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First-passage time statistics of Markov gamma processes
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Journal of the Franklin Institute 350 (2013) 1686–1696 www.elsevier.com/locate/jfranklin
First-passage time statistics of Markov gamma processes Fernando Ramos-Alarcón, Valeri Kontorovichn Electrical Engineering Department, Telecommunications Section, CINVESTAV-IPN., Av. IPN 2508, Colonia San Pedro Zacatenco, C.P. 07360 Mexico City, Mexico Received 12 October 2012; received in revised form 2 February 2013; accepted 19 April 2013 Available online 4 May 2013
Abstract The analysis of the First-Passage Time (FPT) statistics has a relevant importance either in theoretical or practical sense for the signal processing design in communications. This paper introduces a simple approach that allows a rather accurate calculation of an arbitrary number of cumulants of the Probability Density Function (PDF) of the FPT for the relevant case of Markov gamma processes. & 2013 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.
1. Introduction Within the framework of the stochastic signal processing theory for communications, the FirstPassage 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. n
Corresponding author. E-mail address: [email protected] (V. Kontorovich).
0016-0032/$32.00 & 2013 The Franklin Institute. Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jfranklin.2013.04.013
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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 ξðtÞ satisfies the initial condition ξð0Þ ¼ x0 ¼ x, x∈ðw; zÞ, where w and z are respectively the lower and the upper boundaries. The goal is to find the PDF W w;z ðx; tÞ of the First-Passage Time required for the process to attain the boundary z for t40. In order to simplify notations hereafter W w;z ðx; tÞ ¼ Wðx; tÞ. 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 Wðx; tÞ 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 Wðx; tÞ through the cumulants; a simulated version of Wðx; tÞ (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. 2. 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]: 1 ∂2 ∂ ∂ BðxÞ 2 ½Pðx; tÞ þ AðxÞ ½Pðx; tÞ ¼ ½Pðx; tÞ; 2 ∂x ∂t ∂x
ð1Þ
Actually Eq. (1) is a particular case of the backward Kolmogorov equation, however it was used first by Pontryagin in the context of FPT problems. 1
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where Pðx; tÞ is the unknown probability of attaining the upper boundary z at the time t40, being that the initial condition at t ¼ 0 is Pðx; 0Þ ¼ 0 for woxoz, A(x) and B(x) are respectively the drift and diffusion coefficients that in the following are considered, in the Stratonovich form [18]: 2xβ β x ; AðxÞ ¼ α− ; BðxÞ ¼ ð2Þ τ τ β where α40, β40 and τ is the correlation time.2 By taking the time derivative of Pðx; tÞ, it is easy to get an equation for the PDF of the FPT, Wðx; tÞ. Then, multiplying the resulting equation by expðiωtÞ and integrating the latter from 0 to ∞, we get an ordinary differential equation in terms of the characteristic function θðx; iωÞ: 1 d2 d BðxÞ 2 ½θðx; iωÞ þ AðxÞ ½θðx; iωÞ−iωθðx; iωÞ ¼ 0: 2 dx dx
ð3Þ
In order to calculate the cumulants (κ k ) of the FPT, recall that the log-characteristic function is ðiωÞk κ k ðxÞ: k ¼ 1 k! ∞
ln½θðx; iωÞ ¼ ∑
ð4Þ
The boundaries can be either absorbing or reflecting [24,18]. The presence of an absorbing boundary means that once the random process reaches the absorbing boundary it remains there indefinitely and the presence of a reflecting boundary means that once the random process reaches the reflecting boundary (derivative is zero there) it “bounces” in the opposite direction in the next time instant. Thus, assuming an absorbing boundary at z and a reflecting boundary at w ¼ 0, the boundary conditions in terms of the characteristic function become d½θðx; iωÞ θðz; iωÞ ¼ 1; ¼ 0: ð5Þ dx x¼w Eq. (4) shows that the log-characteristic function can be expressed in terms of an infinite set of cumulants [18]. However, if one considers the so-called “curtosis approximation”, it is possible to keep just the first four cumulants (see, for example [18,16], etc.), as the influence of cumulants for the PDF reconstruction diminish as the order of the cumulants increase. From Eq. (4) the curtosis approximation would be 4 ðiωÞk θðx; iωÞ≈exp ∑ κk ðxÞ ; ð6Þ k ¼ 1 k! where κk denotes the k-th cumulant. Substituting Eq. (6) in Eqs. (3) and (5) and regrouping the equations around the terms iωk (ignoring all iωk with k44) one gets finally a set of recurrent linear equations for the first four cumulants of the First-Passage Time: α 1 τ − κ_ 1 ¼ ; κ1 ðzÞ ¼ 0; κ_ 1 ð0Þ ¼ 0 κ€ 1 þ ð7Þ x β βx α 1 − κ_ 2 ¼ −2_κ 21 ; κ 2 ðzÞ ¼ 0; κ_ 2 ð0Þ ¼ 0 κ€ 2 þ ð8Þ x β
2 Note that the stationary solution of Eq. (1) with drift and diffusion coefficients (2) corresponds precisely to a gamma process which is the subject of analysis in this paper. This can be easily verified considering (7.4) in [18] and noting that all improper integrals there have to be considered in the principal value sense.
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α 1 − κ_ 3 ¼ −6_κ 1 κ_ 2 ; κ€ 3 þ x β κ€ 4 þ
κ3 ðzÞ ¼ 0; κ_ 3 ð0Þ ¼ 0
α 1 − κ_ 4 ¼ −8_κ 1 κ_ 3 −6_κ 22 ; x β
κ4 ðzÞ ¼ 0; κ_ 4 ð0Þ ¼ 0:
1689
ð9Þ ð10Þ
These equations can be solved either numerically or analytically through standard and wellknown techniques. Examples of the analytical solutions are presented in the Appendix. The cumulants of interest are particularly the first four Eqs. (20)–(23) and can be evaluated a priori just with knowledge of the parameters of the input Markov gamma process (2) and the boundary conditions. 3. Reconstruction and simulation of Wðx; tÞ The reconstruction of the structure of Wðx; tÞ 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]) ∞
Wðx; tÞ ¼ t a−1 expð−tÞ ∑ cn La−1 n ðtÞ; n¼0
where the coefficients cn are Z ∞ n! cn ¼ La−1 n ðtÞWðx; tÞ dt; Γðn þ aÞ 0
ð11Þ
ð12Þ
and Lna−1 ðtÞ ¼ expðtÞ
t −ða−1Þ dn ðexpð−tÞt nþa−1 Þ; n! dt n
ð13Þ
denote the Laguerre polynomials.3 Evaluation of the coefficients cn [14] requires knowledge of the correspondent initial moments, mn, which can also be expressed in terms of the cumulants (calculated in the previous section) through the well-known relations between moments and cumulants. By making c1 ¼ c2 ¼ 0 the parameters a and b can be evaluated simultaneously and recalling that the first cumulant is the mean and the second cumulant is the variance [16], one easily gets a¼
κ21 ; κ2
b¼
κ2 : κ1
ð14Þ
For each different number of terms, n, in the series (11) one can get different variants for a reconstructed Wðx; tÞ. Each reconstructed Wðx; tÞ will be regarded as a hypothetical distribution for an experimental version of Wðx; tÞ (which is obtained by simulating the input Markov gamma process through the SDE approach [18]) and by means of hypothesis testing it is possible to examine how good the hypothetical distributions fit the experimental Wðx; tÞ (in statistical sense). Three hypothetical (reconstructed) distributions were processed using the series (11) with n¼ 2, 3
As it was pointed out at [18], etc. applying more terms at Eq. (11) not always tends to a better accuracy of this representation for experimental data.
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Fig. 1. Simulation and approximation of Wðx; tÞ for different parameters x, z and α.
n¼ 3 and n¼ 4. Note that the hypothetical distribution using the series (11) with n ¼ 2 requires only c0 ¼ 1=ΓðaÞ and La−1 0 ðtÞ ¼ 1, as long as c1 ¼ c2 ¼ 0, and after a simple change of variable yields a compact closed form representation for Wðx; tÞ which is precisely the gamma distribution : W Γ ðtÞ ¼
t t a−1 ; a exp − b ΓðaÞb
ð15Þ
with parameters a and b given by Eq. (14). The rigorous way to examine the statistical fitness of the hypothetical distributions against the experimental, in our case the simulated, Wðx; tÞ, is to perform the well-known standard statistical goodness of fit tests [10]. In our case we applied the Chi Square, χ 2 , statistical test. The three cases under consideration showed a closed match in relation to the experimental distribution. The case for n ¼ 2 showed and accuracy always greater than 90%. The cases for n¼ 3 and n ¼ 4 were (in that order) always slightly below the case for n ¼ 2, in this way one finds that Wðx; tÞ≈W Γ ðtÞ which is a novel theoretical result. As long as the result for n ¼ 3 was practically the same (graphically and statistically) as that for n ¼ 4, in the following is omitted for simplicity. Fig. 1 shows the comparison of the experimental distribution against the two hypothetical distributions,
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Table 1 Theoretical (with 12 terms in the series) and experimental cumulants. Fig. 1a
κ1 κ2
Fig. 1b
Fig. 1c
Fig. 1d
Theo
Sim
Theo
Sim
Theo
Sim
Theo
Sim
1.45 2.06
1.45 2.02
0.78 0.44
0.78 0.43
1.31 1.23
1.31 1.25
0.99 0.67
0.99 0.68
series (11) with n ¼ 2 and n ¼ 4,4 showing the result of the Chi Square test only for n ¼ 2. Being that Wðx; tÞ≈W Γ ðtÞ, in the following the discussion is focused just on the first two cumulants. Table 1 shows the theoretical equations (20), (21) (using 12 terms in the series representation) and the experimental cumulants related to the plots in Fig. 1.
4. Results For a convergence analysis of Eqs. (20) and (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 analysis. Based on the above, by truncating the series (20) up to the second term one gets for κ1 : 1 0 13 20 z x 6 B C B 7 τ 6B β C B β C C7 : −x@1 þ ð16Þ z @1 þ κ1 ðxÞ ¼ 4 A A 2ðα þ 1Þ 2ðα þ 1Þ 5 αβ Formula (16) shows that β is a scaling factor for the initial point (x) and the final boundary (z). The parameter α drives the shape of the input gamma process and Eq. (16) shows that κ1 is inverse-like proportional to α. This can also be appreciated from the plots of Fig. 1 and Table 1. The same is true for κ2 . This follows from the fact that, while α is growing the initial PDF tends to delta-function. Another interesting issue, which can be intuitively explained, is the proportionality of the cumulants (20)–(23) (not always in a linear way) to non-zero correlation time: if the latter is growing all cumulants shall grow as well and mainly in a non-linear way, as the time scale of the initial gamma process is increasing. It is also interesting to mention that the parameter a drives the shape and the entropy of the PDF Wðx; tÞ≈W Γ ðtÞ and so Table 2 shows its behavior for different scenarios of the problem: α, x, and z with β ¼ 1. Table 2 and Fig. 1 allow to see that the behavior of Wðx; tÞ≈W Γ ðtÞ does not change drastically and it is “centered” around an exponential distribution (a¼ 1). This suggests an exponential bound for Wðx; tÞ or at least for its tails. The hypothetical distribution with n¼ 2 requires two initial moments m1 and m2 that are expressed through the first two cumulants and n¼ 4 requires the first four moments that are expressed through the first four cumulants. 4
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Actually as it is well known [16,8], when the first cumulant κ1 is fixed, the distribution that maximizes the entropy is an exponential one. The same result comes out when one approximates the tails of Wðx; tÞ through the cumulant-based tilted Edgeworth expansion developed by [4] in Table 2 Behavior of Wðx; tÞ≈W Γ ðtÞ through the parameter a for different settings α, x and z with β ¼ 1. α ¼ 3; x ¼ 0:1
α ¼ 3; x ¼ 0:5
α ¼ 1; x ¼ 0:1
α ¼ 1; x ¼ 0:5
z
a
z
a
z
a
z
a
0.15 0.2 0.5 1 2.5 3 6 8 12 15
0.78 1.3 2.51 2.86 2.51 2.33 1.48 1.18 0.89 0.73
– – 0.7 1 2.5 3 6 8 12 15
– – 0.62 1.22 1.96 1.94 1.42 1.16 0.89 0.73
0.2 0.3 0.5 1 2.5 3 6 8 12 15
0.66 0.98 1.27 1.45 1.31 1.25 1.03 0.96 0.79 0.62
– – 0.7 1 2.5 3 6 8 12 15
– – 0.33 0.65 1.10 1.11 1.02 0.96 0.79 0.62
Fig. 2. Exponential approximation of the tails for different Wðx; tÞ.
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the formula (4.45). The plots in Fig. 2 show the exponential bound for the tails of Wðx; tÞ for the same set of plots in Fig. 1. Finally, for asymptotically small x and z ðx⪡1; z⪡1Þ the cumulants (20) and (21) are given by τ z x exp − −exp − κ1 ðxÞ ¼ ; ð17Þ αβ β β 1 x z n n ð−1Þ Γ 2 þ n; − ð−1Þ Γ 2 þ n; − ∞ ∞ 2 τ 2 B β β C B∑ C: − ∑ κ2 ðxÞ ¼ @ n!ð1 þ n þ αÞ n!ð1 þ n þ αÞ A β α n¼0 n¼0 0
ð18Þ
The asymptotic expressions (17), (18) show that even for small “span” of the boundaries, only the first cumulant is proportional to τ. As the cumulants grow, (21)–(23), their dependence on τ becomes more non-linear. 5. An application and final remarks An application of this methodology is introduced by considering a Nakagami fading channel taken out from [20] with m ¼ 0.5, Ω ¼ 1, a Jakes-type Power Spectral Density and a maximum Doppler shift frequency fd ¼ 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 α ¼ 0:881 and β ¼ 0:258 [20]. A Markov gamma distributed process with the same parameters is generated by means of the SDE method. For both experimental processes the FPT PDF is extracted. Using the theoretical approach the first two cumulants are evaluated and with them the parameters a and b for the corresponding gamma distribution are calculated. The three distributions are compared in Fig. 3. Note that the FDD (with gamma distribution) is not rigorously a Markov process. In order to make the comparison as fair as possible the correlation time of the FDD is extracted and
Fig. 3. Application of the approach for Wðx; tÞ in wireless communication channels.
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installed in both the theoretical approach and the SDE simulation. Fig. 3 shows that there are some small differences only on the extreme left side of the distribution which is obviously due to the fact that the FDD is not strictly a Markov process. In general the developed Markovian approach can also be useful out of Markovian framework, as it has been noticed before for other problems (see [1] and the references therein). Let us underline the main differences between the proposed approach and the existing results. Traditional results for the FPT PDF are given in integral form [22]. Applied results in communications provide either some moments of the FPT [6,24,13] or even a recursive form of the FPT PDF [13]. Based on the state transition diagram of the source problem, the mean of the FPT can be obtained [3]. Perhaps the only result that addresses the FPT PDF for a Rayleigh process is presented in [19] but the results are given in integral form. The proposed approach is more attractive and original for several reasons: it is valid for any problem or scenario where the process under study has a gamma distribution with known parameters, it provides series for an arbitrary number of cumulants of the FPT (for higher order cumulants the complexity of the series grows) and gives a simple closed form representation for the PDF of the FPT (which requires just two cumulants). To the best of our knowledge closed forms for the FPT distributions are mainly provided for particular cases, as it is the case of the well-known inverse Gaussian PDF [7] used to describe the distribution of the time a Brownian motion with positive drift takes to reach for the first time a fixed positive level. The Wald distribution [26] is a particular case of the inverse Gaussian and describes the FPT when the sample size tends to infinity in the framework of a sequential analysis. This result allows to back up the finding that Wðx; tÞ≈W Γ ðtÞ through the following reasoning. The Gamma and the inverse Gaussian distribution are two-parameter distributions, they are left skewed with support on ð0; ∞Þ and the most relevant fact in this argument is that through the proper choice of parameter values they can be practically indistinguishable. So, it is interesting to notice that the inverse Gaussian and very close to it the Gamma PDF characterize adequately the FPT distribution of two different processes described by means of SDE. The truncated series and the asymptotic expressions of the cumulants allow seeing how the different parameters α, β and τ and the initial and final boundaries x, z affect the behavior of the cumulants. Moreover, for signal processing applications the different parameters required for the theoretical analysis (α, β, τ and the boundaries x, z) can be extracted from any concrete scenario in a straightforward way. The results presented in this paper are valid for all the particular cases associated to the gamma distribution, i.e. exponential, Erlang, etc. and are not reported neither in the literature or in classic well known handbooks like [5]. Appendix A Using the change of variable κ_ n -yn , it is possible to put the set of Eqs. (7)–(10) into the general form: y_ þ PðxÞy ¼ QðxÞ:
ð19Þ
Solving the equations in the well-known way, the following expressions for the first four cumulants are given by I 1 I 1 ! ∞ ∞ τΓðαÞ z x z ∑ C1 −x ∑ C 1 ð20Þ κ 1 ðxÞ ¼ β β β I1 I1
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τΓðαÞ κ2 ðxÞ ¼ 2 β
τΓðαÞ κ3 ðxÞ ¼ 12 β
I 3 I 3 ! ∞ z x z2 ∑ C 2 −x2 ∑ C2 β β I3 I3
2
τΓðαÞ κ4 ðxÞ ¼ 24 β
∞
I 5 I 5 ! ∞ z x 3 z ∑ C3 −x ∑ C3 β β I5 I5
3
3
∞
1695
ð21Þ
ð22Þ
I 7 I 7 I 7 I 7 ! ∞ ∞ ∞ z z x x 4 4 4 4z ∑ C4 þ z ∑ C5 −4x ∑ C 4 −x ∑ C 5 ; β β β β I7 I7 I7 I7
4
4
∞
ð23Þ where the following notation was used: ∞
∞
∑¼ ∑ In
∞
∞
∑ ⋯ ∑ ;
i1 ¼ 0 i2 ¼ 0
in ¼ 0
I m ¼ i1 þ i2 þ ⋯ þ im
ð24Þ
and C1, C2, C3, C4 and C5 are given by C1 ¼
1 Γð1 þ α þ i1 Þð1 þ i1 Þ
ð25Þ
C2 ¼ C1
Γð1 þ α þ I 2 Þð1 þ i1 Þ Γð2 þ α þ I 3 ÞΓð1 þ α þ i2 Þð2 þ I 3 Þ
ð26Þ
C3 ¼ C2
Γð2 þ α þ I 4 Þð2 þ I 3 Þ Γð3 þ α þ I 5 ÞΓð1 þ α þ i4 Þð3 þ I 5 Þ
ð27Þ
C4 ¼ C3
Γð3 þ α þ I 6 Þð3 þ I 5 Þ Γð4 þ α þ I 7 ÞΓð1 þ α þ I 6 Þð4 þ I 7 Þ
ð28Þ
C5 ¼ C2
Γð3 þ α þ I 6 ÞΓð1 þ α þ i4 þ i5 Þð2 þ I 3 Þ : Γð4 þ α þ I 7 ÞΓð2 þ α þ i4 þ i5 þ i6 ÞΓð1 þ α þ i4 ÞΓð1 þ α þ i5 Þð4 þ I 7 Þ
ð29Þ
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Journal of the Franklin Institute 350 (2013) 1686–1696 www.elsevier.com/locate/jfranklin
First-passage time statistics of Markov gamma processes Fernando Ramos-Alarcón, Valeri Kontorovichn Electrical Engineering Department, Telecommunications Section, CINVESTAV-IPN., Av. IPN 2508, Colonia San Pedro Zacatenco, C.P. 07360 Mexico City, Mexico Received 12 October 2012; received in revised form 2 February 2013; accepted 19 April 2013 Available online 4 May 2013
Abstract The analysis of the First-Passage Time (FPT) statistics has a relevant importance either in theoretical or practical sense for the signal processing design in communications. This paper introduces a simple approach that allows a rather accurate calculation of an arbitrary number of cumulants of the Probability Density Function (PDF) of the FPT for the relevant case of Markov gamma processes. & 2013 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.
1. Introduction Within the framework of the stochastic signal processing theory for communications, the FirstPassage 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. n
Corresponding author. E-mail address: [email protected] (V. Kontorovich).
0016-0032/$32.00 & 2013 The Franklin Institute. Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jfranklin.2013.04.013
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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 ξðtÞ satisfies the initial condition ξð0Þ ¼ x0 ¼ x, x∈ðw; zÞ, where w and z are respectively the lower and the upper boundaries. The goal is to find the PDF W w;z ðx; tÞ of the First-Passage Time required for the process to attain the boundary z for t40. In order to simplify notations hereafter W w;z ðx; tÞ ¼ Wðx; tÞ. 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 Wðx; tÞ 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 Wðx; tÞ through the cumulants; a simulated version of Wðx; tÞ (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. 2. 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]: 1 ∂2 ∂ ∂ BðxÞ 2 ½Pðx; tÞ þ AðxÞ ½Pðx; tÞ ¼ ½Pðx; tÞ; 2 ∂x ∂t ∂x
ð1Þ
Actually Eq. (1) is a particular case of the backward Kolmogorov equation, however it was used first by Pontryagin in the context of FPT problems. 1
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where Pðx; tÞ is the unknown probability of attaining the upper boundary z at the time t40, being that the initial condition at t ¼ 0 is Pðx; 0Þ ¼ 0 for woxoz, A(x) and B(x) are respectively the drift and diffusion coefficients that in the following are considered, in the Stratonovich form [18]: 2xβ β x ; AðxÞ ¼ α− ; BðxÞ ¼ ð2Þ τ τ β where α40, β40 and τ is the correlation time.2 By taking the time derivative of Pðx; tÞ, it is easy to get an equation for the PDF of the FPT, Wðx; tÞ. Then, multiplying the resulting equation by expðiωtÞ and integrating the latter from 0 to ∞, we get an ordinary differential equation in terms of the characteristic function θðx; iωÞ: 1 d2 d BðxÞ 2 ½θðx; iωÞ þ AðxÞ ½θðx; iωÞ−iωθðx; iωÞ ¼ 0: 2 dx dx
ð3Þ
In order to calculate the cumulants (κ k ) of the FPT, recall that the log-characteristic function is ðiωÞk κ k ðxÞ: k ¼ 1 k! ∞
ln½θðx; iωÞ ¼ ∑
ð4Þ
The boundaries can be either absorbing or reflecting [24,18]. The presence of an absorbing boundary means that once the random process reaches the absorbing boundary it remains there indefinitely and the presence of a reflecting boundary means that once the random process reaches the reflecting boundary (derivative is zero there) it “bounces” in the opposite direction in the next time instant. Thus, assuming an absorbing boundary at z and a reflecting boundary at w ¼ 0, the boundary conditions in terms of the characteristic function become d½θðx; iωÞ θðz; iωÞ ¼ 1; ¼ 0: ð5Þ dx x¼w Eq. (4) shows that the log-characteristic function can be expressed in terms of an infinite set of cumulants [18]. However, if one considers the so-called “curtosis approximation”, it is possible to keep just the first four cumulants (see, for example [18,16], etc.), as the influence of cumulants for the PDF reconstruction diminish as the order of the cumulants increase. From Eq. (4) the curtosis approximation would be 4 ðiωÞk θðx; iωÞ≈exp ∑ κk ðxÞ ; ð6Þ k ¼ 1 k! where κk denotes the k-th cumulant. Substituting Eq. (6) in Eqs. (3) and (5) and regrouping the equations around the terms iωk (ignoring all iωk with k44) one gets finally a set of recurrent linear equations for the first four cumulants of the First-Passage Time: α 1 τ − κ_ 1 ¼ ; κ1 ðzÞ ¼ 0; κ_ 1 ð0Þ ¼ 0 κ€ 1 þ ð7Þ x β βx α 1 − κ_ 2 ¼ −2_κ 21 ; κ 2 ðzÞ ¼ 0; κ_ 2 ð0Þ ¼ 0 κ€ 2 þ ð8Þ x β
2 Note that the stationary solution of Eq. (1) with drift and diffusion coefficients (2) corresponds precisely to a gamma process which is the subject of analysis in this paper. This can be easily verified considering (7.4) in [18] and noting that all improper integrals there have to be considered in the principal value sense.
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α 1 − κ_ 3 ¼ −6_κ 1 κ_ 2 ; κ€ 3 þ x β κ€ 4 þ
κ3 ðzÞ ¼ 0; κ_ 3 ð0Þ ¼ 0
α 1 − κ_ 4 ¼ −8_κ 1 κ_ 3 −6_κ 22 ; x β
κ4 ðzÞ ¼ 0; κ_ 4 ð0Þ ¼ 0:
1689
ð9Þ ð10Þ
These equations can be solved either numerically or analytically through standard and wellknown techniques. Examples of the analytical solutions are presented in the Appendix. The cumulants of interest are particularly the first four Eqs. (20)–(23) and can be evaluated a priori just with knowledge of the parameters of the input Markov gamma process (2) and the boundary conditions. 3. Reconstruction and simulation of Wðx; tÞ The reconstruction of the structure of Wðx; tÞ 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]) ∞
Wðx; tÞ ¼ t a−1 expð−tÞ ∑ cn La−1 n ðtÞ; n¼0
where the coefficients cn are Z ∞ n! cn ¼ La−1 n ðtÞWðx; tÞ dt; Γðn þ aÞ 0
ð11Þ
ð12Þ
and Lna−1 ðtÞ ¼ expðtÞ
t −ða−1Þ dn ðexpð−tÞt nþa−1 Þ; n! dt n
ð13Þ
denote the Laguerre polynomials.3 Evaluation of the coefficients cn [14] requires knowledge of the correspondent initial moments, mn, which can also be expressed in terms of the cumulants (calculated in the previous section) through the well-known relations between moments and cumulants. By making c1 ¼ c2 ¼ 0 the parameters a and b can be evaluated simultaneously and recalling that the first cumulant is the mean and the second cumulant is the variance [16], one easily gets a¼
κ21 ; κ2
b¼
κ2 : κ1
ð14Þ
For each different number of terms, n, in the series (11) one can get different variants for a reconstructed Wðx; tÞ. Each reconstructed Wðx; tÞ will be regarded as a hypothetical distribution for an experimental version of Wðx; tÞ (which is obtained by simulating the input Markov gamma process through the SDE approach [18]) and by means of hypothesis testing it is possible to examine how good the hypothetical distributions fit the experimental Wðx; tÞ (in statistical sense). Three hypothetical (reconstructed) distributions were processed using the series (11) with n¼ 2, 3
As it was pointed out at [18], etc. applying more terms at Eq. (11) not always tends to a better accuracy of this representation for experimental data.
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Fig. 1. Simulation and approximation of Wðx; tÞ for different parameters x, z and α.
n¼ 3 and n¼ 4. Note that the hypothetical distribution using the series (11) with n ¼ 2 requires only c0 ¼ 1=ΓðaÞ and La−1 0 ðtÞ ¼ 1, as long as c1 ¼ c2 ¼ 0, and after a simple change of variable yields a compact closed form representation for Wðx; tÞ which is precisely the gamma distribution : W Γ ðtÞ ¼
t t a−1 ; a exp − b ΓðaÞb
ð15Þ
with parameters a and b given by Eq. (14). The rigorous way to examine the statistical fitness of the hypothetical distributions against the experimental, in our case the simulated, Wðx; tÞ, is to perform the well-known standard statistical goodness of fit tests [10]. In our case we applied the Chi Square, χ 2 , statistical test. The three cases under consideration showed a closed match in relation to the experimental distribution. The case for n ¼ 2 showed and accuracy always greater than 90%. The cases for n¼ 3 and n ¼ 4 were (in that order) always slightly below the case for n ¼ 2, in this way one finds that Wðx; tÞ≈W Γ ðtÞ which is a novel theoretical result. As long as the result for n ¼ 3 was practically the same (graphically and statistically) as that for n ¼ 4, in the following is omitted for simplicity. Fig. 1 shows the comparison of the experimental distribution against the two hypothetical distributions,
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Table 1 Theoretical (with 12 terms in the series) and experimental cumulants. Fig. 1a
κ1 κ2
Fig. 1b
Fig. 1c
Fig. 1d
Theo
Sim
Theo
Sim
Theo
Sim
Theo
Sim
1.45 2.06
1.45 2.02
0.78 0.44
0.78 0.43
1.31 1.23
1.31 1.25
0.99 0.67
0.99 0.68
series (11) with n ¼ 2 and n ¼ 4,4 showing the result of the Chi Square test only for n ¼ 2. Being that Wðx; tÞ≈W Γ ðtÞ, in the following the discussion is focused just on the first two cumulants. Table 1 shows the theoretical equations (20), (21) (using 12 terms in the series representation) and the experimental cumulants related to the plots in Fig. 1.
4. Results For a convergence analysis of Eqs. (20) and (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 analysis. Based on the above, by truncating the series (20) up to the second term one gets for κ1 : 1 0 13 20 z x 6 B C B 7 τ 6B β C B β C C7 : −x@1 þ ð16Þ z @1 þ κ1 ðxÞ ¼ 4 A A 2ðα þ 1Þ 2ðα þ 1Þ 5 αβ Formula (16) shows that β is a scaling factor for the initial point (x) and the final boundary (z). The parameter α drives the shape of the input gamma process and Eq. (16) shows that κ1 is inverse-like proportional to α. This can also be appreciated from the plots of Fig. 1 and Table 1. The same is true for κ2 . This follows from the fact that, while α is growing the initial PDF tends to delta-function. Another interesting issue, which can be intuitively explained, is the proportionality of the cumulants (20)–(23) (not always in a linear way) to non-zero correlation time: if the latter is growing all cumulants shall grow as well and mainly in a non-linear way, as the time scale of the initial gamma process is increasing. It is also interesting to mention that the parameter a drives the shape and the entropy of the PDF Wðx; tÞ≈W Γ ðtÞ and so Table 2 shows its behavior for different scenarios of the problem: α, x, and z with β ¼ 1. Table 2 and Fig. 1 allow to see that the behavior of Wðx; tÞ≈W Γ ðtÞ does not change drastically and it is “centered” around an exponential distribution (a¼ 1). This suggests an exponential bound for Wðx; tÞ or at least for its tails. The hypothetical distribution with n¼ 2 requires two initial moments m1 and m2 that are expressed through the first two cumulants and n¼ 4 requires the first four moments that are expressed through the first four cumulants. 4
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Actually as it is well known [16,8], when the first cumulant κ1 is fixed, the distribution that maximizes the entropy is an exponential one. The same result comes out when one approximates the tails of Wðx; tÞ through the cumulant-based tilted Edgeworth expansion developed by [4] in Table 2 Behavior of Wðx; tÞ≈W Γ ðtÞ through the parameter a for different settings α, x and z with β ¼ 1. α ¼ 3; x ¼ 0:1
α ¼ 3; x ¼ 0:5
α ¼ 1; x ¼ 0:1
α ¼ 1; x ¼ 0:5
z
a
z
a
z
a
z
a
0.15 0.2 0.5 1 2.5 3 6 8 12 15
0.78 1.3 2.51 2.86 2.51 2.33 1.48 1.18 0.89 0.73
– – 0.7 1 2.5 3 6 8 12 15
– – 0.62 1.22 1.96 1.94 1.42 1.16 0.89 0.73
0.2 0.3 0.5 1 2.5 3 6 8 12 15
0.66 0.98 1.27 1.45 1.31 1.25 1.03 0.96 0.79 0.62
– – 0.7 1 2.5 3 6 8 12 15
– – 0.33 0.65 1.10 1.11 1.02 0.96 0.79 0.62
Fig. 2. Exponential approximation of the tails for different Wðx; tÞ.
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the formula (4.45). The plots in Fig. 2 show the exponential bound for the tails of Wðx; tÞ for the same set of plots in Fig. 1. Finally, for asymptotically small x and z ðx⪡1; z⪡1Þ the cumulants (20) and (21) are given by τ z x exp − −exp − κ1 ðxÞ ¼ ; ð17Þ αβ β β 1 x z n n ð−1Þ Γ 2 þ n; − ð−1Þ Γ 2 þ n; − ∞ ∞ 2 τ 2 B β β C B∑ C: − ∑ κ2 ðxÞ ¼ @ n!ð1 þ n þ αÞ n!ð1 þ n þ αÞ A β α n¼0 n¼0 0
ð18Þ
The asymptotic expressions (17), (18) show that even for small “span” of the boundaries, only the first cumulant is proportional to τ. As the cumulants grow, (21)–(23), their dependence on τ becomes more non-linear. 5. An application and final remarks An application of this methodology is introduced by considering a Nakagami fading channel taken out from [20] with m ¼ 0.5, Ω ¼ 1, a Jakes-type Power Spectral Density and a maximum Doppler shift frequency fd ¼ 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 α ¼ 0:881 and β ¼ 0:258 [20]. A Markov gamma distributed process with the same parameters is generated by means of the SDE method. For both experimental processes the FPT PDF is extracted. Using the theoretical approach the first two cumulants are evaluated and with them the parameters a and b for the corresponding gamma distribution are calculated. The three distributions are compared in Fig. 3. Note that the FDD (with gamma distribution) is not rigorously a Markov process. In order to make the comparison as fair as possible the correlation time of the FDD is extracted and
Fig. 3. Application of the approach for Wðx; tÞ in wireless communication channels.
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installed in both the theoretical approach and the SDE simulation. Fig. 3 shows that there are some small differences only on the extreme left side of the distribution which is obviously due to the fact that the FDD is not strictly a Markov process. In general the developed Markovian approach can also be useful out of Markovian framework, as it has been noticed before for other problems (see [1] and the references therein). Let us underline the main differences between the proposed approach and the existing results. Traditional results for the FPT PDF are given in integral form [22]. Applied results in communications provide either some moments of the FPT [6,24,13] or even a recursive form of the FPT PDF [13]. Based on the state transition diagram of the source problem, the mean of the FPT can be obtained [3]. Perhaps the only result that addresses the FPT PDF for a Rayleigh process is presented in [19] but the results are given in integral form. The proposed approach is more attractive and original for several reasons: it is valid for any problem or scenario where the process under study has a gamma distribution with known parameters, it provides series for an arbitrary number of cumulants of the FPT (for higher order cumulants the complexity of the series grows) and gives a simple closed form representation for the PDF of the FPT (which requires just two cumulants). To the best of our knowledge closed forms for the FPT distributions are mainly provided for particular cases, as it is the case of the well-known inverse Gaussian PDF [7] used to describe the distribution of the time a Brownian motion with positive drift takes to reach for the first time a fixed positive level. The Wald distribution [26] is a particular case of the inverse Gaussian and describes the FPT when the sample size tends to infinity in the framework of a sequential analysis. This result allows to back up the finding that Wðx; tÞ≈W Γ ðtÞ through the following reasoning. The Gamma and the inverse Gaussian distribution are two-parameter distributions, they are left skewed with support on ð0; ∞Þ and the most relevant fact in this argument is that through the proper choice of parameter values they can be practically indistinguishable. So, it is interesting to notice that the inverse Gaussian and very close to it the Gamma PDF characterize adequately the FPT distribution of two different processes described by means of SDE. The truncated series and the asymptotic expressions of the cumulants allow seeing how the different parameters α, β and τ and the initial and final boundaries x, z affect the behavior of the cumulants. Moreover, for signal processing applications the different parameters required for the theoretical analysis (α, β, τ and the boundaries x, z) can be extracted from any concrete scenario in a straightforward way. The results presented in this paper are valid for all the particular cases associated to the gamma distribution, i.e. exponential, Erlang, etc. and are not reported neither in the literature or in classic well known handbooks like [5]. Appendix A Using the change of variable κ_ n -yn , it is possible to put the set of Eqs. (7)–(10) into the general form: y_ þ PðxÞy ¼ QðxÞ:
ð19Þ
Solving the equations in the well-known way, the following expressions for the first four cumulants are given by I 1 I 1 ! ∞ ∞ τΓðαÞ z x z ∑ C1 −x ∑ C 1 ð20Þ κ 1 ðxÞ ¼ β β β I1 I1
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τΓðαÞ κ2 ðxÞ ¼ 2 β
τΓðαÞ κ3 ðxÞ ¼ 12 β
I 3 I 3 ! ∞ z x z2 ∑ C 2 −x2 ∑ C2 β β I3 I3
2
τΓðαÞ κ4 ðxÞ ¼ 24 β
∞
I 5 I 5 ! ∞ z x 3 z ∑ C3 −x ∑ C3 β β I5 I5
3
3
∞
1695
ð21Þ
ð22Þ
I 7 I 7 I 7 I 7 ! ∞ ∞ ∞ z z x x 4 4 4 4z ∑ C4 þ z ∑ C5 −4x ∑ C 4 −x ∑ C 5 ; β β β β I7 I7 I7 I7
4
4
∞
ð23Þ where the following notation was used: ∞
∞
∑¼ ∑ In
∞
∞
∑ ⋯ ∑ ;
i1 ¼ 0 i2 ¼ 0
in ¼ 0
I m ¼ i1 þ i2 þ ⋯ þ im
ð24Þ
and C1, C2, C3, C4 and C5 are given by C1 ¼
1 Γð1 þ α þ i1 Þð1 þ i1 Þ
ð25Þ
C2 ¼ C1
Γð1 þ α þ I 2 Þð1 þ i1 Þ Γð2 þ α þ I 3 ÞΓð1 þ α þ i2 Þð2 þ I 3 Þ
ð26Þ
C3 ¼ C2
Γð2 þ α þ I 4 Þð2 þ I 3 Þ Γð3 þ α þ I 5 ÞΓð1 þ α þ i4 Þð3 þ I 5 Þ
ð27Þ
C4 ¼ C3
Γð3 þ α þ I 6 Þð3 þ I 5 Þ Γð4 þ α þ I 7 ÞΓð1 þ α þ I 6 Þð4 þ I 7 Þ
ð28Þ
C5 ¼ C2
Γð3 þ α þ I 6 ÞΓð1 þ α þ i4 þ i5 Þð2 þ I 3 Þ : Γð4 þ α þ I 7 ÞΓð2 þ α þ i4 þ i5 þ i6 ÞΓð1 þ α þ i4 ÞΓð1 þ α þ i5 Þð4 þ I 7 Þ
ð29Þ
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