \right) = \gamma p_{1,n}\left( t \right) + \left( \delta p_{0,n + 1}\left( t \right) + \lambda_{2}p_{0,n - 1}\left( t \right) \right)(1 - e^{- \alpha t})\), (55)

\(\left( \lambda_{1} + \lambda_{2} + \delta \right)p_{0,n}\left( t \right) = \gamma p_{m + 1,n}\left( t \right) + \lambda_{1}p_{m - 1,n}\left( t \right) + \lambda_{2}p_{m,n - 1}(t)(1 - e^{- \alpha t})\) (56)

Equation (43) applied to \(X\left( t \right)\)and \(Y(t)\)gives us the following expressions:

\(p_{m}\left( t \right)\mathbb{= P}\left( \left\lbrack X\left( t \right) = m \right\rbrack \right) = \sum_{n = 0}^{\infty}{p_{m,n}\left( t \right) = \left( 1 - \rho_{1} \right)}\rho_{1}^{m}(1 + e^{- \alpha t})\) (57)

\(p_{n}\left( t \right)\mathbb{= P}\left( \left\lbrack X\left( t \right) = n \right\rbrack \right) = \sum_{m = 0}^{\infty}{p_{m,n}\left( t \right) = \left( 1 - \rho_{1} \right)}\rho_{2}^{n}(1 + e^{- \alpha t})\) (58)

By summing and adding member to member on \(m\), equations (53) and (54) we obtain:

\(\left( \lambda_{1} + \lambda_{2} \right)p_{0,0}\left( t \right) + \gamma\sum_{m = 1}^{\infty}{p_{0,0}(t)} = \gamma\sum_{m = 1}^{\infty}{p_{m,0}\left( t \right)} + \lambda_{1}p_{0,0}\left( t \right) + \delta p_{0,1}\left( t \right)(1 - e^{- \alpha t})\)

Let,

\(\lambda_{2}p_{0,0}\left( t \right) = \ \delta p_{0,1}\left( t \right)(1 - e^{- \alpha t})\) \(\forall t \geq 0\) (59)

The member-to-member sum over \(m\)(55) and (56) gives us:

\(\left( \lambda_{1} + \lambda_{2} + \delta \right)p_{0,n}\left( t \right) + \left( \lambda_{1} + \lambda_{2} + \gamma \right)\sum_{m = 0}^{\infty}{p_{m,n}\left( t \right)} = \gamma\sum_{m = 1}^{\infty}{p_{1,n}\left( t \right)} + \left( \delta p_{0,n + 1}\left( t \right) + \lambda_{2}p_{0,n - 1}\left( t \right) \right)\left( 1 - e^{- \alpha t} \right) + \gamma\sum_{m = 0}^{\infty}{p_{m + 1}(t)} + \lambda_{1}\sum_{m = 0}^{\infty}{p_{m - 1,n}(t)} + \lambda_{2}\sum_{m = 0}^{\infty}{p_{m,n - 1}\left( t \right)}(1 - e^{- \alpha t})\)

\(\lambda_{1}\sum_{m = 0}^{\infty}{p_{m,n}(t)} + \lambda_{2}\sum_{m = 0}^{\infty}{p_{m,n}(t)} + \delta p_{0,n}\left( t \right) + \gamma\sum_{m = 0}^{\infty}{p_{m,n}(t)} = \gamma\sum_{m = 1}^{\infty}{p_{1,n}\left( t \right)} + \left( \delta p_{0,n + 1}\left( t \right) + \lambda_{2}p_{0,n - 1}\left( t \right) \right)\left( 1 - e^{- \alpha t} \right) + \lambda_{1}\sum_{m = 0}^{\infty}{p_{m - 1,n}(t)}\) + \(\lambda_{2}\sum_{m = 0}^{\infty}{p_{m,n}(t)} + \delta p_{0,n}\left( t \right) = \left( \delta p_{0,n + 1}\left( t \right) + \lambda_{2}p_{0,n - 1}\left( t \right) \right)\left( 1 - e^{- \alpha t} \right)\) , (60)

By induction on (59), (60) becomes:

\(\lambda_{2}\sum_{m = 0}^{\infty}{p_{m,n}(t)} = \delta p_{0,n + 1}\left( t \right)\left( 1 - e^{- \alpha t} \right)\) (61)

By adding (61) member to member over \(n\) and the fact that \(\sum_{m = 0}^{\infty}{p_{m,n}(t)} = 1\), we obtain:

\(\lambda_{2} = \delta p_{0,n + 1}\left( t \right)\left( 1 - e^{- \alpha t} \right)\) (62)

Firstly,

\(\mathbb{P}\left\lbrack X\left( t \right) = 0,\ Y\left( t \right) > 0 \right\rbrack = \sum_{n = 1}^{\infty}{p_{0,n}(t)} = \frac{\lambda_{2}}{\delta}\left( 1 - e^{- \alpha t} \right) = \rho_{2}(1 - e^{- \alpha t})\), (63)

\(\mathbb{P}\left\lbrack X\left( t \right) = 0 \right\rbrack = \left( 1 - \frac{\lambda_{1}}{\gamma} \right)\left( 1 - e^{- \alpha t} \right) = \left( 1 - \rho_{1} \right)(1 - e^{- \alpha t})\) (64)

On the other hand,

\(p_{0,0,}\left( t \right)\mathbb{= P}\left\lbrack X\left( t \right) = 0,\ Y\left( t \right) = 0 \right\rbrack\mathbb{= P}\left\lbrack X\left( t \right) = 0 \right\rbrack\mathbb{- P}\left\lbrack X\left( t \right) = 0,\ Y\left( t \right) > 0 \right\rbrack = \left( 1 - \rho_{1} \right)\left( 1 - e^{- \alpha t} \right) - \rho_{2}\left( 1 - e^{- \alpha t} \right) = \left( 1 - \rho \right)\left( 1 - e^{- \alpha t} \right)\) (65)

These results above allow us to calculate the performance parameters of non-priority customers.

1.  *Server utilization rate at a time* \(\mathbf{t}\)**,**

\(\mathfrak{u}\left( t \right) = 1 - p_{0,0}\left( t \right) = 1 - \left( 1 - \rho \right)\left( 1 - e^{- \alpha t} \right) = \rho + (1 - \rho)e^{- \alpha t}\) , (66)

2.  *System throughput on a date* \(\mathbf{\text{\ t}}\ \):

> \(d\left( t \right) = \left( 1 - p_{0,0}\left( t \right) \right)\mu = \left( \rho + \left( 1 - \rho \right)e^{- \alpha t} \right)\mu = \lambda + \alpha e^{- \alpha t}\), (67)

3.  > **Theorem** .*The average number of non-priority customers in the system at the moment* \(\ \mathbf{t}\) is given by the expression \({\overline{N}}_{s_{2}}(t) = \frac{\rho_{2}}{1 - \rho}\left( 1 + \frac{\delta}{\gamma}\frac{\rho_{1}}{1 - \rho_{1}} \right)\left( 1 - e^{- \alpha t} \right)\)

> **Proof**
> 
> In the equilibrium state, this number is given by:
> 
> \({\overline{N}}_{s_{2}} = \sum_{m = 0}^{\infty}{\sum_{n = 0}^{\infty}{np_{m,n}(t)}}\) (68)
> 
> Let us note by

\(G_{m} = \sum_{n = 0}^{\infty}{np_{m,n}(t)} = \sum_{n = 1}^{\infty}{np_{m,n}(t)}\) (69)

And so (68) becomes:

\({\overline{N}}_{s_{2}} = \sum_{m = 0}^{\infty}{G_{m} = G_{0} + G_{1} + \ldots}\) (70)

By multiplying equation (55) by \(n\)and summing over \(n\), both sides of the equation, we obtain:

\(\left( \lambda_{1} + \lambda_{2} + \delta \right)\sum_{n = 1}^{\infty}np_{0,n}\left( t \right) = \gamma\sum_{n = 1}^{\infty}{np_{1,n}(t)} + \left( \delta\sum_{n = 1}^{\infty}{np_{0,n + 1}\left( t \right)} + \lambda_{2}\sum_{n = 1}^{\infty}{np_{0,n - 1}\left( t \right)} \right)\left( 1 - e^{- \alpha t} \right)\ \)

\(\left( \lambda_{1} + \lambda_{2} + \delta \right)G_{0} = \gamma G_{1} + \left( \delta G_{0} - \delta\sum_{n = 0}^{\infty}{np_{0,n + 1}\left( t \right)} + \lambda_{2}G_{0} + \lambda_{2}\sum_{n = 0}^{\infty}{np_{0,n - 1}\left( t \right)} \right)\left( 1 - e^{- \alpha t} \right)\)

\(\left( \lambda_{1} + \lambda_{2} + \delta \right)G_{0} = \gamma G_{1} + \delta G_{0} - \delta\left( \frac{\lambda_{2}}{\delta} \right)\left( 1 - e^{- \alpha t} \right) + \lambda_{2}G_{0} + \lambda_{2}(1 - \rho_{1})(1 - e^{- \alpha t})\).

\(\lambda_{1}G_{0} = \gamma G_{1} - \lambda_{2}\left( 1 - e^{- \alpha t} \right) + \lambda_{2}\left( 1 - e^{- \alpha t} \right) - \lambda_{2}\rho_{1}(1 - e^{- \alpha t})\)

Eventually,

\(G_{1} = \rho_{1}G_{0} + \frac{\lambda_{2}}{\gamma}\rho_{1}\left( 1 - e^{- \alpha t} \right)\) (71)

Multiplying (56) by \(n\) and summing to \(n = 1,2,..\)give:

\(\left( \lambda_{1} + \lambda_{2} + \gamma \right)\sum_{n = 1}^{\infty}np_{m,n}\left( t \right) = \gamma\sum_{n = 1}^{\infty}{np_{m + 1,n}(t)} + \lambda_{1}\sum_{n = 1}^{\infty}{np_{m - 1,n}\left( t \right)} + \left( \lambda_{2}\sum_{n = 1}^{\infty}{np_{m,n - 1}\left( t \right)} \right)\left( 1 - e^{- \alpha t} \right)\) ,

Let,

\(\left( \lambda_{1} + \lambda_{2} + \gamma \right)G_{m} = \gamma G_{m + 1} + \lambda_{1}G_{m - 1} + \lambda_{2}\sum_{n = 1}^{\infty}{np_{m,n - 1}\left( t \right)}\left( 1 - e^{- \alpha t} \right)\), (72)

From (57) we know that \(\sum_{n = 0}^{\infty}{p_{m,n}\left( t \right) = \left( 1 - \rho_{1} \right)}\rho_{1}^{m}(1 + e^{- \alpha t})\), in (72) we obtain:

\(\left( \lambda_{1} + \gamma \right)G_{m} = \gamma G_{m + 1} + \lambda_{1}G_{m - 1} + \lambda_{2}\left( 1 - \rho_{1} \right)\rho_{1}^{m}\left( 1 - e^{- \alpha t} \right)^{2}\) (73)

For \(m = 1\), we have:

\(\left( \lambda_{1} + \gamma \right)G_{1} = \gamma G_{2} + \lambda_{1}G_{0} + \lambda_{2}\left( 1 - \rho_{1} \right)\rho_{1}\left( 1 - e^{- \alpha t} \right)^{2}\) (74)

Substituting (70) into (74) we obtain:

\(\left( \lambda_{1} + \gamma \right)\left( \rho_{1}G_{0} + \frac{\lambda_{2}}{\gamma}\rho_{1}\left( 1 - e^{- \alpha t} \right) \right) = \gamma G_{2} + \lambda_{1}G_{0} + \lambda_{2}\left( 1 - \rho_{1} \right)\rho_{1}\left( 1 - e^{- \alpha t} \right)^{2}\)

\(\lambda_{1}\rho_{1}G_{0} + \lambda_{2}\rho_{1}^{2}\left( 1 - e^{- \alpha t} \right) + \gamma\rho_{1}G_{0} + \lambda_{2}\rho_{1}\left( 1 - e^{- \alpha t} \right) = \gamma G_{2} + \lambda_{1}G_{0} + \lambda_{2}\rho_{1}\left( 1 - e^{- \alpha t} \right) - \lambda_{2}\rho_{1}^{2}\left( 1 - e^{- \alpha t} \right)\)

\(\gamma G_{2} = \lambda_{1}\rho_{1}G_{0} + 2\lambda_{2}\rho_{1}^{2}\left( 1 - e^{- \alpha t} \right) + \gamma\rho_{1}G_{0} - \lambda_{1}G_{0}\)

From where

\(G_{2} = \rho_{1}^{2}\left( G_{0} + 2\frac{\lambda_{2}}{\gamma}\left( 1 - e^{- \alpha t} \right) \right)\) (75)

For \(m = 2\)in (73) and by similar reasoning, we obtain:

\(G_{3} = \rho_{1}^{3}\left( G_{0} + 3\frac{\lambda_{2}}{\gamma}\left( 1 - e^{- \alpha t} \right) \right)\) (76)

In general

\(G_{m} = \rho_{1}^{m}\left( G_{0} + m\frac{\lambda_{2}}{\gamma}\left( 1 - e^{- \alpha t} \right) \right)\) (77)

By summing over \(m\)equation (76), we obtain:

\(\sum_{m = 0}^{\infty}G_{m} = {\overline{N}}_{s_{2}} = G_{0}\sum_{m = 0}^{\infty}\rho_{1}^{m} + \frac{\lambda_{2}}{\gamma}(1 - e^{- \alpha t})\sum_{m = 0}^{\infty}{m\rho_{1}^{m}}\) (78)

Knowing that the geometric series \(\sum_{m = 0}^{\infty}\rho_{1}^{m} = \frac{1}{1 - \rho_{1}},\ (\rho_{1} < 1)\)and that the series \(\sum_{m = 0}^{\infty}{m\rho_{1}^{m}} = \sum_{m = 1}^{\infty}{m\rho_{1}^{m}} = \frac{\rho_{1}}{{(1 - \rho_{1})}^{2}}\ \), therefore equation (78) becomes:

\({\overline{N}}_{s_{2}} = G_{0}\frac{1}{1 - \rho_{1}} + \frac{\lambda_{2}}{\gamma}\frac{\rho_{1}}{\left( 1 - \rho_{1} \right)^{2}}\left( 1 - e^{- \alpha t} \right) = \frac{1}{1 - \rho_{1}}\left( G_{0} + \frac{\lambda_{2}}{\gamma}\frac{\rho_{1}}{\left( 1 - \rho_{1} \right)^{}}\left( 1 - e^{- \alpha t} \right) \right)\),

(79)

We observe that \({\overline{N}}_{s_{2}}\)is a function of \(G_{0}\). To do this, let us multiply equation (61) by \(n\)and sum to \(n\), we obtain:

\(\lambda_{2}\sum_{m = 0}^{\infty}{\sum_{n = 1}^{\infty}{np_{m,n}\left( t \right) = \delta\sum_{n = 1}^{\infty}{np_{0,n + 1}(t)(1 - e^{- i\omega t})}}}\) (80)

\(\lambda_{2}\sum_{m = 0}^{\infty}G_{m} = \delta\sum_{n = 0}^{\infty}np_{0,n}\left( t \right) - \delta\sum_{n = 0}^{\infty}\text{np}_{0,n + 1}(t)(1 - e^{- i\omega t})\) (81)

\(\lambda_{2}{\overline{N}}_{s_{2}} = \delta G_{0} - \delta\frac{\lambda_{2}}{\delta}\left( 1 - e^{- \alpha t} \right) = \delta G_{0} - \lambda_{2}(1 - e^{- \alpha t})\)

And so

\(G_{0} = \rho_{2}\left( {\overline{N}}_{s_{2}} + \left( 1 - e^{- \alpha t} \right) \right)\) (82)

Substituting (82) into (79), we obtain:

\({\overline{N}}_{s_{2}} = \frac{1}{1 - \rho_{1}}\left( \rho_{2}\left( {\overline{N}}_{s_{2}} + \left( 1 - e^{- \alpha t} \right) \right) + \frac{\lambda_{2}}{\gamma}\frac{\rho_{1}}{\left( 1 - \rho_{1} \right)^{}}\left( 1 - e^{- \alpha t} \right) \right)\) (83)

\({\overline{N}}_{s_{2}} - \frac{\rho_{2}}{1 - \rho_{1}}{\overline{N}}_{s_{2}} = \frac{1}{1 - \rho_{2}}\left( \rho_{2} + \frac{\lambda_{2}}{\gamma}\frac{\rho_{1}}{1 - \rho_{1}} \right)\left( 1 - e^{- \alpha t} \right)\)

\({\overline{N}}_{s_{2}}\left( 1 - \frac{\rho_{2}}{1 - \rho_{1}} \right) = \frac{1}{1 - \rho_{2}}\left( \rho_{2} + \frac{\lambda_{2}}{\gamma}\frac{\rho_{1}}{1 - \rho_{1}} \right)\left( 1 - e^{- \alpha t} \right)\)

\({\overline{N}}_{s_{2}} = \frac{1}{1 - \rho}\left( \rho_{2} + \frac{\lambda_{2}}{\gamma}\frac{\rho_{1}}{1 - \rho_{1}} \right)\left( 1 - e^{- \alpha t} \right)\)

Eventually :

\({\overline{N}}_{s_{2}}(t) = \frac{\rho_{2}}{1 - \rho}\left( 1 + \frac{\delta}{\gamma}\frac{\rho_{1}}{1 - \rho_{1}} \right)\left( 1 - e^{- \alpha t} \right)\) \(\ avec\ (\rho = \rho_{1} + \rho_{2} < 1)\) (84)

LIttle 's theorem we deduce the following parameters:

4.  > *The average time of a non-priority customer in the system during a period* \(\mathbf{t}\)

> \({\overline{T}}_{s_{2}}(t) = \frac{{\overline{N}}_{s_{2}}}{d(t)} = \frac{\rho_{2}}{1 - \rho}\left( 1 + \frac{\delta}{\gamma}\frac{\rho_{1}}{1 - \rho_{1}} \right)\left( 1 - e^{- \alpha t} \right)\frac{1}{\lambda + \alpha e^{- \alpha t}}\) (85)

5.  *The average time of a non-priority customer in queue for a period* \(\mathbf{\text{t~}}\)**:**

> \({\overline{T}}_{f_{2}}(t) = {\overline{T}}_{s_{2}}(t) - \frac{1}{\mu} = \frac{\rho_{2}}{1 - \rho}\left( 1 + \frac{\delta}{\gamma}\frac{\rho_{1}}{1 - \rho_{1}} \right)\left( 1 - e^{- \alpha t} \right)\frac{1}{\lambda + \alpha e^{- \alpha t}} - \frac{\rho}{\lambda}\).
> 
> \(= \frac{\rho_{2}\left( 1 + \frac{\delta}{\gamma}\frac{\rho_{1}}{1 - \rho_{1}} \right)\left( 1 - e^{- \alpha t} \right) - (1 - \rho)\rho}{(1 - \rho)\lambda}\) (86)

6.  *The average number of non-priority customers in the queue at moment*\(\text{\ t}\).

\({\overline{N}}_{f_{2}} = \lambda{\overline{T}}_{s_{2}}(t) = \frac{\rho_{2}\left( 1 + \frac{\delta}{\gamma}\frac{\rho_{1}}{1 - \rho_{1}} \right)\left( 1 - e^{- \alpha t} \right) - (1 - \rho)\rho}{(1 - \rho)}\) (87)

**Numerical example**

With a commercial bank in Kinshasa (DRC), two categories of customers are served at the single counter of its branch located in Gombe. The priority category called large customers, that is to say having a figure of at least 10 million FC in their account, arrives for operations at the bank following a Poisson rate process \(\lambda_{1} = 0,65\) and the non-priority category called ordinary customers, having less than a million in their respective accounts, arrives to be served according to the same process with a rate \(\lambda_{2} = 0,23\ \); services are exponential in rate \(\mu = 0,94\). Within a 60-minute interval, determine the performance parameters for each category of customers:

\({\overline{N}}_{s_{1}}\left( t \right) = \frac{\rho_{1}(1 - e^{- \alpha t})}{1 - \rho_{1}}\)= \(\frac{0,69\left( 1 - e^{- \left( 0,94 - 0,65 \right)60} \right)}{1 - 0,69} \simeq 2\ \)

\({\overline{T}}_{s_{1}}\left( t \right) = \frac{\rho_{1}(1 - e^{- \alpha t})}{(1 - \rho_{1})(\lambda + \alpha e^{- \alpha t})} = \frac{0,69\left( 1 - e^{- \left( 0,94 - 0,65 \right)60} \right)}{\left( 1 - 0,69 \right)\left( 0,65 + \left( 0,94 - 0,65 \right)e^{- \left( 0,94 - 0,065 \right)60} \right)} = 3,45\ \)

\({\overline{N}}_{s_{2}}\left( t \right) = \frac{\rho_{2}}{1 - \rho}\left( 1 + \frac{\delta}{\gamma}\frac{\rho_{1}}{1 - \rho_{1}} \right)\left( 1 - e^{- \alpha t} \right) = \frac{0,24}{1 - 0,93}\left( 1 + \frac{0,65}{0,23}\ \frac{0,69}{1 - 0,69} \right)\left( 1 - \frac{1}{e^{\left( 0,94 - 0,88 \right)60}} \right) = 11\ \)

\({\overline{T}}_{s_{2}}\left( t \right) = \frac{\rho_{2}}{1 - \rho}\left( 1 + \frac{\delta}{\gamma}\frac{\rho_{1}}{1 - \rho_{1}} \right)\left( 1 - e^{- \alpha t} \right)\frac{1}{\lambda + \alpha e^{- \alpha t}} = \frac{0,244}{1 - 0,93}\left( 1 + \frac{0,69}{1 - 0,69} \right)\left( 1 - e^{- \left( 0,94 - 0,88 \right)60} \right)\frac{1}{0,88 + \left( 0,94 - 0,88 \right)e^{- \left( 0,94 - 0,88 \right)60}} = 13\)

\({\overline{N}}_{f1}\left( t \right) = \frac{{\rho_{1}}^{2} - ({\rho_{1}}^{2} - \rho_{1} + 1)e^{- \alpha t}}{1 - \rho_{1}} = \frac{{(0,69)}^{2} - \left( \left( 0,69 \right)^{2} - 0,69 + 1 \right)e^{- \left( 0,94 - 0,65 \right)60}}{1 - 0,69} = 1\ \ \)

\({\overline{T}}_{f1}\left( t \right) = \frac{{\rho_{1}}^{2} - ({\rho_{1}}^{2} - \rho_{1} + 1)e^{- \alpha t}}{(1 - \rho_{1})(\lambda + \alpha e^{- \alpha t})} = \frac{{(0,69)}^{2} - \left( \left( 0,69 \right)^{2} - 0,69 + 1 \right)e^{- \left( 0,94 - 0,65 \right)60}}{\left( 1 - 0,69 \right)\left( 0,65 + \left( 0,94 - 0,65 \right)e^{- \left( 0,94 - 0,65 \right)60} \right)} = \ 0,42\ \)

\({\overline{N}}_{f_{2}} = \frac{\rho_{2}\left( 1 + \frac{\delta}{\gamma}\frac{\rho_{1}}{(1 - \rho_{1})} \right)\left( 1 - e^{- \alpha t} \right) - (1 - \rho)\rho}{(1 - \rho)\lambda} = \frac{0,24\left( 1 + \frac{0,69}{1 - 0,69} \right)\left( 1 - e^{- \left( 0,94 - 0,88 \right)60} \right) - \left( 1 - 0,93 \right)0,93}{\left( 1 - 0,93 \right)0,88} = \frac{0,687838}{0,07} = 11\ \)

\[{\overline{T}}_{f_{2}}\left( t \right) = \frac{\rho_{2}\left( 1 + \frac{\delta}{\gamma}\frac{\rho_{1}}{(1 - \rho_{1})} \right)\left( 1 - e^{- \alpha t} \right) - (1 - \rho)\rho}{(1 - \rho)\lambda} = \frac{0,24\left( 1 + \frac{0,69}{1 - 0,69} \right)\left( 1 - e^{- \left( 0,94 - 0,88 \right)60} \right) - \left( 1 - 0,93 \right)0,93}{\left( 1 - 0,93 \right)0,88} = 8,64\]

**Graphical representation of performance parameters a function of time**

![](65a293d863222_media/media/image2.png)

Fig.2. Performance parameters a function of time

**Discussion**

The results obtained through this numerical example reflect the differences in the time spent by priority and non-priority customers in the queue. The same is true of their numbers in the system in a given time interval t. In contrast to the results expressed by (Mabela and al, 2021) in the transient regime without priority and Lama steady-state with top priority, this model has the advantage of reducing waiting time for priority customers.

**CONCLUSION**

At the end of this reflection on the performance measurements of a queuing system with absolute priority in a *transient state*, model M/M/1, the main question was to determine the performance parameters of non-priority customers in a transient state. To achieve this, we relied on Kolmogorov equations, state probabilities, Laplace transforms, transition matrices and LIttle's theorem. In the existing literature, nothing similar has been established in a transitional regime with absolute priority. This is what marks the originality and substantial contribution of this reflection in the field of queues. The same reflection will be carried out in a vague environment in the long run.

# References

Baudouin Adia et all, (2022), Analysis of the performance measures of a non-Markovian Fuzzy Queue via Fuzzy Laplace Transforms Method, in *Journal of Computing Research and Innovation (JCRINN)*, vol. 7 no 2 (2022), 304-315.

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