First-passage time statistics of Markov gamma processes
Introduction
Within the framework of the stochastic signal processing theory for communications, the First-Passage Time (FPT) problems are aimed to provide statistical characterization (mean, variance, cumulants, distribution, etc.) of the time intervals required for a given stochastic process to attain, for the first time, certain threshold boundary associated to performance, safety, reliability, etc. of the system. From the practical point of view the FPT statistics are essential to trigger or halt some actions/operations necessary to keep a predetermined working regime of the system. From the theoretical point of view such statistics are the departure point to get other performance statistics.
Besides of its long-time existence, the First-Passage Time (FPT) problems have not been solved yet completely and still are [17], [22], [9], [21] an attractive research topic widely applied in different fields of the stochastic signal processing in communications. In the Phase Locked Loop (PLL) theory the time to first slip or first loss of PLL synchronization is a classical FPT problem [12]. The link reliability in wireless communication channels [27] is also regarded as a FPT problem. The level crossing-based approaches for handoff initiation [25] are also a set of problems which clearly require the FPT statistics. The FPT statistics are found in the sequential analysis of random walks [26], [7] and this analysis is a promising tool in the study of detection techniques in Cognitive Radio and Sensor Networks [11]. The synchronization of block codes [3], the estimated time to get an empty queue [6], an overflowed buffer [24] or the last active sensor [13] are also communications problems that use the FPT approach. This short selection of examples shows the relevance of the FPT statistics as the corner stone for the analysis and processing of signals in communications.
It is worth mentioning that gamma distributed stochastic processes can be found in many signals involved in the information transmission scenario: the instantaneous power of Nakagami fading signals [23] (which underlie in several of the above mentioned applications), the fade and non-fade duration distribution [20], the call duration [2], queuing models [15], the number of information bits in a source for coding [6], etc.
Tractable results for the FPT problems can be achieved essentially under the framework of the theory of Markov processes, and so this is the approach used in this paper. When the statistic of interest is the PDF (stationary or transient) the First-Passage Time problem can be formulated as follows.
Let the behavior of a certain system be characterized by a continuous Markov process described by a one dimensional Stochastic Differential Equation (SDE) and the stable state solution of the SDE be given by a predefined (for our case gamma) PDF, see [18] for instance. The gamma distributed random process satisfies the initial condition , , where w and z are respectively the lower and the upper boundaries. The goal is to find the PDF of the First-Passage Time required for the process to attain the boundary z for . In order to simplify notations hereafter .
There are several methods to solve this problem [2], [4] but here, the approach based on the first Pontryagin equation (see [1]) is followed. Although the straightforward application of the Pontryagin equation to find for Markov gamma processes does not yield a closed form solution, it does help to build a bypass accurate solution that allows to evaluate an arbitrary finite number of cumulants of the First-Passage Time. With these at hand it is possible to “reconstruct” the PDF of the FPT using, for example, the well-known orthogonal series representation (see [19], [20] and the references therein), which will be presented in the following.
In Section 2 the cumulants of the First-Passage Time for a gamma process are analyzed departing from the first Pontryagin equation. In Section 3 orthogonal series are used to reconstruct through the cumulants; a simulated version of (based on the SDE) allows to find that the First-Passage Time distribution can be represented with a high degree of accuracy as a gamma distribution and this is validated using the routine chi-squared goodness of fit statistical test. In Section 4 complementary remarks are discussed and some useful asymptotic expressions for the cumulants are also presented. An application of the proposed approach and some final remarks are presented in Section 5.
Section snippets
Cumulants of the first-passage time problem
Since the classical work of L.S. Pontryagin, it is widely recognized that the first-passage time distribution can be theoretically analyzed using the so-called first Pontryagin equation1 [17]:where is the unknown probability of attaining the upper boundary z at the time , being that the initial
Reconstruction and simulation of
The reconstruction of the structure of might be achieved by decomposing it through an orthogonal series expansion [16]. For the case under study we apply the Laguerre polynomial series given by (see [18], [16], [14])where the coefficients cn areanddenote the Laguerre polynomials.3
Results
For a convergence analysis of Eqs. (20), (21) one might compare the terms needed to get a relaxed precision (say 95%) and the whole terms of the series (100% precision). For Fig. 1a, the 95% precision is achieved with just three terms (per each summation sign on the correspondent series). For Fig. 1d, eight terms are required. In general the number of required terms increases when the difference between x and z grows (which is intuitively natural), but it never exceeds 12 for all cases under
An application and final remarks
An application of this methodology is introduced by considering a Nakagami fading channel taken out from [20] with m=0.5, , a Jakes-type Power Spectral Density and a maximum Doppler shift frequency fd=50 Hz. For a threshold level of −4 dB [20], the channel has a Fade Duration Distribution (FDD) characterized by a gamma distribution with parameters and [20]. A Markov gamma distributed process with the same parameters is generated by means of the SDE method. For both experimental
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