On the opposite, in a scenario with an ideal MAC protocol, this limiting probability does not depend on [8].
Finally, note that the limiting probability (17) equals to zero for any unbalanced configuration with at least one cluster with one sensor.
where π i is the limiting probability that the system is in state i and S is the state space of the Markov chain.
The solutions to some fundamental problems in non-linear systems such as bifurcations, steady state solutions, limiting probability distribution of steady state solutions, domains of attraction and basin boundaries are obtained.
The limiting probability that a node is in zone 1 at any time is given by d 1 d 1 + d 0, where d0 is the length of zone 0. Figure 2 Markov chain model for zone transitions.
In this work we will considered the following practically important conditions: Then the limiting probability function is and the probability where and m = 0, 1, 2,...; p0,0 ≡ p0.
As for the initial phase the limiting probability of a minor outbreak lim n π n is different from the SIR model and equal to as mentioned above.
As is well-known (Diekmann et al. 2012), the offspring distribution is then geometric, and the minor outbreak probability equals / κ and /(κ(1− ε0)), respectively, but, since ε0 may be made arbitrarily small, the limiting probability of a minor outbreak equals / κ.
If p0 > 0, we can define the zero-truncated limiting probability distribution as (30) where m = 1, 2, 3... Using this formula, we can prove the following useful approximation: If a → 0+; θ → 1; b > 0, then (31) where B(b+1, m) is the Beta function.
Proportional hazard models and their extension to include random effects describe the hazard function of each individual λ(t) (i.e., its limiting probability of dying at time t, given it is still alive just prior to t) as the product of a baseline hazard function and a positive (exponential) function of explanatory covariates.
