E-Companion to Fully Sequential Procedures for Large-Scale Ranking-and-Selection Problems in Parallel Computing Environments Jun Luo Antai College of Economics and Management Shanghai Jiao Tong University Shanghai China 200052 L Jeff Hong Department of Economics and Finance and Department of Management Sciences College of Business City University of Hong Kong Kowloon Hong Kong Barry L Nelson Department of Industrial Engineering and Management Sciences Northwestern University Evanston Illinois 60208 Yang Wu Tmall Company the Alibaba Group Hangzhou Zhejiang China 30000 EC Technical Notes EC Derivation of the Asymptotic Independence in Section 23 Because there is only alternative for notational simplicity we omit the (denoting alternative ) in the subscript of Y l and replace Y l by Y l in this section We first derive closed-form expressions for Y l l 2 and their properties Since the m processors are identical and the service time of each customer (replication time) is exponentially distributed with mean µ by the Markovian property we know that the interdeparture time between the (l )th and lth customer denoted by A l l 2 are iid with mean µ µ /m (note that A is the departure time for the first customer which is Y ) At time t 0 0 there are m customers assigned to the server pool; at time t l l 2 the lth customer leaves the system and the next customer in queue is immediately admitted to the empty server (see Figure EC) m 0 A t t 2 A 2 A 3 t 3 A l t l t l t st 2nd 3rd (l )th lth Figure EC The inter-departure time ec
ec2 Notice that Y l is the service time of the lth departing customer in this example for l 2 Moreover the service time of the lth departing customer depends on the time point at which it enters the system For instance the first departure can only enter the system at time t 0 then Y A The second departure can enter the system at times t or t 0 with probabilities m m 2 m(m ) respectively Then m 2 { A2 wp Y 2 m m A + A 2 wp m The third departure can enter the system at time t 2 t or t 0 with probabilities mm m(m ) m 3 m 3 or m(m )(m ) respectively Then m 3 Y 3 A 3 wp m m A 2 + A 3 wp A + A 2 + A 3 wp m 2 (m ) 2 m 2 Similarly like counting the paths in an m-ary tree we can obtain A l wp m m A l + A l wp Y l l d2 A d wp l A d d wp m 2 (m ) l 2 m l (m ) l m l Then based on the closed-form expression for the distribution of Y l we can further derive the mean of Y l and expectation of the sample mean estimator Ȳl(n) respectively as follows l ( ) l d [ ( m E[Y l ] E[A d ] µ ) l ] m m d E [ Ȳ l (n) ] { µ m [ ( ) n ]} n m We next use the moment generating function (MGF) to show the asymptotic independence between Y l and Y as n that is ( lim MYl (t) M (t) M Y Y n l +Y (t)) 0 (EC) Define Y l in a rigorous way Let D l s denote the event that the lth departing customer enters the system at time t s s 0 l Conditioning on D l s Y l can be written as follows Y l {D l s} l ds+ Then the MGF of Y l is M Yl (t) E [E [e ty l D l ]] l P {D s0 l s} E [e t ] l ds+ A d A d or
ec3 Recall that A l are independent exponential random variables with mean µ µ so the MGF of m A d is M Ad (t) ( µt) Furthermore we derive the probability distribution function of D l m(m ) l s 0 P {D l s} m l m s (m ) l s s 2 l m l Plugging the expressions for P {D l s} and M Ad (t) into M Yl (t) and with some algebra we obtain that M Yl (t) mµt mµt ( ) l m mµt m µt Notice that the MGF M Yl (t) is well-defined in the neighborhood of zero Furthermore the nonnegativity of M Yl (t) implies that mµt > 0 which further indicates that m < Then m µt lim l M Yl (t) mµt which is the MGF of an exponential random variable with mean µ mµ By calculating the first order derivative of M Yl (t) at t 0 we also arrive that the conclusion that lim l E[Y l ] µ as in Section 23 Similarly we can derive the closed-form expression of the MGF of Y as M Y (t) mµt mµt ( ) m mµt m µt Then lim n M Y l (t) M Y (t) ( ( mµt) mµt m 2 ( mµt) 2 m ) l (EC2) µt The MGF of Y l + Y is M Yl +Y (t) E [ E [ e t(y l +Y ) D l D ]] l j0 s0 P {D l j D s} E [e t( l dj+ A d + ] ds+ A d ) (EC3) The joint distribution of D l and D seems complicated because it depends on the relations between j and s as well as l and s Since we are interested in the situation that n without loss of generality we assume that n 2 We derive the results for the case that l 3 (in fact the
ec4 case that l or l 2 can be handled in a similar but simpler way as l 3) By enumerating all possibilities we obtain the the joint distribution as follows m(m ) n (m 2) l P {D l j D s} m j 0 s 0 m(m ) n+s (m 2) l s m j 0 s l m s l+ (m ) n+2l s 2 m j 0 s l l + n m(m ) n+j (m 2) l j m j l s 0 m s (m ) n+j s (m 2) l j m j l s j 0 j l s j m j (m ) n+s j (m 2) l s m j l s j + l m s+j l (m ) n+2l s j 2 m j l s l l + n Then Equation (EC3) can be written as the sum of eight parts [ M Yl +Y (t) P {D l 0 D 0} E e t( l d A d + ] d A d ) l + P {D l 0 D s} E [e t( l d A d + ] ds+ A d ) s0 + sl j P {D l 0 D s} E [e t( l d A d + ] ds+ A d ) l + P {D l j D 0} E [e t( l dj+ A d + ] d A d ) l j + P {D l j D s} E [e t( l dj+ A d + ] ds+ A d ) j s l + P {D l j D j} E [e t( l dj+ A d + dj+ A )] d j l + j sj+ l + j l sl P {D l j D s} E [e t( l dj+ A d + ] ds+ A d ) (EC4) (EC5) (EC6) (EC7) (EC8) (EC9) (EC0) P {D l j D s} E [e t( l dj+ A d + ] ds+ A d ) (EC) We now deal with the eight parts in Equation (EC4) (EC) one-by-one The first part is (EC4) P {D l 0 D 0} E [e 2t l d A d +t ] dl+ A d m(m )n (m 2) l m ( 2µt) l ( µt) n
ec5 which converges to zero as n because of the condition m < verified above m µt The second part is l (EC5) P {D l 0 D s} E [e t s d A d +2t l ds+ A d +t ] dl+ A d s0 l s0 m(m ) n+s (m 2) l s m ( µt) s ( 2µt) l s ( µt) n which converges to zero as n because of the condition m < verified above m µt The third part is (EC6) sl sl P {D l 0 D s} E [e t l d A d +t ] ds+ A d m s l+ (m ) n+2l s 2 m ( µt) 2 s m(m ) m ( µt) mµt + mn+ (m ) l m (m )l 0 + m l ( µt) as n l mµt ( µt) l mµt The fourth part is the same as the second part with only exchanging the positions of s and j so (EC7) also converges to zero as n The fifth part is l j (EC8) P {D l j D s} E [e t j ds+ A d +2t l dj+ A d +t ] dl+ A d j s l j j s m s (m ) n+j s (m 2) l j m m(m )n (m 2) l m m(m ) m m(m )n (m 2) l m ( 2µt) l ( µt) 2 ( 2µt) l + ml (m ) n m ( µt) n 2( mµt) 2 0 as n ( 2µt) l j ( µt) n ( mµt) 2 ( µt) n ( µt) n+j s ( mµt) 2 2( mµt) 2 The sixth part (EC9) equals zero because the joint probability equals zero The seventh part is (EC0) l l j sj+ P {D l j D s} E [e t s dj+ A d +2t l ds+ A d +t ] dl+ A d
ec6 l l j sj+ m j (m ) n+s j (m 2) l s m ( 2µt) l s m(m )n (m 2) l m ( 2µt) l 2 ( µt) n ml (m ) n ( 2µt) m ( µt) n 2( mµt) 2 m(m ) m ( µt) 2 0 as n The eighth part is (EC) l j sl l j sl 2( mµt) 2 ( mµt) 2 + ml (m ) n m P {D l j D s} E [e t l dj+ A d +t ] ds+ A d m s+j l (m ) n+2l s j 2 m ( µt) l j ( µt) s ( µt) n+s j m(m ) m ( µt) ( mµt) mn+ (m ) l 2 m ml (m ) n m 0 (m )l m l ( µt) n ( µt) l ( mµt) 2 + m m ( mµt) 2 ( µt) n ( µt) l ( mµt) 0 + as n 2 ( mµt) 2 ( mµt) 2 ( mµt) 2 Combining the limits of the eight parts for (EC4) (EC) and (EC2) yields the equality in Equation (EC) which concludes the desired result EC2 Useful Lemmas Lemma EC Let (x n : n ) be a real-valued sequence such that n n i x i µ as n where µ is finite Then n max in x i 0 as n Proof: Let s n n i x i Then x n n s n n n n s n n µ µ 0 as n Since x n /n 0 it follows that for all ɛ > 0 there exists n n (ɛ) such that for n > n x n /n ɛ For n n n max in x i n max in x i + n n max x i + in n max x i + ɛ in max in +n x i max in +n Then the limsup of the LHS as n is at most 0 + ɛ Since ɛ is arbitrary the result follows x i i
ec7 Lemma EC2 (Fabian (974)) For a fixed triangular continuation region C defined by U(s) max {0 A Bs} and U(s) if B /2 and > 0 then P[B (T ) < 0] 2 e A where T inf{s > 0 B (s) / C} the random stopping time that B () first exits C EC3 Sketch of the Proof of Lemma 2 The detailed proof of Lemma 2 follows exactly the same steps as in Kim et al (2005) so we only provide a sketch of the idea behind it Recall that D[0 ] is the Skorohod space of all right-continuous functions with left limits Let Λ be the set of strictly increasing functions λ mapping the domain [0 ] onto itself such that both λ and its inverse λ are continuous Then the Skorohod metric ρ on D[0 ] can be defined by { } ρ(x Y ) inf λ Λ d : sup λ(t) t d and t [0] sup X(t) Y (λ(t)) d t [0] Besides the definition of the Skorohod metric ρ we also need to define the following two mapping functions (as Definitions 2 and 22 in Kim et al (2005)): Definition EC On the Skorohod space D[0 ] (a) For Y D[0 ] Let T Y (U δ ) inf { t : Y (t) U δ (t) } and define the function p δ : Y D[0 ] p δ (Y ) R by p δ (Y ) Y (T Y (U δ )) (b) For Y D[0 ] Let T Y (U) inf {t : Y (t) U(t)} and define the function p : Y D[0 ] p(y ) R by p(y ) Y (T Y (U)) Notice that the forms of the upper boundaries U δ (t) and U(t) can be generally specified as in Kim et al (2005) however for simplicity we may directly regard U δ (t) U δ j(t) and U(t) as defined in Section 42 To show that Z j (T δ j) B (T j ) in Lemma 2 a key step is to show that p δ (Z j ()) p (B ()) as δ 0 which involves the functional central limit theorem and the generalized continuous mapping theorem (Theorem 344 in Whitt (2002)) The following proposition justifies that the conditions of Theorem 344 in Whitt (2002) can be satisfied Proposition EC If p δ () and p() are as in Definition EC then P[B D p ] 0 (EC2) where D p is the set of x D[0 ] such that p δ (x δ ) p(x) fails for some sequence {x δ } with ρ(x δ x) 0 in D[0 ] as δ 0
ec8 Proposition EC presents the same result as Proposition 3 in Kim et al (2005) and interested readers may refer to the paper for the detailed proof Then with the results in Equations () and (EC2) we can apply the generalized continuousmapping theorem (Theorem 344 in Whitt (2002)) to show that which concludes the proof p δ (Z j ()) p (B ()) as δ 0 EC2 Numerical Results of Test Experiments using Master/Slave When the time to generate one observation is extremely small then the master may not be able to process available observations immediately after they are ready to be sent back to the master We conducted some numerical experiments on our local server to help better understand this issue We consider the SC with k 0 3 0 4 0 5 and implement the APS procedure using Master/Slave structure with m 48 slaves The replication time is simply the time for generating a normal random variable ( 5 0 3 ms) We report the numerical result in Table EC Compared with the result in Table in the paper for the same k 0 3 and m 48 the samples simulated on the server (ie 88 0 5 ) is larger than that generated on the simulator (ie 792 0 5 ) That is because the replication time is relatively short on the server which leads to the situation that the master is not fast enough to process all available samples immediately Thus the available observations are filling the queueing buffer in front of the master waiting for the comparison and elimination decisions This is consistent with the monitoring of the buffer in front of the master during the simulation Secondly we find that the total sample size increases in proportion to the makespan as k increases However most of the time is not devoted to simulation (ie generating samples) but consumed by the master for computing work For instance it only takes about 002 ( 88 0 5 5 0 6 /48) seconds to generate 88 0 5 samples with m 48 slaves but the procedure spends about 90 seconds to select the best This means 978% of the time the slaves are idling either for submitting the result or waiting for the next task which is very close to what we observe during the simulation as shown in Figure EC2 Figure EC2 captures a snapshot of the status of the threads for the Master/Slave structure after starting the APS procedure with 48 slaves From that figure we find that all slaves (denoted as slaves 0 to 47) are working in parallel to generate samples using only a small proportion of the total time Table EC Summary under the SC settings when m 48 Number of alternatives k 0 3 k 0 4 k 0 5 Total samples 88 0 5 2257 0 6 270 0 7 Makespan (seconds) 90 042 3206 PCS 099 099 00
ec9 Figure EC2 A screenshot of the Master/Slave with the number of processors m 48 References Fabian Vaclav 974 Note on anderson s sequential procedures with triangular boundary The Annals of Statistics 2 70 76 Kim S-H B L Nelson James R Wilson 2005 Some almost-sure convergence properties useful in sequential analysis Sequential Anal 24(4) 4 49 Whitt Ward 2002 Stochastic-process limits Springer Series in Operations Research Springer-Verlag New York