A 𝑞-Weibull Counting Process through a Fractional Differential Operator

Abstract

We use the 𝑞-Weibull distribution and define a new counting process using the fractional order. As a consequence, we introduce a 𝑞-process with 𝑞-Weibull interarrival times. Some interesting special cases are also discussed which leads to a Mittag-Leffler form.

1. Introduction

The concept of renewal process has been developed for describing a counting process with the assumption that the times between successive events are independent and identically distributed nonnegative random variables. Count models are used in a wide range of disciplines. For an early application and survey in economics see Cameron and Trivedi [1]; for more recent developments, see Winkelmann [2]; for a comprehensive survey of the literature, see Gurmu and Trivedi [3]. The Poisson process plays a fundamental role in renewal theory (see Muralidharan [4] for more details and the references contained therein).

The data of Jaggia and Thosar [5] on the number of takeover bids received by a target firm after an initial bid illustrate the use of small counts in a Poisson model. For a Poisson count model, there is a general consensus that the inter arrival time follows an exponential distribution. The Poisson count models are valid only when the mean and variance are equal, which gives the condition of equidispersion. But in general, in real life situations, the models are overdispersed which satisfies the heavy-tailedness property. The data of Greene [6] on the number of major derogatory reports in the credit history of individual credit card applicants illustrate over-dispersion, that is, the sample variance is considerably greater than the sample mean, compared to the Poisson which imposes equality of population mean and variance, and excess zeros are present since the observed proportion of zero counts is considerably greater than the predicted probability. An indication of the likely magnitude of underdispersion and over-dispersion can be obtained by comparing the sample mean and variance of the dependent count variable, as subsequent Poisson regression will decrease the conditional variance of the dependent variable. If the sample variance is less than the sample mean, the data will be even more under-dispersed once regressors are included, while if the sample variance is more than twice the sample mean, the data will almost certainly be overdispersed upon inclusion of regressors.

Poisson processes with exponential, gamma, and Weibull, distributions as inter arrival times were developed and their limitations were studied by many authors including Winkelmann [2, 7]. The simplest model for duration data is the exponential, the duration distribution being implied by the pure Poisson process, with density 𝜆𝑒^−𝜆𝑡 and constant hazard rate 𝜆. The restriction of a constant hazard rate is generally not appropriate for econometric data, and we move immediately to the analysis of the Weibull model, which nests the exponential as a special case. Here we consider such a problem with 𝑞-Weibull inter arrival times, which incorporates the Weibull distribution as the pathway parameter tends to unity. This model generalizes the Poisson model with the generalized Weibull as inter arrival times since the 𝑞-Weibull distribution allows a transition to the original Weibull distribution. Also the 𝑞-Weibull distribution nests the exponential, Weibull and 𝑞-exponential distributions as special cases.

In 2005, Mathai introduced a pathway model connecting matrix variate Gamma and normal densities. For the pathway parameter 𝑞>1, −∞<𝑥<∞, 𝑎>0, 𝛼>0, 𝛽>0, 𝛿>0, the following is the scalar version of the pathway model:𝑓₁(𝑥)=𝑐₁|𝑥|^𝛼−11+𝑎𝑞−1|𝑥|^{𝛿−𝛽/(𝑞−1)}, (1.1) where the normalizing constant 𝑐₁ is given by𝑐₁=𝛿𝑎𝑞−1^𝛼/𝛿Γ𝛽/𝑞−1()Γ,𝛽2Γ𝛼/𝛿𝛽/𝑞−1−𝛼/𝛿>𝛼𝑞−1𝛿. (1.2) Observe that, for 𝑞<1, on writing 𝑞−1=−(1−𝑞), the density in (1.1) reduces to the following form:𝑓₂(𝑥)=𝑐₂|𝑥|^𝛼−11−𝑎1−𝑞|𝑥|^{𝛿𝛽/(𝑞−1)}, (1.3) for 𝑞<1, 𝑎>0, 𝛼>0, 𝛽>0, 𝛿>0, 1−𝑎(1−𝑞)|𝑥|^𝛿>0 and the normalizing constant 𝑐₂ being given by𝑐₂=𝛿𝑎1−𝑞^𝛼/𝛿Γ𝛽/1−𝑞+𝛼/𝛿+1()Γ.2Γ𝛼/𝛿𝛽/1−𝑞+1 (1.4) As 𝑞→1, 𝑓₁(𝑥)and 𝑓₂(𝑥) tend to 𝑓₃(𝑥), which is referred to as the “extended symmetric Weibull distribution”, where 𝑓₃(𝑥) is given by𝑓₃(𝑥)=𝛿𝛼𝛽^𝛼/𝛿()|𝑥|2Γ𝛼/𝛿^𝛼−1exp−𝛼𝛽|𝑥|^𝛿,−∞<𝑥<∞,𝑎,𝛼,𝛽,𝛿>0. (1.5)

For different values of the parameters in (1.1), (1.3), and (1.5), we get different distributions like the Weibull, gamma, beta type-1, beta type-2, and so forth. More results are available in Mathai and Haubold [8] and Mathai and Provost [9]. The 𝑞-Weibull distribution is a generalized model for the 𝑞-exponential distribution which facilitates a transition to the Weibull as 𝑞→1 through the pathway parameter 𝑞. Some results in this paper are defined in terms of 𝐻-function which can be defined as follows:𝐻^𝑚,𝑛_𝑝,𝑞(𝑧)=𝐻^𝑚,𝑛_𝑝,𝑞𝑧𝑎₁,𝛼₁,…,𝑎_𝑝,𝛼_𝑝𝑏₁,𝛽₁,…,𝑏_𝑞,𝛽_𝑞=12𝜋𝑖^𝑐+𝑖∞_{𝑐−𝑖∞}𝜙(𝑠)𝑧^−𝑠𝑑𝑠, (1.6) where𝜙(𝑠)=∏^𝑚_𝑗=1Γ𝑏_𝑗+𝛽_𝑗𝑠∏^𝑛_𝑗=1Γ1−𝑎_𝑗−𝛼_𝑗𝑠∏^𝑞_𝑗=𝑚+1Γ1−𝑏_𝑗−𝛽_𝑗𝑠∏^𝑝_𝑗=𝑛+1Γ𝑎_𝑗+𝛼_𝑗𝑠. (1.7) Now we define the fractional order derivative in the Caputo sense, which provides a fractional generalization of the first derivative through the following rule in the Laplace domain:𝐿_𝑡𝐷^𝛽_∗𝑓(𝑡);𝑠=𝑠^𝛽𝑓(𝑠)−𝑠^𝛽−1𝑓0⁺,0<𝛽≤1,𝑠>0. (1.8) The Caputo derivative has been indexed with the subscript * in order to distinguish it from the classical Riemann-Liouville fractional derivative _𝑡𝐷^𝛽. It can be noted from the (1.8) that the Caputo derivative provides a sort of regularization of the Riemann-Liouville derivative at 𝑡=0.

2. Preliminaries on the 𝑞-Weibull Distribution

The 𝑞-Weibull distribution gives a wide range of applications in reliability analysis, statistical mechanics, various engineering fields and so forth. (Picoli et al. [10], Jose et al. [11], Jose and Naik [12], etc.] as a consequence of the pathway model introduced by Mathai [13]. The probability density function (pdf) of the 𝑞-Weibull random variable is given as follows: 𝑓₁(𝑥)=𝛼𝜆^𝛼2−𝑞𝑥^𝛼−11−1−𝑞(𝜆𝑥)^{𝛼1/(1−𝑞)}1,0≤𝑥≤𝜆1−𝑞^1/𝛼,𝛼,𝜆>0. (2.1)

The parameter 𝑞 is known as the pathway parameter (the entropy index in statistical mechanics), which determines the shape of the curves. We call this p.d.f a 𝑞-Weibull distribution of type-1. Observe that, for 𝑞>1, writing 1−𝑞=−(𝑞−1), the density in (2.1) assumes another form which we call it as type-2 𝑞-Weibull model which is defined as follows. For 𝑥≥0 and for 𝛼,𝜆>0, 1<𝑞<2 the probability density function is given by𝑓₂(𝑥)=𝛼𝜆^𝛼2−𝑞𝑥^𝛼−11+𝑞−1(𝜆𝑥)^{𝛼−1/(𝑞−1)},𝑥≥0. (2.2) As 𝑞 approaches to one, we obtain Weibull density.

Figure 1 shows a comparison between the Weibull distribution (𝑞=1) and the 𝑞-Weibull distribution for different values of 𝑞. As 𝑞 goes to 2, we get thicker tailed curves, and as 𝑞 tends to 1 from the right, the curve becomes peaked and slowly moves to the curve corresponding to the Weibull distribution. As 𝑞 moves from the left, the mode decreases, and the curve slowly approaches the Weibull distribution, that is, the curve for 𝑞=1. The main reason for introducing the 𝑞-Weibull model is the switching property of the exponential form to corresponding binomial function. That is,lim_𝑞→1𝐹₀1𝑧𝑞−1;−𝑞−1=lim_𝑞→11+𝑞−1𝑧^{−1/(𝑞−1)}=𝑒^−𝑧. (2.3)

Figure 1 

Weibull and 𝑞-Weibull pdf for different values of 𝑞.

The pathway model includes a wide variety of continuous distributions since one can move from one functional form to another through the pathway parameter. For 𝛼=1, we get the 𝑞-exponential distribution introduced by Tsallis [14]. As 𝛼=1 and 𝑞→1, we get the corresponding exponential distribution. The survival function of the 𝑞-Weibull distribution is given by𝐹₁(𝑡)=1+𝑞−1(𝜆𝑡)^{𝛼(𝑞−2)/(𝑞−1)},for1<𝑞<2,𝐹₂(𝑡)=1−1−𝑞(𝜆𝑡)^{𝛼(2−𝑞)/(1−𝑞)},for𝑞<1. (2.4) As 𝑞→1 both the survival functions converge to the survival function of the extended Weibull distribution. For both 1<𝑞<2 and 𝑞<1, the survival functions are decreasing and tend to 0. The hazard rate function (HRF) of the 𝑞-Weibull distribution is given byℎ₁(𝑡)=𝛼𝜆^𝛼2−𝑞𝑡^𝛼−11+𝑞−1(𝜆𝑡)^𝛼,forℎ𝑡≥01<𝑞<2,₂(𝑡)=𝛼𝜆^𝛼2−𝑞𝑡^𝛼−11−1−𝑞(𝜆𝑡)^𝛼,for|𝑡|<1𝜆1−𝑞^1/𝛼𝑞<1. (2.5)

The 𝑞-Weibull distribution nests three other distributions as special cases. The hazard rate function is nonmonotonic when 𝛼>1 and 𝛽>1.

Remark 2.1. As 𝑞→1 and 𝛼=1, we get the hazard rate function of exponential distribution.

Remark 2.2. As 𝑞→1, we get the hazard rate function Weibull model.

Remark 2.3. If𝑞≠1 and 𝛼=1, we get hazard rate function of the 𝑞-exponential model.

3. A Generalized 𝑞-Counting Process

In this section, we generalize the Poisson process in another direction by introducing a counting process using 𝑞-Weibull inter arrival times. For the Poisson process, the mean and variance are equal, which is not realistic. Now, following McShane et al. [15], we introduce a model which gives more flexibility to the mean-variance relationship.

Let 𝑌_𝑛 be the time from the measurement origin at which the 𝑛th event occurs. Let 𝑋(𝑡) be the number of events that have occurred until the time 𝑡. Then 𝑌_𝑛≤𝑡⇔𝑋(𝑡)≥𝑛. In other words the amount of time at which the 𝑛th event occurred from the time origin is less than or equal to 𝑡 if and only if the number of events that have occurred by time 𝑡 is greater than or equal to 𝑛. Now, the count model, 𝑃_𝑛(𝑡), is given by 𝑃_𝑛(𝑡)[𝑋(𝑡)][𝑋(𝑡)][𝑋(𝑡)]𝑌=𝑃=𝑛=𝑃≥𝑛−𝑃≥𝑛+1=𝑃_𝑛≤𝑡−𝑃𝑌_𝑛+1≤𝑡=𝐹_𝑛(𝑡)−𝐹_𝑛+1(𝑡), (3.1) where 𝐹_𝑛(𝑡) is the CDF of 𝑌_𝑛. When the counting process coincide with the occurrence of an event, then 𝐹_𝑛(𝑡) is the 𝑛-fold convolution of the common inter arrival time distribution. Now we consider the count model for the 𝑞-Weibull distribution by assuming that the inter arrival times are i.i.d 𝑞-Weibull distributed, with cumulative distribution function (CDF), 𝐹(𝑡)=1−1−1−𝑞(𝜆𝑡)^{𝛼(2−𝑞)/(1−𝑞)},10≤𝑡≤𝜆1−𝑞^1/𝛼,11,𝑡≥𝜆1−𝑞^1/𝛼. (3.2) Here we consider the case 𝑞<1. The series expansion for the CDF is given by𝐹(𝑡)=1−^∞_𝑗=0(−1)^𝑗2−𝑞/1−𝑞_𝑗1−𝑞(𝜆𝑡)^𝛼𝑗Γ𝑗+1 (3.3) where (𝛼)_𝑗 is the Pochammer symbol. Then corresponding 𝑓(𝑡) is given by𝑓(𝑡)=^∞_𝑗=0(−1)^𝑗𝛼𝑗2−𝑞/1−𝑞_𝑗1−𝑞^𝑗𝜆^𝛼𝑡^𝛼𝑗−1Γ.𝑗+1 (3.4) We have also the recursive relationship𝑃_𝑛(𝑡)=^𝑡₀𝐹_𝑛−1(𝑡−𝑠)−𝐹_𝑛(𝑡−𝑠)𝑓(𝑠)𝑑𝑠=^𝑡₀𝑃_𝑛−1(𝑡−𝑠)𝑓(𝑠)𝑑𝑠. (3.5) Now, 𝐹₀(𝑡)=1, for all 𝑡 and 𝐹₁(𝑡)=𝐹(𝑡). Also 𝑃₀(𝑡)=𝐹₀(𝑡)−𝐹₁(𝑡)=1−1−𝑞(𝜆𝑡)^{𝛼(2−𝑞)/(1−𝑞)}=^∞_𝑗=0(−1)^𝑗2−𝑞/1−𝑞_𝑗1−𝑞(𝜆𝑡)^𝛼𝑗Γ.𝑗+1 (3.6) Using the recursive formula𝑃₁(𝑡)=^𝑡₀𝑃₀(𝑡−𝑠)𝑓(𝑠)𝑑𝑠=^∞_𝑗=0^∞_𝑖=1(−1)^𝑗+𝑖+12−𝑞/1−𝑞_𝑖2−𝑞/1−𝑞_𝑗1−𝑞^𝑖+𝑗(𝜆𝑡)^{𝛼(𝑖+𝑗)}𝛼𝑖Γ(𝛼𝑖+1)Γ𝛼𝑗+1ΓΓ()Γ𝛼=𝑗+1𝑖+1𝑖+𝑗+1_∞𝑗=1(−1)^𝑗+11−𝑞(𝜆𝑡)^𝛼𝑗Γ𝛼𝑗+1^{𝑗−1𝑖=0}2−𝑞1−𝑞_𝑗−𝑖2−𝑞1−𝑞_𝑖Γ(𝛼𝑖+1)Γ𝛼𝑗−𝑖+1Γ(𝑖+1)Γ𝑗−𝑖+1=^∞_𝑗=1(−1)^𝑗+11−𝑞(𝜆𝑡)^𝛼𝑗𝑎^𝑗_𝑖Γ,𝛼𝑗+1 (3.7) where 𝑎^𝑗_𝑖=^𝑗−1_𝑖=02−𝑞1−𝑞_𝑖2−𝑞1−𝑞_𝑗−𝑖Γ(𝛼𝑖+1)Γ𝛼𝑗−𝑖+1Γ()Γ,𝑖+1𝑗−𝑖+1 (3.8) which suggest a general form𝑃_𝑛+1(𝑡)=^𝑡₀𝑃_𝑛(𝑡−𝑠)𝑓(𝑠)𝑑𝑠=^∞_𝑗=𝑛^∞_𝑖=1(−1)^{𝑗+𝑖+𝑛+1}2−𝑞/1−𝑞_𝑖1−𝑞^𝑖+𝑗(𝜆𝑡)^{𝛼(𝑖+𝑗)}𝑎^𝑛_𝑗Γ(𝛼𝑖+1)Γ𝛼𝑗+1ΓΓ()Γ𝛼.𝛼𝑗+1𝑖+1𝑖+𝑗+1 (3.9)

On simplification, we get𝑃_𝑛+1(𝑡)=^∞_𝑙=𝑛+1(−1)^𝑙+𝑛+11−𝑞(𝜆𝑡)^𝛼𝑙Γ(𝛼𝑙+1)^{𝑙−1𝑘=𝑛}2−𝑞/1−𝑞_𝑙−𝑘𝑎^𝑛_𝑘Γ(𝛼(𝑙−𝑘)+1)Γ(𝛼−𝑘+1)=^∞_𝑙=𝑛+1(−1)^𝑙+𝑛+11−𝑞(𝜆𝑡)^𝛼𝑙𝑎_𝑙^𝑛+1Γ().𝛼𝑙+1 (3.10)

Thus we have the model𝑃[𝑁(𝑡)=𝑛]=^∞_𝑙=𝑛(−1)^𝑙+𝑛1−𝑞(𝜆𝑡)^𝛼𝑙𝑎^𝑛_𝑙Γ(),𝛼𝑙+1,𝑛=0,1,2,… (3.11) where 𝑎⁰_𝑗=((2−𝑞)/(1−𝑞))_𝑗Γ(𝛼𝑗+1)/Γ(𝑗+1), for 𝑗=0,1,… and 𝑎_𝑗^𝑛+1=∑^𝑗−1_𝑖=𝑛((2−𝑞)/(1−𝑞))_𝑗−𝑖Γ(𝛼(𝑗−𝑖)+1)/Γ(𝑗−𝑖+1)𝑎^𝑛_𝑖, for 𝑛=0,1,…, 𝑗=𝑛+1,𝑛+2,…, which is a Mittag-Leffler type distribution. The mean count of this distribution is given by𝐸(𝑁)=^∞_𝑛=1^∞_𝑙=𝑛(−1)^𝑙+𝑛𝑛1−𝑞(𝜆𝑡)^𝛼𝑙𝑎^𝑛_𝑙Γ(𝛼𝑙+1) (3.12) with variance given byVar(𝑁)=^∞_𝑛=2^∞_𝑙=𝑛(−1)^𝑙+𝑛𝑛²1−𝑞(𝜆𝑡)^𝛼𝑙𝑎^𝑛_𝑙Γ()−𝛼𝑙+1^{∞𝑛=1∞𝑙=𝑛}(−1)^𝑙+𝑛𝑛1−𝑞(𝜆𝑡)^𝛼𝑙𝑎^𝑛_𝑙Γ(𝛼𝑙+1)². (3.13) Then the moment generating function can be readily obtained as𝑀_𝑖(𝜃)=𝐸𝑒^𝜃𝑁=^∞_𝑖=0^∞_𝑗=𝑖(−1)^𝑗+𝑖𝑒^𝑖𝑁1−𝑞(𝜆𝑡)^𝛼𝑙𝑎^𝑛_𝑙Γ.𝛼𝑗+1 (3.14)

Now this model generalizes the commonly used Poisson and Weibull count models; we call this new model a 𝑞-Weibull count model. We have already seen the probability density curves for the Weibull, Poisson, and the 𝑞-Weibull count models that are shown in Figure 1. Here we consider the under-dispersed case. In a similar manner, we can construct a heterogeneous 𝑞-Weibull count model with 𝜆 replaced by 𝜆_𝑖 where 𝜆_𝑖 follows gamma distribution with parameters 𝑚 and 𝑝. The p.m.f. of such a process is given by𝑃[𝑁(𝑡)=𝑛]=^∞₀^∞_𝑙=𝑛(−1)^𝑙+𝑛1−𝑞(𝜆𝑡)^𝛼𝑙𝑎^𝑛_𝑙Γ()𝑓𝜆𝛼𝑙+1^𝑖∣𝑚,𝑝𝑑𝜆_𝑖=^∞_𝑙=𝑛(−1)^𝑙+𝑛1−𝑞(𝑡)^𝛼𝑙𝑎^𝑛_𝑙Γ(𝛼𝑙+1)^∞₀1Γ𝑝𝑚^𝑝𝜆_𝑖^{𝑝+𝛼𝑙−1}𝑒^{−𝑚𝜆𝑖}𝑑𝜆_𝑖=^∞_𝑙=𝑛(−1)^𝑙+𝑛𝑒^𝑖𝑚1−𝑞(𝑡)^𝛼𝑙𝑎^𝑛_𝑙Γ𝑝+𝛼𝑙Γ(𝛼𝑙+1)𝑚^𝛼𝑙Γ𝑝. (3.15)

This probability model provides an entirely new class of counting processes derived using 𝑞-Weibull inter arrival times and is an improvement over the traditional Poisson process.

4. Fractional Order Process with 𝑞-Weibull InterArrival Time

Let us consider the 𝑞-counting process with 𝑞-Weibull interarrival times. Here we consider a fractional generalization of the process using the fractional order differential operator instead of the ordinary differentiation. As the fractional order 𝛽 is unity, one can obtain the corresponding process with 𝑞-Weibull interarrival time. If the survival probability for the generalized 𝑞-counting process is Ψ(𝑡)=1+𝑞−1(𝜆𝑡)^{𝛼(𝑞−2)/(𝑞−1)}, (4.1) then it obeys the following ordinary differential equation:𝑑Ψ(𝑡)𝑑𝑡=𝐾𝑡^𝛼−1[Ψ(𝑡)]^1/(2−𝑞),where𝐾=𝛼𝑞−2𝜆^𝛼. (4.2) Now, the generalization is obtained by replacing the first derivative by a fractional derivative (Caputo derivative of order 𝛽∈(0,1]) in (4.2). Thus we have _𝑡𝐷^𝛽_∗Ψ(𝑡)=𝐾𝑡^𝛼−1[Ψ(𝑡)]^1/(2−𝑞)𝐿_𝑡𝐷^𝛽_∗Ψ(𝑡)=𝐾𝐿𝑡^𝛼−1[Ψ(𝑡)]^1/(2−𝑞)𝑝^𝛽𝐿{[Ψ(𝑡)]}−𝑝^𝛽−1=𝐾𝐿𝑡^𝛼−1[Ψ(𝑡)]^1/(2−𝑞). (4.3) To find the Laplace transform involved in the right hand side of (4.3), we have to evaluate the integral𝐼₁𝛼,𝜆,𝑞∶𝑝=^∞₀𝑒^−𝑝𝑥𝑥^𝛼−11+𝑞−1(𝜆𝑥)^{𝛼−1/(𝑞−1)}𝑑𝑥. (4.4)

The integrand can be taken as a product of two integrable functions. Let 𝑥₁ and 𝑥₂ be two independent scalar random variables with probability density functions 𝑓₁(𝑥) and 𝑓₂(𝑥), respectively. Consider the transformation 𝑢=𝑥₁/𝑥₂ and 𝑣=𝑥₂⇒𝑑𝑥₁∧𝑑𝑥₂=𝑣𝑑𝑢∧𝑑𝑣, where ∧ is the wedge product discussed in Mathai [13]. Then the joint density of 𝑢 and 𝑣 is 𝑔(𝑢,𝑣)=𝑣𝑓₁(𝑢𝑣)𝑓₂(𝑣). Now, 𝑔₁(𝑢) is obtained by integrating the joint probability density function 𝑔(𝑢,𝑣) with respect to 𝑣. That is, 𝑔₁(𝑢)=∫_𝑣𝑣𝑓₁(𝑢𝑣)𝑓₂(𝑣)𝑑𝑣. Let 𝑓₁(𝑥)=𝑐₁[1+(𝑞−1)𝜆^𝛼𝑥₁^𝛼]^{−1/(𝑞−1)}, 𝑥₁≥0, and 𝑓₂(𝑥₂)=𝑐₂𝑥₂^𝛼−2𝑒^−𝑝𝑥2, 𝑥₂≥0, 𝛼,𝜆>0, where 𝑐₁ and 𝑐₂ are normalizing constants. These constants can be obtained by integrating 𝑓₁(𝑥) and 𝑓₂(𝑥) with respect to 𝑥. Thus we have 𝑔₁(𝑢)=𝑐₁𝑐₂^∞₀𝑒^−𝑝𝑣𝑣^𝛼−11+𝑞−1(𝜆𝑢𝑣)^{𝛼−1/(𝑞−1)}𝑑𝑣. (4.5) Now𝐸𝑥₁^𝑠−1=𝑐₁^∞₀𝑥₁^𝑠−11+𝑞−1𝜆𝑥₁^{𝛼−1/(𝑞−1)}𝑑𝑥₁=𝑐₁𝛼𝜆^𝑠𝑞−1^{𝑠/𝛼∞0}𝑤^{(𝑠/𝛼)−1}(1+𝑤)^{−1/(𝑞−1)}=𝑐𝑑𝑤₁𝛼𝜆^𝑠𝑞−1^𝑠/𝛼Γ(𝑠/𝛼)Γ1/𝑞−1−𝑠/𝛼Γ11/𝑞−1,ℜ−𝑠𝑞−1𝛼>0, (4.6) where ℜ(⋅) denote the real part of (⋅). Similarly 𝐸𝑥₂^1−𝑠=𝑐₂^∞₀𝑥₂^{𝛼−𝑠−1}𝑒^−𝑝𝑥2𝑑𝑥₂=𝑐₂Γ(𝛼−𝑠)𝑝^𝛼−𝑠. (4.7) Then 𝐸𝑢^𝑠−1=𝐸𝑥₁^𝑠−1𝐸𝑥₂^1−𝑠=𝑐₁𝑐₂𝑝^𝑠−𝛼𝛼𝜆^𝑠𝑞−1^𝑠/𝛼Γ(𝑠/𝛼)Γ1/𝑞−1−𝑠/𝛼Γ(𝛼−𝑠)Γ.1/𝑞−1 (4.8)

Now, the density of 𝑢 is obtained by taking the inverse Mellin transform. The detailed existence conditions for the sMellin and inverse Mellin transforms are available in Mathai [13]. Thus 𝑔₁(𝑢)=𝑐₁𝑐₂𝛼𝑝^𝛼Γ11/𝑞−12𝜋_𝑖^𝑐+𝑖∞_{𝑐−𝑖∞}Γ𝑠𝛼Γ1−𝑠𝑞−1𝛼Γ()𝑢𝛼−𝑠𝑞−1^1/𝛼𝜆𝑝^−𝑠𝑑𝑠, (4.9) and the quantity in (4.4) becomes𝐼₁=𝑐𝛼,𝜆,𝑞∶𝑝¹𝑐₂𝛼𝑝^𝛼Γ𝐻1/𝑞−1^1,22,1𝑞−1^1/𝛼𝜆𝑝,()(),ℜ11−1/𝑞−1,1/𝛼1−𝛼,10,1/𝛼−𝑠𝑞−1𝛼>0. (4.10) Hence (4.3) simplifies to𝑝^𝛽𝐿{[Ψ(𝑡)]}−𝑝⁻¹=𝐾𝛼𝑝^𝛼Γ𝐻1/𝑞−1^1,22,1𝑞−1^1/𝛼𝜆𝑝,()(),ℜ11−1/𝑞−1,1/𝛼1−𝛼,10,1/𝛼−𝑠𝑞−1𝛼>0. (4.11) NowΨ(𝑡)𝜆=1+^𝛼𝑞−2𝑡^{𝛼+𝛽−1}Γ11/𝑞−12𝜋_𝑖^𝑐+𝑖∞_{𝑐−𝑖∞}Γ(𝑠/𝛼)Γ1/𝑞−1−𝑠/𝛼Γ(𝛼−𝑠)𝑞−1^1/𝛼𝜆𝑡^−𝑠Γ𝜆𝛼+𝛽−𝑠𝑑𝑠=1+^𝛼𝑞−2𝑡^{𝛼+𝛽−1}Γ𝐻1/𝑞−1^1,22,2𝑞−1^1/𝛼,()(),𝜆𝑡.1−1/𝑞−1,1/𝛼1−𝛼,10,1/𝛼1−𝛼−𝛽,1 (4.12)

For 𝑡≥0, 𝛼,𝛽>0 and for 1<𝑞<2, the pdf is obtained as 𝑓(𝑡)=2−𝑞𝜆^𝛼𝑡^{𝛼+𝛽−2}Γ𝐻1/𝑞−1^1,22,2𝑞−1^1/𝛼,()(),𝜆𝑡1−1/𝑞−1,1/𝛼1−𝛼,10,1/𝛼2−𝛼−𝛽,1 (4.13) and 0 otherwise. Similar expressions for survival and density functions exist for 𝑞<1.

4.1. Series Expansion for the 𝐻-Function

The 𝐻-function defined in (4.12) can be expanded in a series form by evaluating the contour integral by means of the residue calculus. The poles of the integrand Γ(𝑠/𝛼)Γ(1/(𝑞−1)−𝑠/𝛼)Γ(𝛼−𝑠)((𝑞−1)^1/𝛼𝜆𝑡)^−𝑠/Γ(𝛼+𝛽−𝑠) coming from Γ(𝑠/𝛼), which are 𝑠/𝛼=−𝜈, 𝜈=0,1,… The residue at 𝑠/𝛼=−𝜈, denoted by 𝑅_𝜈, is given by𝑅_𝜈=lim_{𝑠/𝛼→−𝜈}((𝑠/𝛼)+𝜈)Γ(𝑠/𝛼)Γ1/𝑞−1−𝑠/𝛼Γ(𝛼−𝑠)Γ𝛼+𝛽−𝑠𝑞−1^1/𝛼𝜆𝑡^−𝑠=lim_{𝑠/𝛼→−𝜈}((𝑠/𝛼)+𝜈)((𝑠/𝛼)+𝜈−1)((𝑠/𝛼)+𝜈−2)…(𝑠/𝛼)Γ(𝑠/𝛼)Γ1/𝑞−1−𝑠/𝛼()…()Γ()𝑠/𝛼+𝜈−1)(𝑠/𝛼+𝜈−2𝑠/𝛼𝛼+𝛽−𝑠×Γ𝛼−𝑠𝑞−1^1/𝛼𝜆𝑡^−𝑠. (4.14) Hence12𝜋_𝑖^𝑐+𝑖∞_{𝑐−𝑖∞}Γ(𝑠/𝛼)Γ1/𝑞−1−𝑠/𝛼Γ(𝛼−𝑠)Γ𝛼+𝛽−𝑠𝑞−1^1/𝛼𝜆𝑡^−𝑠𝑑𝑠=^∞_𝜈=0𝑅_𝜈=^∞_𝜈=0Γ1/𝑞−1+𝜈Γ(𝛼+𝛼𝜈)Γ()𝛼+𝛽+𝜈𝜈!𝑞−1𝜆𝑡^𝛼𝜈. (4.15) Therefore,Ψ(𝑡)𝜆=1+^𝛼𝑞−2𝑡^{𝛼+𝛽−1}Γ1/𝑞−1^∞𝜈=0(−1)^𝜈Γ1/𝑞−1+𝜈Γ(𝛼+𝛼𝜈)Γ()𝛼+𝛽+𝛼𝜈𝜈!𝑞−1𝜆𝑡^𝛼𝜈,Ψ(𝑡)𝜆=1+^𝛼𝑞−2𝑡^{𝛼+𝛽−1}Γ1/𝑞−1²Ψ₁−𝑞−1(𝜆𝑡)^𝛼(),.𝛼,𝛼1/𝑞−1,1𝛼+𝛽,𝛼 (4.16)₂Ψ₁(⋅) denotes the Wright function introduced by Wright in 1935 which is defined by_𝑝Ψ_𝑞𝑧𝑎_𝑝,𝐴_𝑝𝑏_𝑞,𝐵_𝑞=^∞_𝑛=0∏^𝑝_𝑖=1Γ𝑎_𝑖+𝑛𝐴_𝑖∏^𝑞_𝑗=1Γ𝑏_𝑖+𝑛𝐵_𝑗𝑧^𝑛,𝑛! (4.17) where 𝑎_𝑖,𝑏_𝑗∈𝐶 and 𝐴_𝑖,𝐵_𝑗∈ℜ=(−∞,∞), 𝐴_𝑖,𝐵_𝑗≠0, 𝑖=1,2,…,𝑝, 𝑗=1,2,…,𝑞 with the convergence condition that ∑^𝑞_𝑗=1𝐵_𝑗−∑^𝑝_𝑖=1𝐴_𝑖>−1. The plot of Ψ(𝑡) for various values of parameters is given in Figure 2. It is seen from the graph that Ψ(𝑡) behaves like a survival function.

(a)

(b)

Figure 2 

The plot of Ψ(𝑡) for various values of parameters.

4.2. Some Interesting Special Cases

Note that, for 𝛼=𝛽=1, Ψ(𝑡) reduces to the following form:Ψ(𝑡)=1+𝑞−2𝜆𝑡Γ11/𝑞−12𝜋_𝑖^𝑐+𝑖∞_{𝑐−𝑖∞}Γ(𝑠)Γ1/𝑞−1−𝑠Γ(1−𝑠)Γ(2−𝑠)𝑞−1𝜆𝑡^−𝑠𝑑𝑠. (4.18) Now evaluating the contour integral by the residue calculus, the poles of the integrand Γ(𝑠)Γ(1/(𝑞−1)−𝑠)Γ(1−𝑠)/Γ(2−𝑠)((𝑞−1)𝜆𝑡)^−𝑠 coming from Γ(𝑠) are 𝑠=−𝜈, 𝜈=0,1,…. The residue at 𝑠=−𝜈 denoted by 𝑅_𝜈 is given by𝑅_𝜈=lim_{𝑠→−𝜈}(𝑠+𝜈)Γ(𝑠)Γ1/𝑞−1−𝑠Γ(1−𝑠)Γ(2−𝑠)𝑞−1𝜆𝑡^−𝑠. (4.19) Therefore the corresponding survival function isΨ(𝑡)=1+𝑞−2𝜆𝑡Γ1/𝑞−1^∞𝜈=0(−1)^𝜈Γ1/𝑞−1+𝜈Γ()()𝜈+1𝑞−1𝜆𝑡^𝜈=1+𝑞−1𝜆𝑡^{(𝑞−2)/(𝑞−1)} (4.20) which is the survival function of 𝑞-exponential distribution. For 𝛼=𝛽=1 and as 𝑞→1, then (4.20) will become the survival function of the exponential distribution (interarrival time of the Poisson process). For 𝛼=1 and as 𝑞→1, we get the “Mittag-Leffler type renewal process” introduced by Minardi [16].

Acknowledgments

Both authors thank the referee and the editor for their valuable comments. The first author thanks the University Grants Commission, New Delhi for the financial assistance under the Grant no. F: 34-155/2008 (SR) and Professor A. M. Mathai, Director of CMS Pala for providing research facilities. The second author would like to thank the Department of Science & Technology, Government of India, New Delhi for financial assistance under project no. SR/S4/MS:287/05. Both authors acknowledge the support of Dr. Shanoja Naik, CMS Pala for fruitful discussions and comments while preparing the paper.