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1. Assumptions and notations

Let us use the following notations. When $\mathbf{X}$ is some random variable, its pdf (probability density function) will be noted $x$ , its cdf (cumulative distribution function) will be noted $X$ and its Laplace transform will be noted $\widehat{x}$ . In other words $x\left( u\right) \, du=Pr\left\{ \mathbf{X}\in \left[ u,\, u+du\right] \right\}$ , $X\left( u\right) =\int _{0}^{u}x\left( t\right) \, dt$ , and $\widehat{x}\left( z\right) =\int _{0}^{u}x\left( t\right) \, \exp \left( -zt\right) \, dt$ .

Let $\mathbf{A}$ be the random variable ``inter-arrival duration'' and $\lambda =1/\mathbf{E}\left( \mathbf{A} \right)$ , so that: $a\left( t\right) =\lambda \, \exp \left( -\lambda \, t\right)$ . Let $\mathbf{B}$ be the random variable ``service duration'' and $\mu =1/\mathbf{E}\left( \mathbf{B} \right)$ . This last value is the average throughput during busy periods, and the quantity $\rho =\lambda /\mu$ is the load of the system. It is assumed that no infinite queue happens (therefore $\rho <1$ ).

The random variable $\mathbf{R}$ will denote the sojourn time of a customer in the system, $\mathbf{W}_{0}$ its (perhaps being null) waiting time, and $\mathbf{W}_{1}$ its conditional waiting time, i.e. the waiting time knowing that the customer will wait. All the preceding random variables, and especially the service duration $\mathbf{B}$ , are ``individual variables'', i.e. can be perceived as the result of a process that picks a customer by uniform sorting over their order of arrival, and notes the value associated to this customer.

Another process of selection can be used, that picks a customer by uniform sorting over the instants of observation (that sorting being repeated until a client is eventually served), leading to "temporal" variables. Let $\beta$ be the pdf of $\widetilde{\mathbf{B}}$ (the temporal service duration). Relation $\beta \left( t\right) =\mu \, t\, b\left( t\right)$ is obvious and gives $\mathbf{E}\left( \widetilde{\mathbf{B}} \right) =\left( 1/\mu \right) +\mu \, \mathbf{var}\left( \mathbf{B} \right)$ , while the variance of $\widetilde{\mathbf{B}}$ includes the third moment of $\mathbf{B}$ .

By definition, a customer arriving in a non-empty system finds another customer that is currently being served (and maybe other ones, being waiting). The duration that the incoming customer has to wait until the served customer leaves the system is called the "residual service time", and will be noted $\mathbf{C}$ . The pdf of $\mathbf{C}$ is well-known to be $c\left( t\right) \, dt=\mu \, dt\, \int _{u=t}^{u=\infty }b\left( u\right) du$ , and we have $\mathbf{E}\left( \mathbf{C} \right) =\frac{1}{2}\mathbf{E}\left( \widetilde{\mathbf{B}} \right)$ , while $\mathbf{var}\left( \mathbf{C} \right)$ involves up to the third moment of $\mathbf{B}$ . In what follows, the notations $\overline{x}$ and $\sigma ^{2}$ will always refer to $\mathbf{E}\left( \mathbf{C} \right)$ and $\mathbf{var}\left( \mathbf{C} \right)$ .

Next: 2. Inversion of the Up: Computing Stochastical Bounds for Previous: 2.1 Introduction

douillet@cnam.fr
2000-02-15