Distribution of waiting times

The exponential distribution with parameter X is the distribution of waiting times ( distance in time) between events which take place at a mean rate of X. It is also the distribution of distances between features which have a uniform probability of occurrence (Poisson process), such as the simplest model of faults on a map. The gamma distribution with parameter n and X l, where n is an integer is the distribution of the waiting time between the first and the nth successive events in a Poisson process. Alternatively, the distribution /(t), such as... 

Here we investigate the distribution of waiting times for the event that at least one channel is open for a certain time. We also calculate the probability for a single... 

For every individual simulation process, i.e. for each size of channel cluster, the waiting time for the event at least one channel is open for f/ was detected this was repeated 400 times. Then the distribution of waiting times, w (t) was estimated from binned data. An example of a histogram obtained is seen in Fig. 11.3 for a cluster that contains three channels, with ma = 1.9 ms and ms = 5 ms. 

One of the aspects associated with vacancy-mediated diffusion that differentiates it from the hopping mechanism on this surface is the long waiting time between consecutive jumps. An example of the distribution of waiting times has been plotted for both In and Pd in Fig. 5. 

Figure 5 Distributions of waiting times between subsequent jumps in Cu(0 01), measured for (a) In at 320 K and (b) Pd at 335 K. Both fits are pure exponentials. The time constant t is shown in the graphs for both distributions.

In the special case where there are only two sites, the CTRW procedure, supplemented by the Poisson assumption of Eq. (69), yields the Pauli master equation of Eq. (2). This means that the Pauli master equation is compatible with a random picture, where a particle with erratic motion jumps back and forth from the one to the other state, with a condition of persistence expressed by the exponential waiting time distribution of Eq. (69). Recent fast technological advances are allowing us to observe in mesoscopic systems analogous intermittent properties, with distinct nonexponential distribution of waiting times. This is the reason why in this section we focus our attention on how to derive a v(/(t) with a non-Poisson character. 

In general, three steps are required to create trajectories by means of RW simulations,76,84 (i) determination of an initial orientation/position so that the equilibrium distribution is obeyed, (ii) random selection of a waiting time tw between two subsequent jumps from a suitable distribution and (iii) calculation of the new orientation/position after the jump. After step (i), the steps (ii) and (iii) are performed alternately until a trajectory of sufficient length in time is obtained. While the time scale of the motion is determined by the distribution of waiting times g(tw) in step (ii), the geometry of the motion comes into play in step (iii). For example, the new orientation 0i+ after a -degree-rotational jump of a C-2H bond can be obtained from the old orientation 0, according to... 

In glassy polymeric media, electronic transport can be phenomenologically thought of as a series of discrete steps characterized by a distribution of waiting times i /(t). When the distribution extends into the time scale of observation, the mobility itself will always appear to be time dependent. However, when the distribution does not extend into the time scale of observation, mobility can be characterized by an averaged value for most of the transit event, even though it exhibits thermalization at early times, which may be resolvable under certain experimental conditions (26, 27). Several more microscopically detailed pictures can correspond to this phenomenological description of electronic transport. 

The probability of selecting a particular list is proportional to the rate associated with a given list and the number of events on that list, S. Once a list is selected, an event is selected from the list, at random, and the lattice updated. The distribution of waiting times between subsequent events, St, follows a Poisson distribution and is given by... 

L(t) will denote the load, or number of jobs in the system, waiting or being served, at time t 0. For each model considered in this section, the distribution of L(t) approaches a limit as t — and the limit is referred to as the steady-state distribution of the number in the system. (As we said in Subsection 3.3, we do not distinguish carefully between steady-state and Kmiting distributions.) We denote by L a random variable whose distribution is this steady-state distribution. Let W denote the time-in-system or flow time of the nth arrival to the system. Then, for the models considered here, the distribution of W also has a limit as n —> , which we call the steady-state flow-time distribution. We denote by W a random variable with this distribution. In the same way we can define a random variable Wg whose distribution is the steady-state distribution of waiting time. Flow time and waiting time differ only in that the latter does not include service time. Most of the results given here concern the expected values of the random variables L, W, and Wg for various queues. These... 

Disorder was introduced into this system by postulating a distribution of waiting times. A complementary extension of the theory may be made by considering a distribution of jump distances. It may be shown that, as a consequence of the central limit theorem, provided the single-step probability density function has a finite second moment, Gaussian diffusion is guaranteed. If this condition is not satisfied, however, then Eq. (105) must be replaced by... 

Let us come back now to the question of increasing heterogeneity in the local mobilities upon decreasing the temperature. We have already identified a tendency for immediate back jumps after one torsional transition as the reason for the different temperature dependencies of the mean waiting time between torsional transitions (twait) and the torsional autocorrelation time ttacf)- In a homogeneous system, where every chemically identical torsion shows identical dynamics on the time scales of observation the probability distribution of waiting times should be... 

Figure 9. Distribution of waiting times for a total of 10 torsional transitions per dihedral degree of freedom to occur plotted versus 10 times t/(twait)- The thick line is the Poisson distribution. Upon lowering the temperature the deviation from the Poisson distribution increases.

Another interesting property of the formerly defined stochastic process is the distribution of waiting times between consecutive events. Let p(r)dz be the probability that the next event takes place in the infinitesimal interval [r, x + dx]. 

We need an algorithm to simulate realizations of the production-decay process described in the previous section. As before, let us start by computing the distribution of waiting times. Assume that the system has n molecules at time t and let p z)dr be the probability that the next event (either the synthesis of a new molecule of the degradation of one of the existing ones) takes place in the interval [t + z,t + T + dr]. To compute this probability distribution we need to take into consideration that not only p r)dt is the probability that one event occurs in the interval [t + x,t + x + dr], but also that none of them happens in the interval [t, t + x]. That is... 

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