THE Z-TRANSFORM APPLIED TO BIRTH-DEATH MARKOV PROCESSES

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1 he Zranform applied to birthdeath arov procee HE ZRASOR ALIED O BIRHDEAH ARKO ROCESSES H Department of Electronic and Communication niverity of DareSalaam O Box 3594 DareSalaam anania ABSRAC Birthdeath arov model have been widely ued in the tudy of natural and phyical procee he analyi of uch procee however i motly performed uing time erie analyi In thi report a finite tate birth death arov proce i analyed uing the tranform approach he performance metric of the ytem and their variation with the ytem parameter are then derived and preented Keyword: arov procee birthdeath model frame ynchroniation queuing ytem tranform IRODCIO he tudy of phyical ytem ha relied heavily on the approach of building model and uing the model to both analye and deign the ytem Some of uch model initially developed to tudy problem in mathematic and phyic have turned out to be very novel tool for invetigating and olving problem in other field of tudy (Rojdetveni and Cottam ; Drummond 4 One of uch model i the birthdeath marov model which i ued to characterie procee involving ome ind of population Since many procee can be analyed uing thi model it ha been widely ued to tudy the dynamic of natural and phyical ytem Among the area in which birthdeath model have been employed are communication ytem computer data torage and biological ytem In communication ytem the model have been ued to invetigate pacet tranmiion in CDA baed communication (ere Romero et al 3 diverity in ytem employing receiver diverity (Yang and Alouini 4 bloc error proceing for ytem operating in fading environment (Hueda and Rodrigrue 4 and ynchroniation in high peed communication ytem ( 998; In computer ytem the model have been ued to characterie the torage and flow of information (Kleinroc 975 and the allocation of channel in networ upporting mobile computing (Lee et al 999 In biological ytem the model have been ued to tudy population extinction time (omiu and Loechce 994 and the evolution of gene (Karev et al 4 Birthdeath model have and can therefore be ued to olve a diverity of problem Journal of Science and echnology ol 8 o Augut 8 75

2 he Zranform applied to birthdeath arov procee he approach normally ued to tudy procee exhibiting arovian characteritic i time erie analyi he derivation of the performance metric however can alo be accomplihed by uing the tranform which i quite applicable to cae where the ytem under invetigation exhibit dicrete time dependence (Kleinroc 975 In thi report the tranform i ued to invetigate the performance of a proce exhibiting birth death arovian propertie he performance parameter of the proce are derived and it i then hown how the derived cheme can be ued to characterie variou real life procee SYSE AALYSIS he tranition diagram of the ytem analyed in thi report i given in ig having tate numbered to with the tranition probabilitie between the tate alo hown he tranition diagram can be reduced to that of ig uing tate reduction technique found in Howard (97 In maing the derivation in thi report however the ame notation ued in earlier report by the author (eg have been maintained in order to enure conitency e then introduce the following extra parameter: K n m n and J m with n<m to obtain the partial tranfer function in ig a Q Q Q Q Q Q 5 ig : he tranition diagram of a finite tate birthdeath proce with 5 nm ( nn ( n m mm ( mn ( ig : he reduced tranition diagram of ig nm ( ( ( n n > ( mn Q Q ( ( Q ( m m < (3 nn ( nn nn(k (K nn ( > ( 76 Journal of Science and echnology ol 8 o Augut 8

3 he Zranform applied to birthdeath arov procee mm where ( mm mm (J (J mm ( Q K Q nn(k K (Q Q Q (Q (K Q (K > [(Q (K 3(K ] K> (4 (5 nn ( Q ( ( Q ( ( n n (6 J Q mm(j J ( Q ( 3 ( (J Q (J [ (J Q (J ] J > (7 mm Q ( ( ( Q ( ( m m (8 i( i A( i ( A( i ( i! ( i!! otherwie i ( ( i Q (9 ( and i floor{(i/} ing ( and ( the tranfer function from tate n to m i given by Φnm nm( ( nn( ( Journal of Science and echnology ol 8 o Augut 8 77

4 he Zranform applied to birthdeath arov procee from which we obtain the tranition time from tate n to m a L nm d d ( Φnm e then ue the following notation in the enuing derivation ij ( ij ( ij ij( d d ( (3 and conider two cae: n > and n> > becaue the other cae can be obtained from them hen n and > ( nm ( Q 4( and nn( Q ( (4 (5 herefore Φnm ( (Q ( If we apply ( to (6 and perform ome algebraic manipulation we obtain L nm a Lnm where ( ( ( ( ( ( ( S S S S S S ( ( Q ( ( ( ( ( c c S c S c c c S c c c S c S c c S (6 (7 (8 78 Journal of Science and echnology ol 8 o Augut 8

5 he Zranform applied to birthdeath arov procee when i even and given a S and ( ( ( ( ( S S S S S S S ( ( ( ( ( ( ( c c S c c S c c S c c c c S S c c S (9 when i odd and given a S with ( ( ( (!! (! (!! (! (!! (!!!! ( and (! ( 3! (!! (! (! (!! (! (! (!! ( c floor{/} and c floor{(/} Journal of Science and echnology ol 8 o Augut 8 79

6 he Zranform applied to birthdeath arov procee he elaborate algebraic manipulation have been omitted in deriving (8 and (9 but intereted reader can contact the author for detail ote that when ( and therefore L nm hen n > and > we obtain nm and nn ( ( ( thu giving φnm( Q (Q Q [(Q (KQ (K] ( Q (K Q Q ( ( 3 (K ( [(Q ] (K Q (K ( (Q [ (K Q (K ] o o [(Q (K Q 3(K ] ( ( [ (K (K ] (Q Q o Again uing ( and applying algebraic manipulation to (5 we obtain ( Q K K Lnm K where H (x (x ote that when n and > then [ Q ] Lnm ( [ ( ( ] K H K Q K ( (3 (4 (5 (6 (7 8 Journal of Science and echnology ol 8 o Augut 8

7 he Zranform applied to birthdeath arov procee when n > and then [ ] K Q Q K( H K( K Lnm K and when n then L nm o o (8 (9 he tranition time from tate m to n can be obtained in a imilar manner to thoe for n to m herefore when m and > Φ mn ( ( and therefore Q Q Q L mn Q Q ( Q ( Q Liewie when m < and > we obtain Φ mn ( which give ( Q Q Q ( [( (K Q ] (K (( [ (K Q ] (K [( (K Q (K] 3 ( [ ] (( (K Q (K Q ( Q Q Q Q J J Q Q Q Lmn J rom the above reult when m and then L mn Q Q Q Q Q Q Q Lmn and when m and > ( Q [ (Q Q ( Q ] J H J J (3 (3 (3 (33 (34 (35 Journal of Science and echnology ol 8 o Augut 8 8

8 he Zranform applied to birthdeath arov procee RESLS AD DISCSSIO he reult of the analyi are given in ig 3 to In thee reult it i aumed that tranition between tate tae place at regular interval denoted by and the tranition time Lnm and Lmn are then given a multiple of Alo unle indicated otherwie the parameter have been fixed at Q 5 33 n and m 6 ig 3 how how the tranition time from tate to 6 varie with the tranition probability ( in tate A expected the tranition time i very high at low value of indicating the high reluctance of the ytem to leave tate A increae however the tranition time decreae a expected In ig 4 it i een that the tranition time from tate to 6 decreae very harply a increae implying that the high value of force the ytem to move to tate 6 fater It i alo een that ha a higher effect on the tranition time than ig 5 how that the tranition time increae with hi implie that the ytem ha a higher tendency to tay in any tate a increae ig 6 how the tranition time a a function of n when m the tate to which the ytem i uppoed to tranit to i fixed It i een that the tranition time doe not decreae harply a n approache m a would be expected hi can be attributed to the fact that the ytem pend appreciable time looping in the tate below n and thi increae the tranition time ig 7 how that the tranition time increae a m the tate to which the ytem i to tranit to increae It i alo een that thi curve tae on a hape that i oppoite to that of ig 6 ig 8 how how the tranition time from a higher tate m to a lower one n varie with Q the tranition probability in tate he hape of thi curve reemble the one in ig 3 a expected It i alo een in ig 9 that the tranition time from tate m to n decreae a n approache m hi i expected becaue the ditance between m and n decreae with n inally ig how a expected that the tranition time from tate m to a fixed tate n increae a m increae he behaviour of the ytem ha therefore been well illutrated by the preented plot herea thee reult repreent ome ample behaviour of uch L nm ig 3: he variation of Lnm with at 3 n and m 6 L nm ig 4: he variation of Lnm with at ( / 5 n and m 6 L nm ig 5: he variation of Lnm with at ( / 5 n and m 6 8 Journal of Science and echnology ol 8 o Augut 8

9 he Zranform applied to birthdeath arov procee 4 5 L nm L mn ig 6: he variation of Lnm with n at 33 5 and m 5 n ig 9: he variation of Lmn with n at Q 33 Q 5 and m 4 n L nm L mn m m ig 7: he variation of Lnm with m at 33 5 n and L mn ig 8: he variation of Lmn with Q at Q 33 n m 6 and Q ig : he variation of Lmn with m at 33 Q 5 n and a ytem other behaviour can be obtained by uing different parameter A mentioned earlier ome communication ytem can be repreented by the birthdeath model or example in ome communication ytem a bidirectional counter i employed to implement the ynchroniation algorithm In uch cae the ynchroniation tate of the ytem are repreented by the tate of the counter and the tranition probabilitie between the tate of the counter repreent the probabilitie of receiving either corrupted or uncorrupted ynchroniation information he tranition time Lnm and Lmn then repreent the time it tae the ytem to gain or loe ynchroniation Journal of Science and echnology ol 8 o Augut 8 83

10 he Zranform applied to birthdeath arov procee repectively he model can alo be employed in queuing ytem In thi report however the model i applicable if the ingle queue ingle erver model i employed In uch cae the duration needed for the number of uer to change from n to m or m to n are Lnm and Lmn repectively inally many biological ytem can be repreented by the birth death model In uch cae the tate ued in thi report repreent the population of the biological ytem Since the birth and death rate in a biological ytem are not necearily contant they need to be normalied to the population in each tate thu maing the model invetigated in thi paper applicable he time needed for the population to change from n to m i then Lnm wherea the time needed for the population to change from m to n i Lmn COCLSIO he analyi of a birth death proce in which the birth and death tranition probabilitie are fixed ha been invetigated It ha been hown that the obtained reult repreent the expected behaviour of the ytem and the invetigated model can alo be ued for practical ytem It can for example be ued to invetigate frame ynchroniation ytem that employ bidirectional counter to tore the tate and tatu of ynchroniation queuing ytem in which the arrival and ervice rate are contant and the population dynamic of biological birth death ytem urther reearch in thi area will conider cae where the tranition probabilitie between the tate are not contant ACKOLEDEES he author would lie to than the anonymou reviewer for their contructive comment on the manucript REERECES Drummond D (4 auge oion repreentation for birth/death mater equation Eur hy J B 38:67634 Howard RA (97 Dynamic probability ytem ol I arov model John iley and Son ew Yor 598 pp Hueda R and Rodrigue CE (4 On the relationhip between the bloc error and channeltate arov model in tranmiion over lowfading channel IEEE ran Commun 5(8:6975 Karev olf YI Bereovaya S and Koonin E (4 ene family evolution: An indepth theoretical and imulation analyi of nonlinear birthdeathinnovation model BC Evol Biol 4:3 H (998 he analyi of a tranmiion ytem employing SOElie frame Computer Communication :53537 H ( he analyi of a tranmiion ytem with a birthdeath tructured receiver Eat African Journal of Science 3 (:784 Kleinroc L (975 Queueing Sytem ol: heory Johniley and Son ew Yor Lee C Hu Q and Lee DL (999 A tudy of channel allocation for data diemination in mobile computing environment obile etwor and Application 4:79 ereromero J Aguti R and Sallent O (3 Analyi of ype II Hybrid ARQ trategy in a DSCDA pacet tranmiion environment IEEE ran Commun 5(8:4953 Rojdetveni I and Cottam ( apping of tatitical phyic to information theory with application to biological ytem J heor Biol :4354 omiu J and Loechce (994 On the application of birthdeath model to conervation biology Conervation Biology 8(: Yang HC and Alouini S (4 arov chain and performance comparion of witched diverity ytem IEEE ran Commun 5(7:35 84 Journal of Science and echnology ol 8 o Augut 8

RELIABILITY OF REPAIRABLE k out of n: F SYSTEM HAVING DISCRETE REPAIR AND FAILURE TIMES DISTRIBUTIONS

www.arpapre.com/volume/vol29iue1/ijrras_29_1_01.pdf RELIABILITY OF REPAIRABLE k out of n: F SYSTEM HAVING DISCRETE REPAIR AND FAILURE TIMES DISTRIBUTIONS Sevcan Demir Atalay 1,* & Özge Elmataş Gültekin