Simulating Nonhomogeneous Poisson Point Process Based on Multi Criteria Intensity Function and Comparison with Its Simple Form

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1 ournal of mathematcs and computer Scence 9 (2014) Smulatng Nonhomogeneous Posson Pont Process Based on Mult Crtera Intensty Functon and Comparson wth Its Smple Form Artcle hstory: Receved August 2013 Accepted November 2013 Avalable onlne November 2013 Behrouz Fath-Vajargah Department of Statstcs, Unversty of Gulan, Rasht, Iran fath@gulan.ac.r Hassan Khoshkar-Foshtom Department of Statstcs, Unversty of Gulan, Rasht, Iran hassan.khoshkar@yahoo.com Abstract In ths paper we frst study the general nonhomogeneous Posson pont process based on strct form of an ntensty functon and ts algorthm for generatng t. Then, we employ mult crtera ntensty functon nstead of smple form and establsh a new algorthm, then we compare the effcency of our new algorthm based on ths modfed ntensty functon. Keywords: Intensty functon, Nonhomogeneous Posson pont process, Smulaton. 1. Introducton It s well known that one of the most mportant and famous pont processes s nonhomogeneous Posson process. Let us frst study ths process: 1.1. The nonhomogeneous Posson process From a modelng pont of vew the major weakness of the Posson process s ts assumpton that events are just as lkely to occur n all ntervals of equal sze. A generalzaton, whch relaxes ths assumpton, leads to the nonhomogeneous or non-statonary process. If events are occurrng randomly n tme, and N () t denotes the number of events that occur by tme t, then we say that N ( t ), t 0 consttutes a nonhomogeneous Posson process wth ntensty functon ( t), t 0,f (a) N (0) 0, (b) The numbers of events that occur n dsjont tme ntervals are 133

2 B. Fath-Vajargah, H. Khoshkar-Foshtom/. Math. Computer Sc. 9 (2014) Independent, P exactly 1 event between t and t h (c) lm ( t ) h h0 P 2 or more eventsbetween t and t h (d) h0, lm 0. h In ths step, we brng a bref revew of smulatng the nonhomogeneous Posson pont process [1] that we need t n ths paper. Then, we dscuss on man method whch we desre to use t. Theorem 1: Suppose that events are occurrng accordng to a Posson process havng rate, and suppose that, ndependently of anythng that came before, an event that occurs at tme t s counted wth probablty pt (). Then the process of counted events consttutes a nonhomogeneous Posson process wth ntensty functon ( t ) p( t ) [4] Generatng a nonhomogeneous Posson process (NHPP) Suppose N () t s a nonhomogeneous Posson process wth ntensty () t and that there exsts a such that () t for all t T. Then we can use the followng algorthm, based on above theorem, to smulate N () t. Algorthm 1: The thnnng algorthm for smulatng T tme unts of a NHPP Step 1: t 0, I 0. Step 2: Generate a random number U. 1 Step 3: t t logu. If t T, stop. Step 4: Generate a random number U. () t Step 5: If U, set I I 1, S ( I ) t. Step 6: Go to Step 2. In the above () t s the ntensty functon and s such that () t. The fnal value of I represents the number of events tme T, and S (1),..., S ( I ) are the events tmes. The above procedure, referred to as the thnnng algorthm (because t thns the nonhomogeneous Posson ponts), s clearly most effcent, n the sense of havng the fewest number of rejected events tmes, when () t s near throughout the nterval[5]. Thus, an obvous mprovement s to break up the nterval nto subntervals and then use the procedure over each subnterval. Now, wth regards to the above explanatons we establsh a new algorthm based on mult crtera ntensty functon and we try to nvestgate on the effcency of ths algorthm. 2. Method Here, we generate frst T tme unts of nonhomogeneous Posson process based on mult crtera ntensty functon. That s, determne approprate values k, 0 t 0 t1 t 2... t k t k 1 T, 1,..., k 1 such that 134

3 B. Fath-Vajargah, H. Khoshkar-Foshtom/. Math. Computer Sc. 9 (2014) ( s) f t 1 s t, 1,..., k 1 (1) Now, we generate the nonhomogeneous Posson process over the nterval 1 by generatng exponental random varables wth rate, and acceptng the generated event occurrng at tme s, ( s ) s ( t 1, t), wth probablty. Because of the memoryless property of the exponental and the fact that the rate of an exponental can be changed upon multplcaton by a constant, t follows that there s no loss of effcency n gong from one subnterval to the next. That s, f we are at, whch s such that t ( t 1, t) and generate X, an exponental wth rate [ X ( t t )] we can use ( t, t ) t X t, then as the next exponental wth rate 1.We thus have the followng algorthm for generatng the frst T tme unts of a nonhomogeneous Posson process wth ntensty functon () s when the relatons (1) are satsfed. In the algorthm t represents the present tme, the present nterval (. e., j when t j1 t t j), I the number of events so far, and S (1),..., S ( I ) the events tmes. Algorthm 2: Smulaton the frst T tme unts of a NHPP Step 1: t 0, 1, I 0. 1 Step 2: Generate a random number U and set X log U. Step 3: If t X t, go to Step 8. Step 4: t t X. Step 5: Generate a random number U. () t Step 6: If U, set I I 1, S ( I ) t. Step 7: Go to Step 2. Step 8: If k 1, stop. ( X t t ) Step 9: X, t t, 1. 1 Step 10: Go to Step 3. ( t, t ) Suppose now that over some subnterval 1 we have that 0, where nf { ( s) : t 1 s t } In such a stuaton we should not use the thnnng algorthm (algorthm1) drectly but rather should frst smulate a Posson process wth rate over the desred nterval and then smulate a nonhomogeneous Posson process wth the ntensty functon ( s) ( s) when s ( t, t ) 1. (The fnal exponental generated for the Posson process, whch carres one beyond the desred boundary, need not be wasted but can be sutably transformed so as to be reusable.) The superposton (or mergng) of the two processes yelds the desred process over the nterval. The reason for dong t, ths way s that t saves the need to generate unform random varables for a Posson dstrbuted number, wth mean ( t t ) 1, of the event tmes. For example, consder the case where

4 B. Fath-Vajargah, H. Khoshkar-Foshtom/. Math. Computer Sc. 9 (2014) ( s) 10 s, 0 s 1 would generate an expected number of 11 events, each of Usng the thnnng method wth 11 whch would requre a random number to determne whether or not t should be accepted. On the other hand, to generate a Posson process wth rate 10 and then merge t wth a nonhomogeneous Posson process wth rate ( s) s, 0 s 1 (generated by the thnnng algorthm wth 1), would yeld an equally dstrbuted number of event tmes but wth the expected number needng to be checked to determne acceptance beng equal to1. 3. Numercal result Now, we brng an example based on algorthm 2 and present ts results n the followng fgures. Example: We wrte a program that uses algorthm 2 to generate the frst 10 tme unts of a nonhomogeneous Posson process wth ntensty functon () t such as: t 0 t 5 () t 5 1 5( t 5) 5 t 10 We consder ths example for two ndvdual cases: 1- Drectly based on the ntensty functon () t. 2- Based on probablty functon pt (). In the frst case the followng graph has been concluded t Fgure 1. ( [, ], t [5,10], ( t ) [,15( t 5)]) Based on our new procedure we obtan as supremum of the case study and snce k 1 then are equal to 5, 135 t, t equal to 5,10, respectvely. 1, 2 2 2, also we have

5 B. Fath-Vajargah, H. Khoshkar-Foshtom/. Math. Computer Sc. 9 (2014) Wth regards to the above graph durng 60 seconds the fnal value of I s equal to 37 and S( I) In the case 2, the correspondng fgure s gven below: t 2 Fgure 2. ( [, ], t [5,10], p( t ) [ ( ), (1 5( t 5))]) In ths case based on theorem 1, p ( t ), p ( t ) are obtaned as below: t 5 2 t 1 ( t) ( ( )) 0 t p1 ( t) ( (1 5( t 5)) ) ( t ) 15( t 5) 5t 10 2 p () t As we can see from the above graph, the fnal value of I s equal to 341 and S( I) durng 15 seconds. As we can see n Fg 2, there s ncreasng relatonshp between the number of events I and the events tmes S( I ). Moreover, based on the presented graph, comparng the rate of convergence of the above cases, we conclude that n case 2 we has much better smoothness n convergence, accuracy and t has hgher speed to convergence. By the other word, when we employ the probablty functon the ncrementng the events tmes behavor more regular and has faster convergence to tends to the desred events. Now, f we compare the effcency of the new algorthm wth the thnnng or random samplng method and ts correspondng algorthm as presented n [1], we conclude that when we employ 2 137

6 B. Fath-Vajargah, H. Khoshkar-Foshtom/. Math. Computer Sc. 9 (2014) mult crtera ntensty functon the events tmes have grown better than the case we use the smple form ntensty functon. 4. Concluson We have presented performances of the new algorthm for smulatng the nonhomogeneous Posson pont processes based on mult crtera () t. We concluded that ths algorthm has provded more smoothness and speed to converge to desred events whch we have desred to reach t. 5. References [1] B. Fath-Vajargah and H. Khoshkar-Foshtom, On smulatng pont processes based on effcent algorthms, ournal of world appled programmng, Accepted. (2013). [2] P. A. W. Lews and G. S. Shedler, Smulaton of nonhomogeneous Posson processes by thnnng, Nav. Res. Log. Quart. 26, (1979). [3] B.. T. Morgan, Elements of smulaton. Chapman and Hall, London, (1983). [4] S. M. Ross, Stochastc processes, 2 nd edton, Wley, New York, (1996). [5] S. M. Ross, Introducton to probablty models, 8 th edton, Academc Press, San Dego, (2003). [6] B. D. Rpley, Computer generaton of random varables: A Tutoral, Inst. Statst. Rev. 51, (1983). [7] B. D. Rpley, Stochastc smulaton. Wley, New York, (1986). [8] R. Y. Rubensten, Smulaton and the Monte Carlo method. Wley, New York, (1981). [9] B. W. Schmeser, Random varate generaton, a Survey, Proc Wnter Smulaton Conf, Orlando, FL; pp , (1980). [10] D. Snyder and M. Mller, Random pont processes n tme and space. New York: Sprnger- Verlag, (1991). 138