Cumulative Reward Distribution

The tool calculates the distribution of the cumulative reward $CR(t)$ in the following cases: (1) when the random variable $CR(t)$ is not bounded, (2) when the random variable is bounded by $L_{bound}$ and $U_{bound}$.

One measure that can be obtained with this algorithm is the transient queue length distribution (and from that, the packet loss ratio as a function of time). Let $b(t)$ be the number of packets stored in a limited buffer and $M = \{ M(t), t \geq 0 \}$ the process that models the traffic source (Markov reward model). It is not difficult to see that if $C$ is the channel capacity and if we associate to state $s$ a rate reward $r_{c(s)} = \lambda_s - C$, then the random variable $CR(t)$ is equal to the buffer size at $t$ provided that $CR(t)$ is limited between $0$ and the maximum buffer size $B$.

The interface for this method is presented in Figure .

Figure: Cumulative Reward Distribution Interface

The input parameters: , , , and Precision have to be specified as in the Uniformization technique. To choose the Reward Name, the user have to click on the little button on the right hand side of the box with the Reward Name. Then, another window with the name of all rewards, specified by the user will appear and the user will be able to select one of them. The probabilities will be computed for the Reward Levels given. For example, if we give as reward levels $1000$ and $1500$, the tool outputs will be $P[CR(t) > 1000]$ and $P[CR(t) > 1500]$ plus the probabilities calculated for the lower and upper bound, provided that bounds are given for the reward $CR(t)$.