Analysis of an M/M/1 Queue Using Fixed Order of Search for Arrivals and Service

egp.ex .tex Analysis of an M / M / 1 Queue Using Fixed Order of Searc h for Arriv als and Service 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 a nalyze an M / M / 1 queue with a se rvice discipline in which customers , up on ar riving when the server is busy , search a se q uence of stations for a v a cant statio n at whic h to wait, and in whic h the server, upon b ecoming free when one or more customers are waiting, searches the stations in the same order for a s tation o ccupied b y a customer to serve. W e show how to find complete as y mptotic expans ions for all the mo men ts of the w aiting time in the heavy traffic limit. W e show in particular that the v ariance of the waiting time for this discipline is mo re similar to that of last-come-first-ser ved (which has a p ole of order three a s the arriv al ra te appro aches the ser vice r a te) than t hat o f fir st-come-first- s erved (whic h has pole of order t wo). Keyw ords: Queueing theor y , Lambert ser ie s, asymptotic expansio ns. Sub ject Classification: 60 K26, 9 0B22 1. In tro duction W e consider the M / M / 1 queue (with indep endent expo nen tially distributed interariv al times, indep en- dent exp onentially distributed service times, a nd a single server) with v ar ious s e r vice disc iplines. W e shall be interested mainly in the “heavy traffic” limit, λ → 1, where λ is the arriv al rate (measured in units o f the service rate), and all asymptotic statements in this pap er refer to this limit. It is well known (see Little [L]) that the av erage waiting time E x[ W ] ( W is the length o f in terv a l from arriv al to commencement of service) does not depend on t he service discipline (the rule used to deter mine which waiting customer is served next when the server b ecomes free). W e hav e Ex[ W ] = λ 1 − λ ∼ 1 1 − λ . (1 . 1) The v ar iance of W , how e ver, (and more gener ally its higher moments) do es dep end on the s ervice discipline. Kingman [K ] has shown that, among all service disciplines, “first-c ome-first-ser ved” (FCFS) minimizes the v ar iance of W . W e hav e V ar[ W FC FS ] = λ (2 − λ ) (1 − λ ) 2 ∼ 1 (1 − λ ) 2 (1 . 2) (see for example Rior da n [R2, pp.1 0 2–103 ]). T ambouratzis [T] has shown that “la st-come-firs t-served” (LCFS) ma ximizes this v ariance. W e have V ar[ W LCFS ] = λ (2 − λ + λ 2 ) (1 − λ ) 3 ∼ 2 (1 − λ ) 3 (1 . 3) (see for example Riorda n [R2, pp. 10 6–109 ]; LCFS w as firs t analyzed by V aulot [V2]). W e note that the difference betw een F CFS and LCFS is qua litative, in that V ar[ W LCFS ] has a pole of order three at λ = 1, whereas V ar [ W FC FS ] has a po le of o rder only t wo there. Another service discipline that has been studied is “ random-or der-of-ser vice” (ROS), first successfully analyzed by V a ulot [V1]. One of the motiv a tio ns for studying R O S was sta ted by Riorda n [R1]: “In many s witc hing s y stems it is not feasible to fully r ealize this ethical ideal of firs t come, first s erved, and it ha s lo ng b een o f interest to deter mine delays on a nother basis . The co n trasting assumption is of calls pick ed a t r andom, which is again an idealization but in la rge office s app ear s to b e called for, as a b ound fo r the s e rvice actually given.” This s ta temen t sug gests that (1) in pra c tical systems of that er a (around 195 3) it was not p ossible to keep tra ck of the o r der of arriv al, (2 ) ROS w as analyzed as a substitute for the serv ic e discipline actually implemen ted, and (3) it was hop ed that the perfor mance o f R OS would appr oximate that of the discipline actually implement ed. F or ROS, we hav e V a r[ W R OS ] = λ (4 − 2 λ + λ 2 ) (2 − λ ) (1 − λ ) 2 ∼ 3 (1 − λ ) 2 (1 . 4) (see for example Riordan [R2, pp. 103– 106]). Comparing (1.4) with (1.2) and (1.3), we s ee that R O S is qualitatively “more like” F CFS than LCFS, in that V ar[ W R OS ] has a pole o f order t w o rather than three, though its coefficient is larg er than that of FCFS by a factor of three. 1 In the early 1950s, telepho ne s witching sys tems were elec tr omechanical and did not employ random- ization beyond that present in the arriv al and service pro c e sses. In this paper w e shall analy ze a service discipline that w e shall call “fixed-order-o f-search” (FOS). F or this discipline, there is an infinite sequence W 1 , W 2 , . . . of “waiting stations”, each of which can b e either “v acant” or “ o ccupied”. A cus tomer a rriving when the server is busy searches these stations in increasing order of their indices a nd occupies the first v acant station it finds. When the server beco mes free and one or more custo mers are w aiting, it s earches the statio ns in the same or der and serves the customer waiting at the first o ccupied station it finds, thereby v acating that statio n. This discipline was introduced b y E s chenf eldt, Gross and Pippenger [E]. Like FCFS and LCFS (and unlike ROS), it do es no t employ ra ndomization b eyond that pres en t in the arriv al a nd service pro cesses. If the serv er were to search the stations in the rev erse or der to that us ed by arriving customers (serving the customer waiting at the o c cupied station with the la rgest, rather than the smalles t, index), the result would b e LCFS, with its concomitant maximum v ariance for the waiting time. The c hoice of the same order of sea rch for b oth customers and the server thus repr e s ent s an attempt to improve upon LCFS, while still using a fixed order of search in each case. W e shall give an exact formula for V ar[ W FOS ]: V a r[ W FOS ] = λ (6 + λ + λ 2 ) (1 − λ ) 3 − 4 ψ (1) λ (1) (1 − λ ) log 2 λ , (1 . 5) where ψ (1) q ( x ) is the q -trig amma function (defined b elow). W e s hall indicate ho w s imilar, but increasing ly more co mplica ted, formulas can b e derived for the higher moments of W FOS . W e shall also give an asymptotic formula for V ar[ W FOS ]: V a r[ W FOS ] ∼ 8 − 4 ζ (2 ) (1 − λ ) 3 (1 . 6) where ζ ( s ) = P n ≥ 1 1 /n s is the Riemann zeta function and ζ (2) = π 2 / 6 (see for example Whit taker and W a tson [W2, pp. 265– 280]). W e shall als o indicate how complete asymptotic e x pansions (with error terms of the form O  (1 − λ ) R  for any R ) can b e derived for the v ariance o f W FOS , as well as for the higher mo men ts. Comparing (1.6) with (1.2) and (1.3 ), we see that FOS is qualitatively “more like” LCFS than FCFS, in that V a r[ W FOS ] has a p ole of o rder thre e ra ther than tw o , though its co efficient is smaller than that o f LCFS by a fa ctor of 4 − 2 ζ (2) = 4 − π 2 / 3 = 0 . 7101 . . . . Eschenfeldt, Gross and Pipp enger [E ] initia ted the study of F OS, determining the distr ibution of the index I of the station W I at which a ne wly arriving customer waits (where I = 0 if the server is idle at the time of the arriv al): w e have Pr[ I ≥ 0] = 1 a nd Pr[ I ≥ i ] = (1 − λ ) λ i 1 − λ i (1 . 7) for i ≥ 1. T he moments of I can be expressed in terms of the sums T l ( λ ) = X i ≥ 1 i l λ j 1 − λ i , (1 . 8) for which we hav e the asymptotic formulas T 0 ( λ ) ∼ 1 1 − λ log 1 1 − λ (1 . 9) 2 and T l ( λ ) ∼ l ! ζ ( l + 1) (1 − λ ) l +1 (1 . 10) for l ≥ 1 , and the ex act for m ulas T 0 ( λ ) = ψ λ (1) + log(1 − λ ) log λ (1 . 11) and T l ( λ ) = ψ ( l ) λ (1) log l +1 λ (1 . 12) for l ≥ 1 . Here ψ q ( x ) = ∂ log Γ q ( x ) /∂ x is the q -dig amma function, the log arithmic deriv ative o f the q -gamma function Γ q ( x ) = (1 − q ) 1 − x Q n ≥ 0  (1 − q n +1 ) / (1 − q n + x )  (see for example Gasp er and Rahma n [G, p. 16]), and ψ ( l ) q ( x ) = ∂ l ψ q ( x ) /∂ x l is the l -th q -p olygamma function. (The sums T l ( λ ), whic h a re called Lambert series (see for example Hardy and W right [H, p. 257]), ar e the generating functions T l ( λ ) = P n ≥ 1 σ l ( n ) λ n for the sums σ l ( n ) = P d | n d l of th e l - th p ow ers o f the divisors of o f n (see for example Hardy and W rig h t [H, p. 239]).) In terms of the T l ( λ ), we hav e Ex[ I m ] = (1 − λ ) X 0 ≤ l ≤ m − 1  m l  ( − 1) m − 1 − l T l ( λ ) . In Section 2 we sha ll determine the moment genera ting M W ( s ) function for W FOS (whic h in what follows we sha ll denote simply W ). In Section 3, we sha ll der ive the exact fo r mu la (1.5) and the asymptotic fo r mu la (1.6). W e s hall also indicate how similar exact and a symptotic for mu las ca n b e found for the higher mo men ts of W . Finally , w e shall indicate how thes e asymptotic formulas can b e extended to complete asymptotic expansions (with erro r terms of the form O  (1 − λ ) R  for any R ) for these qua ntities. 2. The Gene rating F unctions Consider the ra ndom proc e s s whose state v ar iable J denotes the n um be r of customers in the system. The ra ndom v aria ble J is zer o during an idle p erio d (in terv al of time during which the ser ver is idle). It is incremented whenever a customer ar rives, and decremented whenever a customer departs (that is, at the termination of a service interv al). Arriv als o ccur in a P oisso n proces s with rate λ . During a busy per io d (in terv a l of time during which J ≥ 1 ), de pa rtures o ccur in an indep endent Poisson pro cess at rate 1. Thus, during a bus y p erio d, “trans itions”, by which we mean arriv als and departures together, o ccur in a Poisson pro cess at rate 1 + λ . F urthermore, dur ing a bus y per io d, the probability that the next transition will be an arr iv al is p = λ/ (1 + λ ), a nd the pr obability that it will be a departure is q = 1 / (1 + λ ). In this section we sha ll study the distribution of the random v ariable N , defined as the num b er of transitions that occ ur betw een the ar riv al of a cus to mer (excluded) and the departur e that initia tes its service interv al (included). Spec ific a lly , we shall determine the probability g enerating function g ( t ) = P n ≥ 0 Pr[ N = n ] t n for N . W e hav e N = 0 if the arr iv al initiates a busy p erio d, and N ≥ 1 if it o ccurs during a busy p erio d. The index I of the station W I at which a newly arr iving c ustomer waits has the distribution giv en b y (1.7). As a first step to determining g ( t ), w e shall determine the co nditional gener ating function g i ( t ) = P n ≥ 0 Pr[ N = n | I = i ] t n . W e b egin with tw o sp ecial cases. If i = 0, the arr iv al initiates a busy perio d, so N = 0 and g 0 ( t ) = 1. If i = 1, the custo mer waits at W 1 and will b e ser ved a s so on as the nex t depa rture o ccurs. The n umber of transitions preceding and including this next departure has a ge ometric dis tr ibution, and g 1 ( t ) = P n ≥ 1 p n − 1 q t n = q t/ (1 − pt ). 3 F o r the general cas e , w e consider the random proces s whose state v ariable K denotes the num ber of v acant sta tions among W 1 , . . . , W i . Since the customer in question w aits at station W i , we have K = 0 immediately after the arriv al of tha t customer. F urthermor e, the first time ther eafter at which K = i coincides with the beg inning of the se rvice interv al for that customer , and th us o ccurs a fter exactly N transitions hav e o ccurred. When 1 ≤ K ≤ i − 1 , the random v ariable K is incremen ted b y a departure (beca use the next customer ser ved will b e waiting at o ne of the sta tions under co nsideration) and decremented by an arr iv al (b ecause the a rriving customer will wait at o ne of thes e stations). If howev er K = 0 , a depar ture will incremen t K , but an arr iv al will leave K unchanged (because it will ha v e to wait at a station beyond W i ). Thu s the pro cess determining N given I = i is a r andom walk with one “reflecting barrier ” (at K = 0) and o ne “absor bing barrier” (a t K = i ), a s shown in Figure 1. start / / GFED @ABC 0 p   q + + GFED @ABC 1 q * * p k k · · · q , , p k k ONML HIJK i − 1 q / / p j j _^]\ XYZ[ WVUT PQRS stop Figure 1. The pro ces s deter mining N , given that I = i . Non-terminal states are lab eled with v alues o f the ra ndom v ariable K . The num b er N of steps to abso rption in this pro cess ha s b een studied by W e e sakul [W1], w ho shows that the generating function is given by g i ( t ) = q i t i  Q ( t ) − P ( t )  Q ( t ) i +1 − P ( t ) i +1 − pt  Q ( t ) i − P ( t ) i  , (2 . 1) where P ( t ) = 1 − p 1 − 4 pq t 2 2 and Q ( t ) = 1 + p 1 − 4 pq t 2 2 . W e note tha t fo r t = 1 w e hav e P (1) = p , Q (1) = q a nd g i (1) = 1. W e can now express the unconditional genera ting function g ( t ) by using summation by par ts: g ( t ) = X i ≥ 0 g i ( t ) Pr[ I = i ] = X i ≥ 0 g i ( t ) (Pr[ I ≥ i ] − Pr[ I ≥ i + 1]) = g 0 ( t ) Pr[ I ≥ 0] + X i ≥ 1  g i ( t ) − g i − 1 ( t )  Pr[ I ≥ i ] = 1 + X i ≥ 1  g i ( t ) − g i − 1 ( t )  Pr[ I ≥ i ] , bec ause g 0 ( t ) = Pr[ I ≥ 0] = 1 . Thus, using (1.7), we hav e g ( t ) = 1 + (1 − λ ) X i ≥ 1 ( ∇ g ) i ( t ) λ i 1 − λ i , (2 . 2) 4 where ( ∇ g ) i ( t ) = g i ( t ) − g i − 1 ( t ) denotes the backward difference o f g i ( t ). W e note that for t = 1 we hav e ( ∇ g ) i (1) = 0, so g (1) = 1. W e are now rea dy to dr ive the mo ment gener a ting function M W ( s ) = Ex[ e sW ] for W . Each intertran- sition time X is exp onentially distr ibuted with mean 1 / (1 + λ ), so the mo ment generating function for X is M X ( s ) = (1 + λ ) / (1 + λ − s ). The moment gene r ating function fo r the sum P 1 ≤ k ≤ n X k of n indep endent int ertrans itio n times X 1 , . . . , X n is M X ( s ) n =  (1 + λ ) / (1 + λ − s )  n . Th us the waiting time W , which is the sum of the random num ber N of indepe nden t intertransition times has the moment g e nerating function M W ( s ) = X n ≥ 0 Pr[ N = n ] M X ( s ) n = g  M X ( s )  (2 . 3) = g  1 + λ 1 + λ − s  . 3. The M omen ts In this section we shall derive the mean and v ariance for N and W , and indicate ho w to derive the higher moments as well. F o r the mea n of N , we use the formula Ex[ N ] = g ′ (1). W e hav e g ′ i (1) = (1 + λ )  i (1 − λ ) − λ (1 − λ i )  (1 − λ ) 2 , so ( ∇ g ′ ) i (1) = (1 + λ ) (1 − λ i ) 1 − λ . Thu s we hav e Ex[ N ] = g ′ (1) = (1 − λ ) X i ≥ 1 ( ∇ g ′ ) i (1) λ i 1 − λ i = (1 − λ ) X i ≥ 1 (1 + λ ) (1 − λ i ) 1 − λ λ i 1 − λ i = (1 + λ ) λ 1 − λ . (3 . 1) F o r the mean of W , we use the formula fo r the ex p ectatio n of the sum of a ra ndom num be r N of indep en- dent , identically distributed r andom v ar iables X , X 1 , X 2 , . . . : Ex h P 1 ≤ k ≤ N X k i = Ex[ N ] Ex[ X ]. Since the int ertrans itio n time X satisfies Ex [ X ] = 1 / (1 + λ ), we hav e Ex[ W ] = E x[ N ] Ex[ X ] = (1 + λ ) λ 1 − λ 1 1 + λ = λ 1 − λ , in accorda nce with (1.1). 5 F o r the v ar iance o f N , we b egin by using the fo rmula fo r the second factoria l moment: Ex[ N ( N − 1)] = g ′′ (1). W e hav e g ′′ i (1) = (1 − λ ) 2 (1 + λ ) 2 i 2 − (1 − λ )(1 + λ )(1 − 10 λ − 3 λ 2 ) i (1 − λ ) 4 −  6 λ (1 − λ )(1 + λ ) 2 i + 2 λ (1 + λ )(1 + 4 λ − λ 2 ) − 2 λ 2 (1 + λ ) 2 λ i  (1 − λ i ) (1 − λ ) 4 , so ( ∇ g ′′ ) i (1) =  2(1 − λ )(1 + λ ) 3 λ i + 6 i (1 − λ ) 2 (1 + λ ) 2 − 2(1 − λ ) 3 (1 + λ )  (1 − λ i ) − 4 i (1 − λ ) 2 (1 + λ ) 2 (1 − λ ) 4 . Thu s we hav e Ex[ N ( N − 1)] = 2(1 + λ ) 3 (1 − λ ) 2 X i ≥ 1 λ 2 i + 6(1 + λ ) 2 1 − λ X i ≥ 1 i λ i − 2(1 + λ ) X i ≥ 1 λ i − 4(1 + λ ) 2 1 − λ X i ≥ 1 1 λ i 1 − λ i = 2 λ 2 (1 + λ ) 2 (1 − λ ) 3 + 6 λ (1 + λ ) 2 (1 − λ ) 3 − 2 λ (1 + λ ) 1 − λ − 4(1 + λ ) 2 1 − λ T 1 ( λ ) , . where we hav e used the definition (1.8) to ev aluate the la st sum. It follows that V a r[ N ] = Ex[ N ( N − 1)] + Ex[ N ] − Ex[ N ] 2 = 2 λ 2 (1 + λ ) 2 (1 − λ ) 3 + 6 λ (1 + λ ) 2 (1 − λ ) 3 − λ (1 + λ ) 1 − λ − λ 2 (1 + λ ) 2 (1 − λ ) 2 − 4(1 + λ ) 2 1 − λ T 1 ( λ ) , . = λ (1 + λ ) 2 (6 + λ + λ 2 ) (1 − λ ) 3 − λ (1 + λ ) 1 − λ − 4(1 + λ ) 2 1 − λ T 1 ( λ ) , . where w e have used (3.1), then combined the first, second and fourth terms. F or the v a riance o f W , we use the formula for the v ariance of a ra ndom num b er N of indep endent, identically distributed random v aria bles X , X 1 , X 2 , . . . : V ar h P 1 ≤ k ≤ N X k i = V ar[ N ] Ex[ X ] 2 + E x[ N ] V ar[ X ]. Since the intertransition time X satisfies Ex[ X ] = 1 / (1 + λ ) and V ar[ X ] = 1 / (1 + λ ) 2 , we hav e V a r[ W ] = V ar[ N ] Ex[ X ] 2 + Ex[ N ] V ar[ X ] =  λ (1 + λ ) 2 (6 + λ + λ 2 ) (1 − λ ) 3 − λ (1 + λ ) 1 − λ − 4(1 + λ ) 2 1 − λ T 1 ( λ )  1 (1 + λ ) 2 + λ (1 + λ ) 1 − λ 1 (1 + λ ) 2 = λ (6 + λ + λ 2 ) (1 − λ ) 3 − 4 1 − λ T 1 ( λ ) . Ev alua ting T 1 ( λ ) using (1.12) yields V a r[ W ] = λ (6 + λ + λ 2 ) (1 − λ ) 3 − 4 ψ (1) λ (1) (1 − λ ) lo g 2 λ , confirming (1.5), whereas using (1.10) yields V a r[ W ] ∼ λ (6 + λ + λ 2 ) (1 − λ ) 3 − 4 ζ (2 ) (1 − λ ) 3 , 6 confirming (1.6). It is straightforw ard to gener alize the deriv ations of the mean a nd v ariance of W given ab ov e to the higher moments. W e b egin by indicating how to derive the higher fac to rial moments of N . W e hav e Ex[ N ( N − 1) · · · ( N − m + 1)] = g ( m ) (1) = (1 − λ ) X i ≥ 1 ( ∇ g ( m ) ) i (1) λ i 1 − λ i . After differentiating g i ( t ) with resp ect to t ( m times), then ev aluating the r esult at t = 1, and finally differencing with res pect to i , the re s ult is a biv ariate p olynomia l P ( i, u ) (with co efficients that a re rational functions of λ ) in the v a r iables i a nd u = λ i . Dividing this p olynomial b y 1 − u = 1 − λ i , we obtain P ( i , u ) = Q ( i, u )(1 − u ) + R ( i ), with quo tien t Q ( i, u ) and remainder R ( i ). W e then have Ex[ N ( N − 1) · · · ( N − m + 1)] = (1 − λ ) X i ≥ 1 Q ( i, λ i ) λ i + (1 − λ ) X i ≥ 1 R ( i ) λ i 1 − λ i . (3 . 2) The first sum in (3.2 ) can be expr essed as a linear c o m bination (with co efficients that are ratio nal functions of λ ) of sums of the form S l,k ( λ ) = X i ≥ 1 i l λ ki . These s ums are themselves rational functions of λ : S l,k ( λ ) = A l ( λ k ) (1 − λ k ) l +1 , where A l ( x ) = P 0 ≤ k ≤ l a ( l , k ) x k is the l -th Euler ian p olynomial and the a ( l , k ) are the E ulerian num bers , with g enerating function P l,k ≥ 0 a ( l , k ) z k y l /l ! = z (1 − z ) / ( e y (1 − z ) − z ) (s e e Comtet [C, p. 245]). The second sum in (3.2) ca n b e expressed as a linear co m bination (a gain with co efficient s that are rational functions of λ ) of the sums T l ( λ ) given by (1.8), with asymptotic formulas given by (1.9 ) and (1.10), and with e xact formulas given by (1.11) and (1.12). W e are now ready to obtain the moments of W . Different iating the identit y (2.3) m times, we obtain M ( m ) W ( s ) = F m  M (1) X ( s ) , . . . , M ( m ) X ( s ); g (1) ( M X ( s )) , . . . , g ( m ) ( M X ( s ))  , where F m ( x 1 , . . . , x m ; y 1 , . . . , y m ) is the p olyno mial F m ( x 1 , . . . , x m ; y 1 , . . . , y m ) = m ! X 1 ≤ l ≤ m x l X e 1 ,e 2 ,...,e m Y 1 ≤ k ≤ m  y k k !  e k , and the inner sum is ov er all e 1 , e 2 , . . . , e m such that e 1 + e 2 + · · · + e m = l and e 1 + 2 e 2 + · · · + m e m = m (see for example Comtet [C, p. 1 37]). E v aluating at s = 0 and using M X (0) = 1 and M ( l ) X (0) = 1 / (1 + λ ) l for l ≥ 1 , we have M ( m ) W (0) = F m  g (1) (1) , . . . , g ( m ) (1); 1 1 + λ , . . . , 1 (1 + λ ) m  = 1 (1 + λ ) m X 1 ≤ l ≤ m g ( l ) (1) X e 1 ,e 2 ,...,e m  m e 1 , e 2 , . . . , e m  = 1 (1 + λ ) m X 1 ≤ l ≤ m g ( l ) (1) n m l o , 7 where the  m l  are the Stirling num ber s 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 [C, pp. 206– 207]). Th us we hav e Ex[ W m ] = M ( m ) W (0) = 1 (1 + λ ) m X 1 ≤ l ≤ m n m l o g ( l ) (1) , where the g ( l ) (1) are the fa ctorial moments of N determined in the pr eceding para graph. The asymptotic formulas given ab ov e for the moments o f W can b e ex tended to complete asymptotic expansions, with erro r ter ms of the for m O  (1 − λ ) R  for any R . Any rational function of λ has a Laur ent series around λ = 1, which will serve a n an asymptotic expansio n as λ → 1 a s well. Thu s the only remaining problem is to find asy mptotic expansio ns for the sums T l ( λ ). These expansio ns have b een given by Eschenfeldt, Gross and Pipp enger [E]. W e have T 0 ( λ ) ∼ 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)! , where γ = 0 . 5772 . . . is Euler’s co nstant, B r is the r -th Bernoulli nu mber, defined b y t/ ( e t − 1) = P r ≥ 0 B r t r /r ! (see fo r ex a mple Roma n [R3, p. 9 4], a nd h = − log λ ha s the expa nsion h = − log  1 − (1 − λ )  = X r ≥ 1 (1 − λ ) r r , so that its recipro ca l has the expansion 1 h = 1 1 − λ X r ≥ 0 ( − 1) r C r (1 − λ ) r r ! , where C r is the r -th Berno ulli nu mber of the second kind, defined by t/ log(1 + t ) = P r ≥ 0 C r t r /r ! (see for example Roman [R3, p. 116 ]). (These num ber s are also ca lled the Ca uch y num ber s of the firs t kind, a nd are given by C r = R 1 0 x ( x − 1) · · · ( x − r + 1) dx ; s ee for exa mple Comtet [C, pp. 29 3–294 ].) F o r l ≥ 1, we hav e T l ( λ ) ∼ X r ≥ 0 ( − 1) r + l − 1 B r B r + l h r − l r ! ( r + l ) . W e note tha t, if l is o dd, then this expansion ha s only finitely many terms (be cause B r = 0 for o dd r ≥ 3). 5. Ackno wledg m en t The r esearch re p or ted here was suppo rted by Gra nt CCF 09 17026 from the Na tional Science F oundation. 8 6. R eferences [C] L. Comtet, A dvanc e d Combinatorics: The Art of Finite and Infinite Exp ansion , D. Reidel Publishing Co., Do rtrech t, 197 4. [E] P . Es c henfeldt, B. Gr oss a nd N. Pipp enger , “Sto chastic Service Systems, Ra ndom Interv al Graphs a nd Search Algorithms”, arXi v:1107 .4113v2 [ma th.PR] . [G] G. Gas per and M. Rahman, Basic Hyp er ge ometric Series , Cambridge Universit y Pr ess, Cambridge, 1990. [H] G. H. Hardy and E . M. W right , Int r o duction to the The ory of Nu mb ers (5th e dition) , Cla rendon Press, Oxford, 1979 . [K] J. F. C. King man, “The Effect o f Q ueue Discipline on W aiting Time V a riance”, Math. Pr o c. Cambridge Phil. So c. , 58 :1 (196 2) 163 – 164. [L] J. D. C. Little, “A Pro of for the Queuing F o rmula: L = λW ”, Op er. R es. , 9 (1961) 383– 387. [R1] J. Riordan, “Delay Curves for Calls Served at Rando m” , Bel l System T e chnic al Journ al , 32 (1953 ) 100-1 19. [R2] J. Riordan, Sto chastic Servic e S ystems , J o hn Wiley a nd So ns, New Y ork, 196 2 . [R2] S. Roman, The Umbr al Calculus , Academic P ress, New Y ork, 1 984. [T] D. G. T am b o uratzis, “On a Prop erty o f the V ariance of the W aiting Time o f a Queue” , J. Appl. Pr ob. , 5:3 (1968) 702–7 03. [V1] ´ E. V a ulot, “D´ elais d’attent e des app els t´ el´ ephoniques trait´ es au ha s ard”, Comptes R endus A c ad. Sci. Paris , 222 (1946) 2 68–26 9. [V2] ´ E. V a ulot, “D´ elais d’attent e des app els t´ el´ ephoniques dans l’o rdre inverse de leurs arriv´ ee”, Comptes R endus A c ad. Sci. Paris , 238 (1954 ) 1188– 1189. [W1] B. W eesakul, “The Rando m W alk b et ween a Re flec ting and a n Absorbing Barr ier”, Ann. Math. Statist. , 32:3 (1 961) 765– 769. [W2] E. T. Whittaker and G. N. W atson, A Course of Mo dern Analysis (4th e dition) , Cambridge Universit y Press, London, 1927. 9