BRANCHING PROCESSES AND THEIR APPLICATIONS: Lecture 5: Crump-Mode-Jagers processes and queueing systems with processor sharing June 7, 5 Crump-Mode-Jagers process counted by random characteristics We give here only an informal description of the Crump-Mode-Jagers process counted by random characteristics or, what is the same, of the general branching process counted by random characteristics. A particle, say, x, of this process is characterised by three random processes ( x ; x (); x ()) which are iid copies of a triple (; (); ()) and whose components have the following sense: if a particle was born at moment x then x is the life-length of the particle; x (t x ) - is the number of children produced by the particle within the time-interval [ x ; t); x (t x ) = if t x < ; x (t x ) is a stochastic process subject to changes ONLY within the time-interval [ x ; x + x ) while outside the interval it has the form 8 < if t x < x (t x ) = : x ( x ) if t x x (it is NOT assumed that x (t) is a nondecreasing function in t ): The stochastic process Z (t) = X x x (t x ) where summation is taken over all particles x born in the process up to moment t is called the general branching process counted by random characteristics.
Examples: ) (t) = I ft [; )g in this case Z (t) = Z(t) is the number of particles existing in the process up to moment t; ) (t) = ti ft [; )g + I f < tg then Z (t) = Z(u)du; ) (t) = I ft g then Z (t) is the total number of particles born up to moment t: Classi cation. E() <; =; > - subcritical, critical and supercritical, respectively. Let v () v () ::: v (n) ::: be the birth moments of the children of the initial particle. Then (t) = fn : v (n) tg is the number of children born by the initial particle up to moment t: We have Z (t) = (t) + X x (t x ) = (t) + X Zn (t v (n)) x6= where Zn () ; n = ; ; ::: are iid copies of Z (): Hence it follows that EZ (t) = E(t) + E 4 X Zn (t v (n)) 5 = E(t) + E 4 X = E(t) + E 4 X E [Zn (t v (n)) jv () ; v () ; :::; v (n) ; :::] 5 E [Z n (t v (n)) jv (n)] 5 = E(t) + E 4 X E [Z (t u)] ( (u) (u )) 5 ut = E(t) + EZ (t u) E(du): Thus, we get the following renewal-type equation for A (t) = EZ (t) and (t) = E(t) : A (t) = E(t) + A (t u)(du): ()
Malthusian parameter: a number is called the Malthusian parameter of the process if e t (dt) = () (such a solution not always exists). For the critical processes = ; for the supercritical processes > ; for the subcritical processes < (if exists). If the Malthusian parameter exists we can rewrite () as C (t) = e t E(t) + C (t Z u u)d e y (dy) where C (t) = e t A (t): In view of () and given that, say, e t E(t) is directly Riemann integrable and e t E(t)dt < ; te t (dt) < we can apply the key renewal theorem to conclude that if the measure is non-lattice then M(t) = e y (dy) lim t! C (t) = lim e t A (t) = t! e t E(t)dt te (dt) t : In particular, if G(t) is the life-length distribution of particles and (t) = I ft [; )g we get E(t) = P ( > t) = G(t) and lim e t EZ (t) = t! if the respective integrals converge. R e t ( G(t)) dt R te t (dt) MjGj system with processor sharing discipline The model: a Poisson ow of customers with intensity comes to a system with one server which has unit service intensity. The service time distribution of a particular customer is (if there are no other customers in the queue) B(u): If there are M customers in the system at some moment T they are served simultaneously with intensity M each. Let W l;:::;l N (l N )
be the waiting time for the end of service of a customer which arrived to the queue at the moment when the queue had N customers with remaining service times l ; :::; l N : The question is to study the properties of the random variable W l;:::;l N (l N ) when l N! : To solve this problem we construct an auxiliary general branching process. Construction of the branching process. Consider a general branching process in which initially at time t = there are N particles with remaining life-lengths l ; :::; l N ; l N and which constitute the zero generation of this process. The life-length distribution of any newborn particle x is P ( x u) = B(u); the reproduction process x (t) of the number of children produced by a particle up to moment t has the probability generating function Es x (t) = e (s )u db(u) + e (s )t ( B(t)) that is, this is an ordinary Poisson ow with intensity stopped when the particle dies: Es x (t) = Es P oi(t^x) : Let Z(t; l ; :::; l N ; l N ) denote the number of particles in the process at moment t with the mentioned initial conditions. We use a simpli ed notation Z(t) if at moment t = there is only one particle of zero age in the process. We will consider also the process with immigration X(t; l ; :::; l N ; l N ) which has the same initial conditions and development as Z(t; l ; :::; l N ; l N ) but, in addition, given X(t; l ; :::; l N ; l N ) = it starts again by one individual of zero age after a random time r i having distribution P (r i u) = e u (if the process dies out for the i th time). X(t) is used if we initially start by the process Z(t): Now let x x :::be the sequential moments of jumps of the process X(t; l ; :::; l N ; l N ). We construct by the general branching process the following queueing system with S(T ) being the number of customers in the queue at moment T : ) the queue has N customers at T = with remaining service times l ; :::; l N ; l N ; ) the moment T i of the i th jump of the queue size S() is speci ed as T i = Z xi X(y; l ; :::; l N ; l N )dy + Z xi I fx(y; l ; :::; l N ; l N ) = g dy: ) the service discipline is such that at each moment T the number of customers in the queue and their remaining service times coincide with the number of individuals and the remaining life-lengths of individuals in the branching process at moment t(t ) where T = (T ) X(y; l ; :::; l N )dy + (T ) 4 I fx(y; l ; :::; l N ) = g dy:
Thus, t $ T is a random change of time. Theorem. The described queueing system is a processor-sharing system with service time of customers B(u) and a Poisson ow of customers with intensity of arrivals. Proof. Let S(T ) be the number of customers in the queue at time T and let ; ; ::: be the moments of changes the size of the queue. Let us show that the evolution of the constructed queue coincides with the evolution of a queueing system with processor sharing discipline. It is enough to show that this is true for T [; ] and then, using the memoryless property of the Poisson ow to show in a similar way that this is true for T [ ; ] and so on. To demonstrate this it is enough to check that: ) = Nl ^ ::: ^ Nl N ^ d where P (d u) = e u ; ) If = Nl i then at this moment the i-th customer comes out of the queue; if = d then one new customer arrives; ) at any moment T [; ] the remaining service times of the initial N customers are l N T; :::; l N N T: Let be the rst moment of change of X(t; l ; :::; l N ). Clearly, = l ^ ::: ^ l N ^ d ^ ::: ^ d N where P (d i u) = e u and where the sense of d i is the birth of an individual by the initial particle labelled i: On the interval u [; ] the processing time of the queueing system T and the time t passed from the start of the evolution of the general branching process are related by T = Nt: Hence ) is valid. Further, = N (l ^ ::: ^ l N ^ d ^ ::: ^ d N ) = Nl ^ ::: ^ Nl N ^ (N(d ^ ::: ^ d N )) and P (N(d ^ ::: ^ d N ) y) = This proves ). Point ) is evident. Corollary. S(T ) = X(t(T ); l ; :::; l N ): Corollary. W l;:::;l N (l N ) = Z ln e y=n N = e y : Z(y; l ; :::; l N )dy: More detailed construction: Let L be the life-length of a particle and let () () ::: be the birth moments of her children. Denote (t; L) = fn : (n) tg : Then the process generated by this particle can be treated as a process with immigration stopped at moment L where Es (t;l) = e (s ) min(t;l) ; 5
and, since each newborn particle generates an ordinary process without immigration, we see that the o spring size of new particles at moment t in the process is Z (u;l) (t u)(du; L) where Z i (y) are independent branching processes initiated by one individual of zero age. Thus, Z(y; l ; :::; l N ) = I fl yg + +::: + I fl N yg + and, in particular, we have Z (u;l)(y u)(du; l ) Z (u;ln )(y u)(du; l N ) W l;:::;l N (l N ) = = Z ln Z(y; l ; :::; l N )dy NX NX min(l N ; l k ) + Z ln dy Z (u;lk )(y u)(du; l k ): Since the birth moments of new particles constitute a Poisson ow with intensity we have E [(u; l)jl] = min (u; l) : Hence Z y E dy Z (u;lk )(y u)(du; l k ) E Z (u;lk )(y u)(du; l k ) j(u; l k ); u l k E Z (u;lk )(y u) j(u; l k ); u l (du; l k ) E [Z(y u)] (du; l k ) E [Z(y u)] E [(du; l k )jl k ] E [Z(y u)] du E [Z(u)] du : Hence EW l;:::;l N " X N NX (l N ) = E min(l N ; l k ) + Z lk dy E [Z(u)] du : 6
One can prove also that if = El N = and < then for xed l ; :::; l N lim W l ;:::;l N (l N ) = l N! udb(u) < almost surely (in particular, if it comes to an empty system). 7