Analysis of batch Bernoulli process subject to discrete-time renewal generated binomial catastrophes

Abstract

With the evolution of internet, there has been increased interest in the use of computer-communication networks and digital communication systems. But unusual events, like virus attack, that often results in abrupt change in the state of the system poses a major threat to these systems. The occurrence of a virus may cause immediate removal of some elements (packets) from the system. Such type of situations can be well represented as discrete-time catastrophe model where interruptions due to various types of virus attacks are referred to as catastrophe. In this paper we study a discrete-time catastrophe model in which population grows according to batch Bernoulli process and catastrophes occur according to discrete-time renewal process. When a catastrophe occurs, an element (individual or packet) of the population survives with probability p or dies with probability $1-p$. The analysis of the model has been done using supplementary variable technique and steady-state probability generating function of the distribution of population size at post-catastrophe, arbitrary and pre-catastrophe epochs are obtained under late arrival system with delayed access. In order to make the model more useful for practitioners, a step-wise computing process for evaluation of distribution of population size at various epochs for commonly used inter-catastrophe time distributions viz. geometric, deterministic and arbitrary are given. We also present recursive formulae to compute factorial moments of the population size at various epochs. Further to study the effect of critical model parameters, numerical results and some graphs are included.

Appendices

Appendix 1: Evaluation of A(s)

$$\begin{aligned} a_n= & {} P[n ~\hbox {elements arrive in ICT}], ~~ n\ge 0, \\= & {} \sum _{k=1}^{\infty } P[n ~\hbox {elements arrive } / S=k] P[S=k], \\= & {} \sum _{k=1}^{\infty } P\left[ \sum _{j=1}^{k} X_j=n \right] b_k, \\ a_n= & {} \sum _{k=1}^{\infty } \left( {\begin{array}{c}k\\ n\end{array}}\right) \lambda ^n \{g_n\}^{k*}(1-\lambda )^{k-n}b_k,~~ n\ge 0, \end{aligned}$$

where $\{g_n\}^{k*}$ is the kth order convolution of $g_n$.

Multiplying the above expression by $s^n$ and summing over the range of n from 0 to $\infty$, we get

$$\begin{aligned} A(s)= & {} \sum _{k=1}^{\infty } (1-\lambda +\lambda G(s))^k b_k ~=~ B(1-\lambda +\lambda G(s)). \end{aligned}$$

Appendix 2: Evaluation of $A_e(s)$

$$\begin{aligned} a_n^e= & {} P[n ~\hbox {arrives in remaining ICT}], ~~ n\ge 0, \\= & {} \sum _{k=1}^{\infty } P[n ~\hbox {arrives} / R=k] P[R=k],~~~~~~~~~~ \hbox {where}~R~\hbox {is the remaining ICT}, \\= & {} \sum _{k=1}^{\infty } P\Big [\sum _{j=1}^{k} X_j=n \Big ] P[R=k], \\ a_n^e= & {} \sum _{k=1}^{\infty } \left( {\begin{array}{c}k\\ n\end{array}}\right) \lambda ^n \{g_n\}^{k*}(1-\lambda )^{k-n} ~ \frac{1}{b}\left( 1-\sum \limits _{i=1}^{k}b_i\right) ,~~ n\ge 0. \end{aligned}$$

Multiplying the above expression by $s^n$ and summing over the range of n from 0 to $\infty$, we get

$$\begin{aligned} A_e(s)= & {} \sum _{k=1}^{\infty } (1-\lambda +\lambda G(s))^k ~\frac{1}{b}\left( 1-\sum \limits _{i=1}^{k}b_i\right) ~=~ \frac{1-B(1-\lambda +\lambda G(s))}{b \lambda (1-G(s))}. \end{aligned}$$

Appendix 3: Convergence of infinite product

We prove the convergence of infinite product appearing in (21). Using a comparison argument, it is clear that the infinite product $\prod \nolimits _{k=1}^{\infty }A(1-p^k+p^ks)$ converges, when $|s| \le 1$, if the product $\prod \nolimits _{k=1}^{\infty }A(1-p^k)$ converges (see Economou 2004).

In order to prove the above result, we use following theorem:

Theorem 3

If, for every k, $a_k \ge 0$, then the product $\prod \nolimits _{k=1}^{\infty }(1-a_k)$ is also convergent iff $\sum \nolimits _{k=1}^{\infty } a_k$ converges [see Theorem 4, pg. 220 Knopp (1951)].

Now using the above theorem and considering $a_k=1-A(1-p^k)$, it is sufficient to prove that

$$\begin{aligned} \sum _{k=1}^{\infty }(1-A(1-p^k)) < \infty . \end{aligned}$$

(58)

We know that every pgf $C(s)=\sum \nolimits _{n=0}^{\infty }c_n s^n,~~ |s| \le 1$, can be written as

$$\begin{aligned} \frac{1-C(s)}{1-s}=\sum _{l=0}^{\infty }\sum _{n=l+1}^{\infty } c_n s^l. \end{aligned}$$

(59)

Setting $C(s)=A(s)$ and $s=1-p^k$ in (59), we get

$$\begin{aligned} \frac{1-A(1-p^k)}{1-(1-p^k)}= & {} \sum _{l=0}^{\infty }\sum _{n=l+1}^{\infty } a_n (1-p^k)^l, \\ 1-A(1-p^k)= & {} p^k \sum _{l=0}^{\infty }\sum _{n=l+1}^{\infty } a_n (1-p^k)^l, \\< & {} p^k \sum _{l=0}^{\infty }\sum _{n=l+1}^{\infty } a_n, \\= & {} p^k \sum _{n=1}^{\infty } n a_n = p^k {a} < \infty , \end{aligned}$$

where ${a}=a_{(1)}= b \lambda {\bar{g}}$ is the mean number of arrivals between two consecutive catastrophes. Now we have

$$\begin{aligned} \sum _{k=1}^{\infty } (1-A(1-p^k))< \sum _{k=1}^{\infty } p^k {a} = \frac{p {a}}{1-p} < \infty . \end{aligned}$$

Hence the infinite product converges.