On The Mean First Exit Time For A Compound Poisson Process

If   ) Y ( E i , 2 i ) Y ( V   , ..., , 3 , 2 , 1 i  the expected value and variance of t X are    t ) X ( E t , ) ( t ) X ( V 2 2      , respectively. In particular, if i Y , ..., , 3 , 2 , 1 i  are Poisson distributed in (1),   0 t , X t  is called as a Neyman type A process and if i Y , ..., , 3 , 2 , 1 i  are distributed according to the binomial distribution,   0 t , X t  is called as a Neyman type B process. Similarly, if i Y , ..., , 3 , 2 , 1 i  are geometric distributed,   0 t , X t  is called as a Polya-Aeppli process [1].


Introduction
Concerning applications, the first exit time can be defined as the length of the busy period for a M/G/1 queue in queuing theory [3].In risk analysis, the distribution function of the first exit time is that of the first time at which the accumulated total claims from an insurance company exceeds its capital.Besides these, the first exit time is defined as the first time until either outdating or total depletion of stock for perishable items in inventory theory [4].
Laplace transforms of the distribution function of the first exit time with two parallel boundaries were derived by Dvoretzky et al. [5] for a Poisson process.Explicit formulas for the distribution function of the first exit time for the Poisson process were obtained by Delucia and Poor [6] and Rolski et al. [7].The Laplace Stieltjes transforms of the distribution function of the first exit time with positive jumps were given by   The mean first exit time is obtained for the compound Poisson process with an upper horizontal boundary and positive integer-valued jump sizes.The paper is organized as follows.We start by introducing in Section 2 the probability function of t X and the distribution function of the first exit time for the compound Poisson process.In Section 3, we derive the mean first exit time for compound Poisson process with an upper horizontal boundary and positive integer-valued jump sizes.Finally, an application to the traffic accidents is presented in Section 4. The conclusion is given in Section 5.

Some Preliminary Results On The Compound Poisson Process
where However, it is not easy to yield the explicit probabilities of t X from (2) since it needs infinite sum [7].Therefore, a recursive algorithm was derived by Panjer [10]


The probabilities in ( 5) can be used if from (5) and presented in Figures 1-3 X t  is also called as a Neyman type A process in Figure 1 and  process with different values of  and 5 t  .
The distribution function of  is obtained as given below if the cumulative probabilities in ( 7) substituted into (6)

Mean First Exit Time
In this section, mean the first exit time E( )  is obtained for the compound Poisson process where 3,..., 1,2, i , Y i  are discrete random variables.

Let E( )
 be the mean first time at which a given compound Poisson process hits a given subset of the state space.Let  be the upper horizontal boundary for (6), we obtain from (6) , j 1, 2,...  According to the mean first exit time E( )  , the right-hand side terms depend on how  can be partitioned into different forms with using integers 1, 2, ... To obtain E( )  , a new algorithm is prepared in R. A summary outline of the basic steps and operations for the algorithm is given as Step 1 Determine the value of parameter  and initial parameters of the algorithm (t,  , and Step 2 Multiply each terms of E( ) for each expected value E( )  for 1  and keep the numerical values of this step.
Step 3 Form the new j  , j 2,3,...  terms beginning from E( )  for 2  to E( )  for

 
/2    and keep the numerical values of this step.

Numerical Example
Meintanis [13] obtained a new goodness of fit test for certain bivariate distributions based on accident data and fatalities in The Netherlands.The data were obtained from the database of BRON of the Ministry of Transport, The Netherlands.In particular, total accidents and fatalities recorded on Sundays of each month over the period 1997-2004 in the region of Groningen are given in Table 1.In this study the same data is used to show applicability of the compound Poisson process and the mean first exit time.For the construction of a model to explain the total number of fatalities from the accidents the following random variables are defined:

Mean exit time E( )
 is obtained from ( 11

CONCLUSION
We conclude with the comment that the mean first exit time E( )  can be computed easily Then, an application to traffic accident data is presented to illustrate the usage of the mean first exit time for the compound Poisson process.

ACKNOWLEDGEMENT
The author is grateful to the anonymous referee for his valuable suggestion that improved the presentation.

Figure 4 :
Figure 4:The distribution function of  with 10   for several values of t where i Y , ..., , 2 , 1 i  have

Figure 6 :
Figure 6: The distribution functions of  with 10   for several values of t where i Y , ..., , 2 , 1 i 

tN:
The number of accidents which occur in Groningen between years1997-2004;   i Y : The number of fatalities of i th accident such that ,value = 0.57), we have seen that the number of the accidents which occur in Groningen between years 1997-2004, defined as  

Figure 7 :
Figure 7: The occurrence probabilities of traffic accidents within

Figure 9 :
Figure 9: The probability of total fatality number which will occur within 3 , 2 , 1 t  months jump sizes are discrete random variables and the boundary is upper horizontal.
 is called as a Polya-Aeppli process[1].The statistical significance of the compound Poisson process arises from its applicability in real life situations, where the researcher often observes only the total amount t Poisson process with parameter  taking several values.The distribution function of  is given

Table 1 :
Total Sunday accidents (left entry) and the corresponding number of fatalities (right entry) recorded in the region Groningen for each month during the years1997-2004.

Fit of the Poisson, Binomial or Geometric Distributions to Fatalities:
To decide the best distribution between Poisson distribution, binomial distribution and geometric distribution for the number of fatalities, the goodness of fit tests were performed and the results are presented in Table2.It is seen in Table2that the chi-square values are less than the critical table values for each distribution at the 5% level of significance.This means that the Poisson

Table 2 :
Comparison of fit of Poisson and binomial distributions to observed frequency for fatalities

Table 3 :
Mean exit time E( )