Stochastic Service Systems, Random Interval Graphs and Search Algorithms

egp.pu b.rere v.arxiv.tex Sto c hastic Service Systems, Random In terv al Graphs and Searc h Algorithms Patric k Eschenfeldt pesche nfeldt @hmc.edu Ben Gr oss bgross @hmc.e du Nic holas Pipp enger njp@ma th.hmc .edu Department of Mathematics Harvey Mudd College 1250 Dartmouth Av enue Claremont, CA 91711 Abstract: W e consider several stochastic service systems, and study the asymptotic b ehavior of the moments of v arious quantities that hav e applicatio n to mo dels for random interv al gra phs and a lgorithms for sea rching for an idle server o r empt y waiting sta tion. I n tw o c ases the moments turn o ut to inv olv e La m b ert series for the gener ating functions for the sums of pow ers of divisors of p ositive integers. F or these ca ses we ar e able to o btain complete asymptotic expansions for the moments of the quantities in ques tio n. Keyw ords: Q ueueing theo ry , interv al graphs , Lamber t ser ie s, asymptotic ex pa nsions. Sub ject Cl assification: 60 K26, 9 0B22 1. In tro duction Our goal in this pap er is the study of sto chastic service s ystems, with an eye to tw o applica tions: mo dels for rando m interv al g r aphs and the ana lysis of se a rch a lgorithms. The sys tems we study ar e traditionally designated M / M / 1 (indep endent exp onentially distributed int erar riv a l times, indep enden t exp onentially dis- tributed ser vice times a nd a single ser ver) and M / M / ∞ (independent exp onentially distr ibuted interarriv al times, indep enden t exp onentially distributed ser vice times and infinitely many ser vers). In a previo us paper , Pipp enger [P ] introduced mo dels fo r random interv al g r aphs ba sed on sto chastic service systems, and analyzed, among others, a mo del based on the M / M / ∞ system. F or this sys tem, id le p erio ds (in terv als of time dur ing which all servers are idle) alternate with bu sy p erio ds (interv als of time dur ing whic h at leas t one server is busy). A ra ndom gr aph can b e co nstructed b y co nsidering a busy per io d, letting the vertices cor r esp ond to custo mers ser ved during this busy p erio d, and putting an edge betw een tw o vertices if the service interv a ls of the corre spo nding customer s ov erlap. Since edg es corr esp ond to intersections b et ween interv a ls in a totally or dered set, the res ulting graph is an interv al gr aph, so this pro cedure yields a mo del for random interv al graphs. (Other mo dels for random in terv a l graphs have b een considered by Scheinerman [S1, S2] and by Go dehardt and Jaw orsk i [G2].) Suppo se that in the sy s tem M / M / ∞ customers arrive at ra te λ a nd ar e served at ra te 1. Pipp enger [P] showed that the num ber N of vertices in the c o rresp onding rando m g raph (which corr esp onds to the nu mber of cus to mers served during the busy p erio d) is such that the distribution of N /e λ tends to that of an exp onential with mean 1 as λ → ∞ . F urthermore, the chromatic num ber K of the g raph (which for an int erv al g raph e q uals the num ber of vertices in the lar gest clique of the gra ph, a nd c o rresp onds to the la rgest nu mber of customers simultaneously in the system during the busy perio d) is such that K/ eλ tends to 1 in probability as λ → ∞ . Our firs t goa l in this pap er is to study the corresp onding ra ndo m interv al gra ph mo del for the M / M / 1 system. In this case we m ust hav e λ < 1 to ensure that the busy p erio d is finite with probability one, a nd that the num b er of custo mer s served during the busy p erio d has finite ex pecta tion, and we shall b e interested in asymptotics as λ → 1 . When there is o nly o ne server, custo mer s who arrive when the s erver is busy must wait for se r vice, a nd a servic e discipline (whic h deter mines which o f the waiting customers will be ser ved next when the curr en t ser vice interv al ends) must b e sp ecified. As a result, the co rresp onding interv al graph will dep end on the ser vice discipline us e d. Consider, for e x ample, a busy p erio d consisting of six service int erv als, with t wo new customers a r riving during the firs t in terv al and one new custo mer arr iv ing during each of the second, third and four th interv als. Then if the ser vice discipline is “fir st-come-firs t-served”, the resulting graph will be 1 3 5 2 = = = = = = = =         4 = = = = = = = =         6 , > > > > > > > whereas if the s ervice discipline is “last-co me-first ser ved”, the resulting gra ph will b e 1 2 3 4 5 6 . O O O O O O O O O O O O O O ? ? ? ? ? ? ?        o o o o o o o o o o o o o o 1 Nevertheless, the num ber of vertices a nd the chromatic n umber the res ulting gr aph (in this ex ample, 6 and 3, re spec tiv ely) are indep endent of the ser vice discipline. In Section 2, we shall derive asy mptotic e xpansions for the moments of these quantities. The leading terms of these expansions ar e Ex[ N m ] ∼ 2 m − 1 (2 m − 3)!! λ m − 1 (1 − λ ) 2 m − 1 (1 . 1) for m ≥ 1, where (2 m − 3)!! = (2 m − 3) · (2 m − 5) · · · 3 · 1 and ( − 1)!! = 1, Ex[ K ] ∼ lo g 1 1 − λ , (1 . 2) and, for m ≥ 2 , Ex[ K m ] ∼ m ! ζ ( m ) (1 − λ ) m − 1 , (1 . 3) where ζ ( m ) = P n ≥ 1 1 /n m is the Riemann zeta function. It will b e noted that Ex[ K ] g rows quite slowly as λ → 1 . If the random v ariable J denotes the num b er of customers in the system in eq uilibrium, then Ex[ J ] = λ/ (1 − λ ), which gr ows m uch mor e rapidly (see Co hen [C2], p. 181 ). It may app ear pa radoxical that the maximum num ber of custo mers in the system grows mor e slowly than the mean num ber of custo mers, but it must b e b orne in mind that Ex[ K ] is a n av erage ov er busy per io ds, whereas Ex [ J ] is an av erage ov er time. Indeed, the ma jority of busy per io ds have K = 1: after the ar riv a l initia ting the busy per io d, the next even t determines whether K = 1 (if that even t is a ser vice termination) or K > 1 (if that even t is another ar riv a l). Because λ < 1, the former (with proba bilit y 1 / (1 − λ )) is mo re likely than the la tter (with pr obability λ/ (1 − λ )). W e also note that, since ζ (2) = π 2 / 6, we hav e V ar[ K ] = Ex[ K 2 ] − Ex[ K ] 2 ∼ π 2 / 3(1 − λ ), which grows much more r a pidly than Ex[ K ] (or even Ex[ K ] 2 ). F or the chromatic num ber , corresp onding to the maximum num b er o f custo mers in the system simultaneously dur ing the bus y p er io d, the res ults inv olve Lambert s eries that are gener a ting functions for arithmetical functions arising in num ber theory: the sums of p ow ers of the divisors of p ositive integers. These are S l ( λ ) = X n ≥ 1 σ l ( n ) λ n , (1 . 4) where σ l ( n ) denotes the sum of the l -th p owers o f the diviso rs of n (se e Har dy and W right [H, p. 23 9 ]). Our second goal in this pap er is to analyze search alg orithms connected with sto chastic s e rvice s y stems. Consider an infinite sequence S 1 , S 2 , . . . of se r vers in a n M / M / ∞ system. Supp ose that each newly arriv ing customer sca ns this sequence in orde r and eng ages the first currently idle server. W e are interested in the index L of the se r ver S L engaged by a newly arr iving customer in equilibrium. This s ystem has b e en extensively studied by Newell [N2], who suggests that L “ is approximately uniformly distributed ov er the int erv al” [1 , λ ], ba sing this assertion o n the appr oximation Pr[ L > l ] ≈ ( 1 − l λ , if l < λ , 0 , if l > λ . (1 . 5) But no er r or b ounds a r e given for this or o ther approximations stated b y Newell, and not even the fact that the first moment has the asymptotic b ehavior Ex[ L ] ∼ λ 2 (1 . 6) 2 that it would have under the uniform distribution is established rig orously . In Section 3 we sha ll g ive a rigoro us version of (1.5) that will suffice to establish not only (1.6), but also the next term, Ex[ L ] = λ 2 + 1 2 log λ + O (1) , (1 . 7) and mo re g enerally Ex[ L m ] = λ m m + 1 + m λ m − 1 log λ 2 + O  λ m − 1  (1 . 8) for m ≥ 1. In particular, we have V ar[ L ] = Ex[ L 2 ] − Ex[ L ] 2 = λ 2 12 + λ log λ 2 + O ( λ ) . Since the int erv al [0 , 1] is b ounded, formula (1 .8) s hows that the m -th moment of L/λ tends to 1 / ( m + 1) as λ → ∞ for all m ≥ 1, and thus s uffices to show that the distr ibution o f L/λ tends to the uniform distribution on the interv al [0 , 1]. W e note that a pr o blem that is in a sense dua l to ours (finding the la rgest index of a busy ser ver, rather than the sma llest index o f an idle server) has b een treated by Coffman, K adota and Shepp [C1]. Our final results co ncern the analogue of the pre ceding sear ch problem for the M / M / 1 sys tem. Here there is only a s ingle server, but a n infinite sequence W 1 , W 2 , . . . of waiting stations . A customer a rriving when the server is busy s cans this sequence in order and waits at the first v a cant station. When the s erver bec omes free and there is at least one customer waiting, it too scans this sequence in order , and serves the customer waiting a t the first o ccupied station. W e ar e interested in the index I of the sta tion W I at which a newly ar riving custo mer waits in equilibrium (taken to b e zero if the s erver is idle a t the time o f a rriv al). (The index of the fir st station w hich the ser ver finds o ccupied (taken to b e zer o if the ser vice interv a l initiates a busy p erio d) ha s, of cour se, the same distr ibution as I .) W e sha ll show that the distr ibution of the r andom v ar iable I is closely related to that o f the random v aria ble K s tudied in Sec tio n 2 , with Ex[ I m ] = λ Ex[ K m ]. This fact a llows asy mptotic expansio ns for the moment s o f I to b e obtained from those of the mo ments of K in a straightforward way , with the result that the lea ding ter ms ar e the same. 2. Random Int erv al Graphs Our g oal in this se ction is to determine a symptotic expa ns ions for the mo men ts of the size (num be r of v ertices) and chromatic num ber (num ber of vertices in the lar gest clique) for the r andom interv al graph corres p onding to the M / M / 1 system. Thes e quantities corr esp ond to the num be r N o f c ustomers served during the busy p erio d and the maximum num b er K of customers in the system simultaneously dur ing the busy p erio d. The random v ariable N ha s a Catalan distribution: P r [ N = n ] = 1 2 n − 1  2 n − 1 n  p n − 1 q n , (2 . 1) where p = λ/ (1 + λ ) and q = 1 / (1 + λ ) (see for ex a mple Cohen [C2 , pp. 19 0 –191], or Rior dan [R1, pp. 64–65 ]). This distribution can b e derived as follows. Let J denote the num ber of customers in the system. When the busy p erio d b egins, J = 1. During the busy p erio d, J is incremented whenever a new custo mer arrives, and 3 J is decremented whe ne ver a s ervice in terv a l e nds and a customer depar ts, un til J = 0, at which time the busy p erio d ends (see Figur e 1 ). start   _^]\ XYZ[ WVUT PQRS stop GFED @ABC 1 p * * q o o · · · p + + q k k ONML HIJK j p * * q j j · · · q k k Figure 1. State transition diagram for determining the nu mber of customers served during the busy p erio d in M / M / 1. When J ≥ 1, the probability that the next transition is a n arriv al is p = λ/ (1 + λ ), and the pr obability that the next transitio n is a departure is q = 1 / (1 + λ ). If n customers ar e ser ved during the busy p erio d, there must b e n − 1 further ar riv a ls (be yond the one that b egan the busy p erio d) and n departures, a nd these m ust o ccur in such an order that J = 0 for the first time immediately after the la st of these n departures. The num ber o f s uc h o rders is A n =  2 n − 1 n  / (2 n − 1) =  2 n − 2 n − 1  /n , the n -th Cata lan num be r (see for example Comtet [C3, p. 53 ]). Thus we have (2.1). Since the ge ner ating function for the Ca talan num bers is a ( z ) = P n ≥ 1 A n z n =  1 − √ 1 − 4 z  / 2, the probability generating function g N ( z ) for N is g N ( z ) = X n ≥ 1 Pr[ N = n ] z n = a ( pq z ) p = 1 − √ 1 − 4 pq z 2 p . Since for m ≥ 1, w e hav e d m a ( z ) / dz m = 2 m − 1 (2 m − 3)!! / (1 − 4 z ) (2 m − 1) / 2 , the factorial moments o f N are given by Ex[ N ( N − 1) · · · ( N − m + 1 )] = 1 p d m dz m 1 − √ 1 − 4 pq z 2     z =1 = p m − 1 q m 2 m − 1 (2 m − 3)!! √ 1 − 4 pq (2 m − 1) = 2 m − 1 (2 m − 3)!! λ m − 1 (1 − λ ) 2 m − 1 , bec ause √ 1 − 4 pq = q − p = (1 − λ ) / (1 + λ ). Since x m = P 0 ≤ l ≤ m  m l  x ( x − 1) · · · ( x − l + 1), where the  m l  are the Stirling n umbers of the second kind, with the genera ting function P m ≥ l ≥ 0  m l  y l z m l ! = e y ( e z − 1) (see for example Comtet [C3, pp. 206– 207]), and  m 0  = 0 for m ≥ 1, the o rdinary moments o f N are given by Ex[ N m ] = X 1 ≤ l ≤ m n m l o Ex[ N ( N − 1) · · · ( N − l + 1)] = X 1 ≤ l ≤ m n m l o 2 l − 1 (2 l − 3)!! λ l − 1 (1 − λ ) 2 l − 1 . Since  m m  = 1 , this gives the asymptotic formula (1.1) as λ → 1. 4 W e turn now to the ra ndom v a riable K , the maximum n umber of customers in the system simultaneously during the busy p erio d (countin g the customer b eing served, so that K ≥ 1). The distribution of K is given by Pr[ K > k ] = (1 − λ ) λ k 1 − λ k +1 (2 . 2) (see for example Cohen [C 2 , pp. 19 1–193 ]). This distr ibutio n can b e derived as follows. Consider a game play ed b etw een tw o play ers: P , who b egins with v dollars, a nd Q who b egins with w dollar s. At each step of the game, a bias s ed coin is toss ed; P wins with probability p , in which c a se Q pays P o ne dollar, a nd Q wins with the complementary proba bilit y q = 1 − p , in which case P pays Q o ne dollar. The game contin ues un til one of the play ers is ruined (that is, has no money left). It is known that (1) with pro ba bilit y o ne, either P or Q is even tually r uined, and (2), if p 6 = q , then the probability that Q is ruined is Pr[ Q r uined] = ( q /p ) v − 1 ( q /p ) v + w − 1 (2 . 3) (see for example F e lle r [F, p. 3 45]). Now co ns ider a busy p erio d of the M / M / 1 queue. The successive even ts of arr iv als and terminations of service in terv a ls during the busy p erio d co rresp ond to steps in the g ame describ ed ab ove. The wealth of play er P will c o rresp ond to the num b er J of customers in the system, so v = 1. An arriv al will corresp ond to a win by play er P , so p = λ/ (1 + λ ), and the ter mina tion of a ser vice in terv al will c o rresp ond to a win by play er Q , so q = 1 / (1 + λ ). Suppose that play er Q b egins with w = k dollar s. Then the even t K > k will corres p ond to Q being ruined. Substituting these v alues in (2.3 ) yields (2 .2) (see Figure 2). start   WVUT PQRS ONML HIJK K ≤ k GFED @ABC 1 p * * q o o · · · p + + q k k ONML HIJK j p * * q j j · · · p + + q k k ONML HIJK k p / / q j j WVUT PQRS ONML HIJK K >k Figure 2. State tr a nsition dia gram for deter mining whether K ≤ k o r K > k . This corresp ondence also shows what happens for λ ≥ 1. F or λ = 1 (in which case the busy pe r io d is finite with probability one, but its exp ected length is infinite), we have take p = q = 1 / 2 , and hav e Pr[ Q ruined] = v v + w . This result yields Pr[ K > k ] = 1 k + 1 , so that Ex[ K ] = X l ≥ 0 Pr[ K > k ] (2 . 4) diverges log arithmically . Of course, for λ > 1 (in which case the busy p erio d is infinite with po sitive probability), (2.3) shows that (2.4) diverges linear ly . 5 Our next goal is to determine the moments of K : Ex[ K m ] = X k ≥ 0 k m Pr[ K = k ] . W riting ∆ m ( k ) = k m − ( k − 1) m = X 0 ≤ l ≤ m − 1  m l  ( − 1) m − 1 − l k l for the backw ard differences of the m -th p ow ers, and se tting T m ( λ ) = X n ≥ 1 n m λ n 1 − λ n , (2 . 5) summation b y parts yields Ex[ K m ] = X k ≥ 0 k m Pr[ K = k ] = X k ≥ 0 ∆ m ( k + 1) Pr[ K > k ] = (1 − λ ) X k ≥ 0 ∆ m ( k + 1) λ k 1 − λ k +1 = 1 − λ λ X j ≥ 1 ∆ m ( j ) λ j 1 − λ j = 1 − λ λ X j ≥ 1 X 0 ≤ l ≤ m − 1  m l  ( − 1) m − 1 − l j l λ j 1 − λ j = 1 − λ λ X 0 ≤ l ≤ m − 1  m l  ( − 1) m − 1 − l T l ( λ ) . (2 . 6) Since Ex[ K m ] is a linear combination of the T l ( λ ), it will suffice to deter mine the asy mpto tic b ehavior of the sums T m ( λ ). T he sums T l ( λ ) are in fact the Lambert series S l ( λ ) given b y (1.4); we hav e T l ( λ ) = X j ≥ 1 j l λ j 1 − λ j = X j ≥ 1 j l X i ≥ 1 λ ij = X i ≥ 1 X j ≥ 1 j l λ ij = X n ≥ 1 λ n X d | n d l (2 . 7) = X n ≥ 1 σ l ( n ) λ n , = S l ( λ ) where the inner sum in (2.7) is ov er integers d dividing n . 6 W e note that the s ums T l ( λ ) can b e expressed in terms of known (alb eit exotic) functions of ana lysis. W e define the q -gamma function by Γ q ( x ) = (1 − q ) 1 − x Y n ≥ 0 1 − q n +1 1 − q n + x (see for example Gasp er and Rahman[G1, p. 1 6]). (This function gets its name from the fact that lim q → 1 Γ q ( x ) = Γ( x ), where Γ( x ) is the E ule r g amma function; see for example Whittaker and W atson [W, pp. 235–2 64].) If we define the q -digamma function ψ q ( x ) as the logar ithmic deriv ativ e ψ q ( x ) = ∂ ∂ x log Γ q ( x ) = − log(1 − q ) + log q X n ≥ 0 q n + x 1 − q n + x of the q -ga mma function, then we hav e T 0 ( λ ) = ψ λ (1) + log(1 − λ ) log λ . T o go further , we define the l -th q -p olyg amma function ψ ( l ) q as the l -th deriv ative ψ ( l ) q ( x ) =  ∂ ∂ x  l ψ q ( x ) of the q -diga mma function. If we set z = q n + x , then  z ∂ ∂ z  =  1 log q ∂ ∂ x  . Since X i ≥ 1 i l z i =  z ∂ ∂ z  l z 1 − z , we hav e X i ≥ 1 i l q i ( n + x ) = 1 log l q  ∂ ∂ x  l q n + x 1 − q n + x . Summing ov er n ≥ 0 yields X i ≥ 1 i l q ix 1 − q ix = X i ≥ 1 i l X n ≥ 0 q i ( n + x ) = 1 log l q  ∂ ∂ x  l X n ≥ 0 q n + x 1 − q n + x = 1 log l q  ∂ ∂ x  l ψ q ( x ) + log(1 − q ) log q . Thu s for l ≥ 1 w e have T l ( λ ) = ψ ( l ) λ (1) log l +1 λ . 7 Our nex t goa l is to derive the leading terms (1.2 ) and (1 .3) of the a s ymptotic expansion of the moment s of K . T o es ta blish (1.2 ) and (1.3 ), we beg in b y deriving T 0 ( λ ) ∼ 1 1 − λ log 1 1 − λ (2 . 8) and, for l ≥ 1 , T l ( λ ) ∼ l ! ζ ( l + 1) (1 − λ ) l +1 . (2 . 9) Once these fo rmulas are established, it will b e clear that the s um in (2.6) is dominated by the term fo r which l = m − 1, so that E x[ K m ] ∼ m S m − 1 ( λ ), and (1.2) a nd (1.3) follow from (2.8) and (2.9), resp ectively . Our strategy for proving (2.8) and (2.9) will b e to approximate the sums S l ( λ ) by int egra ls I l ( λ ) = Z ∞ 1 x l λ x dx 1 − λ x , then then to show that the differ ence S l ( λ ) − I l ( λ ) is neg ligible in compar ison with I l ( λ ). It will b e conv enien t to wr ite λ = e − h . The limit λ → 1 then corr esp onds to h → 0 . W e hav e h = lo g 1 λ = lo g 1 1 − (1 − λ ) ∼ 1 − λ. (2 . 10) F or l = 0, we have I 0 ( λ ) = Z ∞ 1 λ x dx 1 − λ x = Z ∞ 1 X l ≥ 1 λ lx dx = X l ≥ 1 Z ∞ 1 e − hlx dx = X l ≥ 1 e − hl hl = 1 h X l ≥ 1 λ l l = 1 h log 1 1 − λ . Substituting (2.10 ) in this result yields I 0 ( λ ) ∼ 1 1 − λ log 1 1 − λ . (2 . 11) W e b ound | S 0 ( λ ) − I 0 ( λ ) | by the total v ariation of f ( x ) = λ x / (1 − λ x ). Since f ( x ) decreases monotonically from λ/ (1 − λ ) to 0 a s x incr eases from 1 to ∞ , we hav e | S 0 ( λ ) − I 0 ( λ ) | ≤ λ/ (1 − λ ) ∼ 1 / (1 − λ ). Since this difference is negligible in compar is on with (2.11), we obtain (2.8). 8 F or l ≥ 1, we have I l ( λ ) = Z ∞ 1 x l λ x dx 1 − λ x = Z ∞ 0 ( y + 1) l λ y +1 dy 1 − λ y +1 = Z ∞ 0 X 0 ≤ i ≤ l  l i  y i λ y +1 dy 1 − λ y +1 = Z ∞ 0 X 0 ≤ i ≤ l  l i  y i X j ≥ 1 λ j ( y +1) dy = Z ∞ 0 X 0 ≤ i ≤ l  l i  y i X j ≥ 1 e − hj ( y +1) dy = X 0 ≤ i ≤ l  l i  X j ≥ 1 Z ∞ 0 y i e − hj ( y +1) dy = X 0 ≤ i ≤ l  l i  X j ≥ 1 i ! e − hj ( hj ) i +1 = X 0 ≤ i ≤ l  l i  i ! h i +1 X j ≥ 1 λ j j i +1 = X 0 ≤ i ≤ l  l i  i ! h i +1 Li i +1 ( λ ) , (2 . 12) where Li k ( λ ) = P n ≥ 1 λ n /n k is the k -th p olylo garithm. Since Li 1 ( λ ) = log  1 / (1 − λ )  and Li k ( λ ) → ζ ( k ) as λ → 1 for k ≥ 2, the sum in (2.12) is dominated by the term for which i = l , and we hav e I l ( λ ) ∼ l ! ζ ( l + 1) h l +1 ∼ l ! ζ ( l + 1) (1 − λ ) l +1 (2 . 13) W e b ound | S l ( λ ) − I l ( λ ) | by the total v ar iation of f ( x ) = x l λ x / (1 − λ x ) for 0 ≤ x < ∞ . As x increa ses, f ( x ) increases monotonically from 0 to a maximum, then decreas es monotonica lly to 0. Thus the total v ariation of f ( x ) is t wice the maximum. This maximum is max 0 ≤ x< ∞ f ( x ) = max 0 ≤ x< ∞ x l e − hx 1 − e − hx = max 0 ≤ x< ∞ x l e hx − 1 = 1 h l max 0 ≤ y < ∞ y l e y − 1 . F urther more, y l / ( e y − 1) ≤ l !, b ecause e y − 1 = P n ≥ 1 y n /n ! ≥ y l /l !. Thu s | S l ( λ ) − I l ( λ ) | ≤ 2 max 0 ≤ x< ∞ f ( x ) ≤ 2 l ! /h l ∼ 2 l ! / (1 − λ ) l . Since this differ ence is negligible in compar ison with (2.13), we obtain (2.9). W e shall now show how asymptotic expa nsions, w ith erro r ter ms of the form O  (1 − λ ) R  for any R , can b e derived for all of the momen ts Ex[ K m ]. The ess ence of the a rgument is to use the Euler-Mac laurin 9 formula to estimate the differ e nc e b etw een S l ( λ ) and I l ( λ ). This is mos t conv eniently done using a result of Za g ier [Z]. Indeed, for l ≥ 1, Z agier g ives the e x pansion for S l ( λ ), in terms of the pa rameter h = − log λ rather than 1 − λ . All that remains for us to do is substitute a n ex pa nsion for h in terms of 1 − λ . F or l = 0, the expansion for S 0 ( λ ) in ter ms o f h has b een g iven by Eg ger (n´ e Endres) a nd Steiner [E1, E2], again using the res ult of Zagier . W e shall pro ceed differently , to obtain an ex pansion inv olving − log(1 − λ ) rather than − log h . Pr op osition: (Zagier [Z , p. 3 18]) Let f ( x ) be ana lytic at x = 0 , with p ower s e r ies f ( x ) = P r ≥ 0 b r x r ab out x = 0. Supp ose that R ∞ 0 | f ( r ) ( x ) | dx < ∞ for a ll r ≥ 0, where f ( r ) ( x ) denotes the r - th deriv ative of f ( x ). Define F = R ∞ 0 f ( x ) dx . Let g ( x ) = P n ≥ 1 f ( nx ) . Then g ( x ) has the a s ymptotic expa nsion g ( x ) ∼ F x + X r ≥ 0 b r B r +1 ( − 1) r x r ( r + 1) , (2 . 14) where B r is the r -th Bernoulli num ber, defined by t/ ( e t − 1) = P r ≥ 0 B r t r /r !. This result is prov ed by using the Euler - Maclauren formula, Z N 0 f ( y ) dy = f (0) 2 + X 1 ≤ n ≤ N − 1 f ( n ) + f ( N ) 2 + X 1 ≤ r ≤ R − 1 ( − 1) r B r +1 ( r + 1)!  f ( r ) ( N ) − f ( r ) (0)  + ( − 1) R Z N 0 f ( R ) y B R ( { y } ) R ! dy , where B r ( y ) is the r -th Bernoulli p olyno mial, defined by te y t / ( e t − 1) = P r ≥ 0 B r ( y ) t r /r !, and { y } = y − ⌊ y ⌋ denotes the fra ctional part o f y . (F or the Euler-Ma clauren formula, the Ber no ulli num b ers and the Ber noulli po lynomials, see for example Whittaker and W atson [W, pp. 125–1 28], where, howev er, the indexing of the nu mbers and po lynomials is different.) The condition R ∞ 0 | f ( r ) ( y ) | dy < ∞ allows us to le t N → ∞ , obta ining Z ∞ 0 f ( y ) dy = X n ≥ 1 f ( n ) − X 0 ≤ r ≤ R − 1 ( − 1) r B r +1 ( r + 1 )! f ( r ) (0) + ( − 1) R Z ∞ 0 f ( R ) ( y ) B R ( { y } ) R ! dy . If we now write f ( xy ) instea d of f ( y ), we o bta in Z ∞ 0 f ( x y ) dy = X n ≥ 1 f ( nx ) − X 0 ≤ r ≤ R − 1 ( − 1) r B r +1 ( r + 1 )! f ( r ) (0) x r + ( − 1) R x R Z ∞ 0 f ( R ) ( xy ) B R ( { y } ) R ! dy . Changing the v a riable of integration fr om y to y /x then yields 1 x Z ∞ 0 f ( y ) dy = X n ≥ 1 f ( nx ) − X 0 ≤ r ≤ R − 1 ( − 1) r B r +1 ( r + 1)! f ( r ) (0) x r + ( − 1) R x R − 1 Z ∞ 0 f ( R ) ( y ) B R ( { y /x } ) R ! dy . The integral on the left-hand side is F , the first sum o n the r ight-hand side is g ( x ), f ( r ) (0) = r ! b r , and the last term on the right-hand side is O ( x R − 1 ). Thus F x = g ( x ) − X 0 ≤ r ≤ R − 1 b r B r +1 ( − 1) r x r ( r + 1 ) + O ( x R − 1 ) , which yields the expansion (2.14). 10 F or l ≥ 1, we define f ( x ) = x l e x − 1 . Then f ( x ) is analytic at x = 0 with the T aylor series f ( x ) = X r ≥ 0 B r x r + l − 1 r ! and the integral F = Z ∞ 0 x l e − x dx 1 − e − x = l ! ζ ( l + 1) (see for example Whittak er and W a ts o n [W, p. 26 6]). F urthermor e , f ( r ) ( x ) is a r ational function of x a nd e x , in which the deg ree o f the n umerator in e x is r , while the denominator is ( e x − 1) r +1 . Thus f ( x ) satisfies the conditions of the prop ositio n, and we hav e the asymptotic expansio n g ( x ) ∼ l ! ζ ( l + 1) x + X r ≥ 0 ( − 1) r + l − 1 B r B r + l x r + l − 1 r ! ( r + l ) . Recalling that λ = e − h , so that h = − log λ , we therefore have T l ( λ ) = X n ≥ 1 n l e − hn 1 − e − hn = 1 h l X n ≥ 1 ( nh ) l e nh − 1 = 1 h l X n ≥ 1 f ( nh ) = g ( h ) h l ∼ l ! ζ ( l + 1) h l +1 + X r ≥ 0 ( − 1) r + l − 1 B r B r + l h r − 1 r ! ( r + l ) . (2 . 15) W e no te that, if l is odd, then this expansion has only finitely man y terms (b ecause B r = 0 for odd r ≥ 3). T o obta in an asy mptotic expans ion in terms o f 1 − λ , we must substitute the expansion for 1 /h : 1 h = 1 − log λ = − 1 log  1 − (1 − λ )  = 1 1 − λ − (1 − λ ) log  1 − (1 − λ )  = 1 1 − λ X r ≥ 0 ( − 1) r C r (1 − λ ) r r ! , (2 . 16) where C r is the r -th Bernoulli num ber of the seco nd kind, defined by t/ log(1 + t ) = P r ≥ 0 C r t r /r ! (see fo r example Roman [R2, p. 116 ]). (These num ber s are a lso ca lled the Cauchy num bers of the first kind, and are given by C r = R 1 0 x ( x − 1) · · · ( x − r + 1 ) dx ; see for example Comtet [C2, pp. 293– 294].) 11 F or l = 0, we must pro ceed differently , b ecause f ( x ) = 1 e x − 1 has a p ole at x = 0. W e define f ∗ ( x ) = f ( x ) − e − x x = 1 e x − 1 − e − x x . Then f ∗ ( x ) is analytic a t x = 0 with the T aylor series f ∗ ( x ) = X r ≥ 0  B r +1 − ( − 1) r +1  x r ( r + 1 )! and the integral F ∗ = Z ∞ 0  1 e x − 1 − e − x x  dx = γ (see for example Whittak er and W atso n [W, p. 246]). F urthermore , f ∗ ( R ) ( x ) is a r ational function of x and e x , in which the degree o f the numerator in e x is R , while the denominator is  ( e x − 1) x  R +1 . Thus f ∗ ( x ) satisfies the conditions of the prop osition, and we hav e the asy mpto tic expans ion g ∗ ( x ) ∼ γ x + X r ≥ 0 ( − 1) r B r +1  B r +1 − ( − 1) r +1  x r ( r + 1 ) ( r + 1)! . W e ther e fore hav e T 0 ( λ ) = X n ≥ 1 e − nh 1 − e − nh = X n ≥ 1 1 e nh − 1 = X n ≥ 1 e − nh nh + X n ≥ 1 1 e nh − 1 − e − nh nh = 1 h log 1 1 − λ + X n ≥ 1 f ∗ ( nh ) = 1 h log 1 1 − λ + g ∗ ( h ) ∼ 1 h log 1 1 − λ + γ h + X r ≥ 0 ( − 1) r B r +1  B r +1 − ( − 1) r +1  h r ( r + 1 ) ( r + 1)! . (2 . 17) T o obtain as ymptotic expans io ns for the moments o f K , we substitute (2.16) into (2.15) a nd (2 .1 7), then substitute the results into (2.6), using the expansio n 1 − λ λ = 1 − λ 1 − (1 − λ ) = X r ≥ 1 (1 − λ ) r . 12 Retaining only terms that do not v anish as λ → 1, we obtain Ex[ K ] = log 1 1 − λ + γ + O  (1 − λ ) log 1 1 − λ  and Ex[ K 2 ] = π 2 3 1 1 − λ + log 1 1 − λ + ( γ − 1) + O  (1 − λ ) log 1 1 − λ  for the first tw o moments. Thus we hav e V ar[ K ] = Ex[ K 2 ] − Ex[ K ] 2 = π 2 3 1 1 − λ − log 2 1 1 − λ + (1 − 2 γ ) log 1 1 − λ − γ 2 + O  (1 − λ ) log 2 1 1 − λ  . 3. Searc h Algo ri thms In this sectio n we sha ll a nalyze sear ch algorithms for M / M / ∞ and M / M / 1 systems. W e b egin b y studying the distribution of the random v ariable L , defined as the index of the fir st idle ser ver S found by an arr iving cus tomer in the M / M / ∞ system. Our go al is to prov e (1.8), which gives the first tw o terms in the a symptotic expansio ns of the moments of L . The key to our res ults is the probability P r[ L > l ], which is simply the pr obability that the fir st l servers S 1 , . . . , S l are all busy . It is well known that this probability is g iven b y the Erla ng loss formula Pr[ L > l ] = λ l /l ! P 0 ≤ k ≤ l λ k /k ! = 1 D l , where D l = X 0 ≤ k ≤ l l ! ( l − k )! λ k (3 . 1) (see for example Newell [N, p. 3]). The sum D l can b e ex pressed as an in tegral, D l = Z ∞ 0  1 + x λ  l e − x dx (see for exa mple Newell [N, p. 7]), and mo st of Newell’s analysis is bas e d on such a r epresentation. But we shall work directly with the expressio n of D l as the sum in (3.1). W e shall par tition the v alues of l into t wo r a nges. The first, which we shall call the “bo dy” of the distribution, will b e 0 ≤ l ≤ l 0 = λ − s , where s = √ λ . The s e cond, which we s hall ca ll the “ta il”, will b e l > l 0 . W e b egin with the b o dy . W e s hall es ta blish the estima te Pr[ L > l ] = (1 − l /λ ) + 1 λ (1 − l/ λ ) + O  1 λ  + O  1 λ 2 (1 − l /λ ) 3  (3 . 2) for l ≤ l 0 = λ − s , where s = √ λ . W e be g in by us ing the princ iple of inclusion-e x clusion to derive b ounds on the deno minator D l . 13 W e b egin with a lower b ound. Since l ( l − 1) · · · ( l − k + 1) ≥ l k −   X 0 ≤ j ≤ k − 1 j   l k − 1 = l k −  k 2  l k − 1 , we hav e D l = X 0 ≤ k ≤ l l ( l − 1) · · · ( l − k + 1) λ k ≥ X 0 ≤ k ≤ l  l λ  k − 1 λ X 0 ≤ k ≤ l  k 2   l λ  k − 1 . F or the fir st sum we have X 0 ≤ k ≤ l  l λ  k = 1 + O  ( l/ λ ) l  1 − l /λ . W e note that the log arithm of ( l /λ ) l has a no n-negative seco nd deriv ativ e for l ≥ 1. Thus ( l/λ ) l assumes its maximu m in the in terv a l 0 ≤ l ≤ l 0 for l = 0 , l = 1 o r l = l 0 . Its v alues there ar e 0 , 1 /λ a nd (1 − s/ λ ) λ − s = (1 − 1 / √ λ ) λ − √ λ ≤ e − √ λ +1 , resp ectively . As λ → ∞ , the large s t of these v alues is 1 /λ , so we hav e O  ( l/ λ ) l  = O (1 /λ ) for 0 ≤ l ≤ l 0 . Thus the fir s t sum is X 0 ≤ k ≤ l  l λ  k = 1 + O (1 / λ ) 1 − l /λ . F or the s e cond sum we have X 0 ≤ k ≤ l  k 2   l λ  k − 1 = 1 + O  l 2 ( l/ λ ) l  (1 − l /λ ) 3 . The log a rithm of l 2 ( l/ λ ) l has a non-negative s econd deriv ativ e for l ≥ 3, so an ar gument similar to that used for the first s um shows that O  l 2 ( l/ λ ) l  = O (1 /λ ) for 0 ≤ l ≤ l 0 . Thus we have X 0 ≤ k ≤ l  k 2   l λ  k − 1 = 1 + O (1 / λ ) (1 − l /λ ) 3 and the lower bo und D l ≥ 1 + O (1 / λ ) 1 − l /λ − 1 + O (1 / λ ) λ (1 − l /λ ) 3 . (3 . 3) F or a n upp er b ound, we hav e l ( l − 1) · · · ( l − k + 1) ≤ l k −   X 0 ≤ j ≤ k − 1 j   l k − 1 +   X 0 ≤ i l ], we take the recipro c a l of D l : Pr[ L > l ] =  1 + O (1 / λ ) 1 − l /λ − 1 + O (1 / λ ) λ (1 − l /λ ) 3 + O  1 λ 2 (1 − l /λ ) 5  − 1 =  1 + O (1 / λ )  (1 − l /λ )  1 − 1 λ (1 − l/ λ ) 2 + O  1 λ 2 (1 − l /λ ) 4  − 1 =  1 + O (1 / λ )  (1 − l /λ )  1 + 1 λ (1 − l/ λ ) 2 + O  1 λ 2 (1 − l /λ ) 4  =  1 + O (1 / λ )   (1 − l /λ ) + 1 λ (1 − l /λ ) + O  1 λ 2 (1 − l /λ ) 3  . Observing that O (1 /λ ) (1 − l /λ ) = O (1 /λ ) and O (1 /λ ) /λ (1 − l /λ ) = O  1 /λ 2 (1 − l /λ ) 3  , we obtain (3.2). W e turn no w to the tail. W e shall establish the estimate Pr[ L > l ] = O ( e − λ λ l /l !) (3 . 4) for l ≥ λ − s , whe r e s = √ λ . T o obtain an upp er b ound on P r[ L > l ], w e obtain a low er b ound on D l . W e hav e D l = X 0 ≤ k ≤ l l ! ( l − k )! λ k ≥ l ! ⌊ λ − s ⌋ ! λ l −⌊ λ − s ⌋ + · · · + l ! ⌊ λ − 2 s ⌋ ! λ l −⌊ λ − 2 s ⌋ , (3 . 5) bec ause l − ⌊ λ − s ⌋ ≥ l − ( λ − s ) ≥ 0 by ass umption and ⌊ λ − 2 s ⌋ ≥ 0 for a ll sufficiently la rge λ . There are ⌊ λ − 2 s ⌋ − ⌊ λ − 2 s ⌋ + 1 ≥ s terms in the sum (3 .5). F ur thermore, the smallest of these terms is the last, b ecause its denominator contains factor s of λ where the preceding terms contain factors sma lle r than λ . Thus we hav e D l ≥ s l ! ⌊ λ − 2 s ⌋ ! λ l −⌊ λ − 2 s ⌋ . F or the factoria l in the denominator of this bo und, we shall us e the estimate n ! ≤ e √ n e − n n n , which holds for all n ≥ 1 (b ecause the trap ezoida l rule under estimates the in tegral R n 1 log x dx o f the c oncav e function log x ). This estimate y ie lds D l ≥ s l ! e ⌊ λ − 2 s ⌋ e p ⌊ λ − 2 s ⌋ ⌊ λ − 2 s ⌋ ⌊ λ − 2 s ⌋ λ l −⌊ λ − 2 s ⌋ . (3 . 6) 15 W e have e ⌊ λ − 2 s ⌋ ≥ e λ − 2 s − 1 , ⌊ λ − 2 s ⌋ ⌊ λ − 2 s ⌋ ≤ ( λ − 2 s ) ⌊ λ − 2 s ⌋ = λ ⌊ λ − 2 s ⌋ (1 − 2 s/λ ) ⌊ λ − 2 s ⌋ ≤ λ ⌊ λ − 2 s ⌋ (1 − 2 s/λ ) λ − 2 s − 1 ≤ λ ⌊ λ − 2 s ⌋ e ( − 2 s/λ )( λ − 2 s − 1) ≤ λ ⌊ λ − 2 s ⌋ e − 2 s +4 s 2 /λ +1 ≤ λ ⌊ λ − 2 s ⌋ e − 2 s +5 and p ⌊ λ − 2 s ⌋ ≤ s. Substituting these b ounds into (3.6) yields D l ≥ l ! e λ e 7 λ l . T a king the r ecipro cal o f this b ound yields (3.4). W e sha ll now use (3 .2) a nd (3.4) to prove (1.8 ). W e write ∆ m ( l ) = l m − ( l − 1 ) m = m l m − 1 + O ( l m − 2 ) for the backw ard differences of the m -th p ow ers of l . Then partial summation yields Ex[ L m ] = X l ≥ 0 l m Pr[ L = l ] = X l ≥ 0 ∆ m ( l ) P r[ L > l ] = X l ≥ 0 m l m − 1 Pr[ L > l ] + O   X l ≥ 0 l m − 2 Pr[ L > l ]   (3 . 7) This formula shows that we should ev a luate sums of the form U n = X l ≥ 0 l n Pr[ L > l ] . (3 . 7) W e sha ll show that U n = λ n +1 ( n + 1)( n + 2) + λ n log λ 2 + O ( λ n ) . (3 . 8) Substitution of this formula into (3.7) will then yield (1.8 ). W e sha ll break the r ange of s umma tio n in (3.7) at l 0 = λ − s , where s = √ λ , using (3.2) for 0 ≤ l ≤ l 0 and (3 .4 ) for l > l 0 . Summing the first term in (3 .2), we have X 0 ≤ l ≤ l 0 l n (1 − l /λ ) = 1 λ X 0 ≤ l ≤ l 0 ( λ l n − l n +1 ) = 1 λ  λ l n +1 0 n + 1 + O ( l n 0 )  −  λ n +2 n + 2 + O ( l n +1 0 )  = 1 λ  λ ( λ n +1 − ( n + 1) λ n s ) n + 1 + O ( λ n )  −  λ n +2 − ( n + 2) λ n +1 s n + 2 + O ( λ n +1 )  = λ n +1 ( n + 1)( n + 2) + O ( λ n ) . 16 Summing the second term in (3.2), we hav e X 0 ≤ l ≤ l 0 l n λ − l = X s ≤ k ≤ λ ( λ − k ) n k = X s ≤ k ≤ λ  λ n k + O ( λ n − 1 )  = λ n log λ s + O ( λ n ) = λ n log λ 2 + O ( λ n ) , where we have used P 1 ≤ k ≤ n 1 /k = lo g n + O (1). Summing the third term in (3.2) of course yields O ( λ n ). Summing the last term in (3.2), we hav e λ X 0 ≤ l ≤ l 0 l n ( λ − l ) 3 = λ X s ≤ k ≤ λ ( λ − k ) n k 3 ≤ λ n +1 X s ≤ k ≤ λ 1 k 3 ≤ λ n +1 X k ≥ s 1 k 3 = λ n +1  2 s 2 + O  1 s 3  = O ( λ n ) , where we hav e used P k ≥ n 1 /k 3 = 2 /n 2 + O (1 / n 3 ). Co m bining these estimates, we obtain X 0 ≤ l ≤ l 0 l n Pr[ L > l ] = λ n +1 ( n + 1)( n + 2) + λ n log λ 2 + O ( λ n ) . (3 . 9) Finally , summing (3.4 ) we hav e X l>l 0 l n e − λ λ l l ! ≤ X l ≥ 0 l n e − λ λ l l ! = O ( λ n ) , bec ause the summation o n the right-hand side is the n - th moment o f a Poisson random v ariable with mea n λ , which is a p oly no mial o f de g ree n in λ . Thus X l>l 0 l n Pr[ L > l ] = O ( λ n ) . Combining this estimate with (3.9) yields (3.8) and completes the pro of of (1.8). Our final go al is to study the distribution of the r a ndom v ariable I , defined as the index of the first v aca n t waiting station W I found by a customer in the M / M / 1 sys tem, when the server, up on b ecoming free when at least o ne customer is waiting, ser ves the customer a t the first o cc upied waiting s tation. W e shall show that the distribution of I is g iven b y Pr[ I > i ] = (1 − λ ) λ i +1 1 − λ i +1 . (3 . 10) 17 The even t I > i is simply the even t tha t the fir st i waiting stations W 1 , . . . , W i are all o ccupied; we hav e I = 0 when the server is idle. Letting the ra ndom v ariable J denote the num ber o f customers in the system, as b efore , we o bs erve that the even t J > i is necessar y for the even t I > i : if statio ns W 1 , . . . , W i are o ccupied and the server is busy , there ar e a t least i + 1 custo mers in the sys tem. Thus we can wr ite Pr[ I > i ] = X j >i Pr[ I > i | J = j ] Pr[ J = j ] . W e sha ll show that Pr[ I > i | J = j ] = 1 − λ 1 − λ i +1 (3 . 11) for all j > i . Since P r[ J > i ] = λ i +1 , it will then follow tha t Pr[ I > i ] = 1 − λ 1 − λ i +1 X j >i Pr[ J = j ] = 1 − λ 1 − λ i +1 Pr[ J > i ] = (1 − λ ) λ i +1 1 − λ i +1 , confirming (3.10) Before proving (3.1 1 ) in the genera l ca se, it will be helpful to consider tw o sp ecial cas es. If i = 0 , then the ev ent J > i is sufficient a s well a s neces sary for the even t I > i : if the server is busy , an arriving customer m ust wait. Thus P r[ I > i | J = j ] = 1 for all j > 0, c onfirming (3 .11) in this cas e. F o r i = 1, we assume that J = j > 1 and a sk for the conditiona l probability that W 1 is o ccupied. If the cur rent ar riv a l o ccurs at time t 0 , we cons ider the la test trans ition in the embedded Ma rko v chain for J that precedes t 0 . Suppo se this previous transition occur s at time t 1 . If this pre vious tr ansition was a n a rriv al, then W 1 will be occ upied by it (if it was not alrea dy o ccupied) a nd thus will b e o ccupied at t 0 . If o n the other hand this previous transition was a departure, then W 1 will be v acated by it (if it w as not a lready v acant) a nd thus will b e v a c ant at t 0 . Th us we m ust determine the pro ba bilit y that this previous tra nsition w as an ar riv a l. W e claim that the pr evious transitio n was an arriv al is q = 1 / (1 + λ ) and the proba bilit y that it was a departure is p = λ/ (1 + λ ). T o prove this claim, we note that the Marko v chain fo r J is r eversible ; that is, if a movie is made of its tra nsitions, the movie r un backw ard is sto chastically indistinguisha ble from the movie run for w ard. (Reversibilit y follows from the fact that in this Marko v chain, trans itions o ccur only b et ween adjacent states; that is, J is incremented by a n a rriv al and decr emen ted by a depa rture.) The previous transition was a n arriv al if and only if it app e ars as a depar ture when the movie is run backw ard, and by reversibilit y this event oc c urs with probability q = 1 / (1 + λ ) provide that the trans ition do es not involv e the state J = 0. (When J = 0 , the next transition is an a rriv al with proba bilit y o ne , ra ther than with probability p = λ/ (1 + λ ).) But since J > 1 a t t 0 , we have J > 0 at t 1 . This prov es our claim. W e therefore hav e Pr[ I > i | J = j ] = q = 1 / (1 + λ ) = (1 − λ ) / (1 − λ 2 ), again confirming (3.11). W e are now ready to pr ov e (3.11) in the ge ne r al case. W e assume that J = j > i and a sk for the conditional pro bability that W 1 , . . . , W i are all occupied. T o determine whether this even t o ccur s, we shall again trace ba ckw ard in time through the transitions preceding the current a rriv al a t time t 0 . In this case we may hav e to trace ba c k through arbitra rily ma ny transitions. As we tra ce backw ard, we keep track of the difference d ( t ) betw een the n umber of a rriv als and the n umber of departures in the interv a l [ t, t 0 ). W e shall contin ue tracing as long as − 1 < d ( t ) < i , stopping at the la test time t 1 such that d ( t 1 ) = − 1 or d ( t 1 ) = i . 18 First, w e claim that if d ( t 1 ) = i , then W 1 , . . . , W i are all o ccupied at time t 0 . T o prove this claim, w e match each departure in [ t 1 , t 0 ) with a later arriv al “like parentheses”. That is, we as so ciate each depa rture in this interv al with a left pare nthesis and each arriv al with a rig h t parenthesis. Since d ( t ) ≥ 0 for t 1 ≤ t ≤ t 0 , there a re at lea s t as many right par ent heses as left par ent heses in any suffix of the re sulting str ing , so all the left parentheses can b e matched to right parentheses, leaving d ( t 1 ) = i r ight parentheses unmatched. W e thus have i unmatched arr iv als . Thes e a rriv als o ccupy the stations W 1 , . . . , W i (if they were not alr eady o ccupied), and an y stations that are v acated by subsequent departures ar e r eo ccupied by the matching arriv als, so these stations all rema in o ccupied at time t 0 . This prov es o ur fir st cla im. Next, we claim that if d ( t 1 ) = − 1, then at least one of the stations W 1 , . . . , W i is v a c a nt a t time t 0 . T o prov e this cla im, we define m = max { d ( t ); t 1 ≤ t ≤ t 0 } . W e hav e m < i . W e define t 2 = max { t : t 1 < t ≤ t 0 and d ( t ) = m } . Since d ( t ) ≤ m for t 1 ≤ t ≤ t 2 , we ca n ma tc h arr iv als with later depa rtures in this int erv al, leaving at least m + 1 departures unmatched. These departures will v acate stations W 1 , . . . , W m +1 (if they were not already v ac a nt ), a nd any of thes e statio ns that are o ccupied by subs e quent a rriv als in this int erv al will b e rev acated by the matching depar tures, s o that these s tations will all b e v aca nt at time t 2 . F or each of these m + 1 stations W j , let e ( j ) denote the difference b etw een the num b e r of arriv als that o ccupy W j and the num b er of depar tures that v acate W j . W e have 0 ≤ e ( j ) ≤ 1 , b ecause arriv als that o ccupy a given s tation alternate with departur es that v acate it. Since d ( t 2 ) = m , we hav e P 1 ≤ j ≤ m +1 e ( j ) = m . It follows that e ( j ) = 0 for at le ast o ne v alue o f j , a nd so W j remains v acant at time t 0 for this v alue of j . This prov es our second claim. Finally , we c la im that a s we tr a ce backw ard, the probability that the next tra nsition is an arriv al is alwa ys q = 1 / (1 + λ ), and the pro bability tha t the next tra nsition is a departure is always p = λ/ (1 + λ ). T o prov e this cla im, we observe that there are at least i + 1 customers in the sy stem at time t 0 , and thus a t least i + 1 − d ( t ) customers in the system at a n y time t for t 1 < t < t 0 . F urthermore i + 1 − d ( t ) ≥ 2 , b ecause d ( t ) < i for t 1 < t < t 0 . Thus there are at lea st t wo customers in the system a t every time t ∈ ( t 1 , t 0 ). It follows that as we trace ba ckward, none o f the transitio ns we encounter involv e the state J = 0. Th us as we trace backw ard, the probability that the next transition is an arr iv al is alwa ys q = 1 / (1 + λ ), and the probability that the next transition is a depar tur e is always p = λ/ (1 + λ ). This pr ov es o ur third claim. These three claims show that the pr o cess o f deter mining whether I > i (given that J = j > i ) is as shown in Figure 3. start   ONML HIJK GFED @ABC I ≤ i GFED @ABC 0 q * * p o o · · · q + + p k k GFED @ABC d q * * p j j · · · q , , p k k ONML HIJK i − 1 q / / p j j ONML HIJK GFED @ABC I >i Figure 3. State tr ansition diagram for determining whether I ≤ i o r I > i (given that J = j > i ). Compariso n with Figure 2 shows that this pro cess is the sa me at the pro cess of deter mining whether K > k , except that the roles of p and q are exchanged. This exc hange is equiv alen t to the substitution of 19 1 /λ for λ , so Pr[ I > i | J = j ] is obtained by making this substitution in the e x pression (2.2) fo r Pr[ K > k ]; Pr[ I > i | J = j ] = (1 − 1 /λ ) (1 /λ ) i  1 − (1 /λ ) i +1  = 1 − λ 1 − λ i +1 . This again confirms (3 .11), which completes the pro of of (3.10). It fo llows that the mo ments o f I differ from those o f K by a factor of λ = 1 − (1 − λ ). This fact allows asymptotic expansions for the moments of I to be obtained fro m those o f K , with the result that the leading terms are the sa me. 4. Conclusion W e o bserve that the search alg orithm describ ed in the pr eceding section for the M / M / 1 system defines a deter ministic service discipline distinct fro m b oth first-come-fir st-served and last-come-first- s erved. It would b e of interest to determine the distribution o f the waiting time W exper ienced by a customer for this discipline, or even the moments of this distribution. W e hope to address this q ue s tion in a future pap er . 5. Ac kno wledgmen t The r esearch repor ted her e was suppo rted by Gra nt CCF 0 91702 6 from the National Science F oundation. 6. References [C1] E. G. Coffman, Jr., T. T. Kadota a nd L. A. Shepp, “A Sto chastic Mo del of F ragmentation in Dynamic Storage Allo cation”, SIAM J. Comput. , 14:2 (1985 ) 416– 425. [C2] J. W. 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