The M/M/Infinity Service System with Ranked Servers in Heavy Traffic

egp.ii i.arxi v.tex The M / M / ∞ Service System with Rank ed Serv ers in Hea vy T raffic 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 Colleg e 1250 Dartmouth Aven ue Claremont, CA 9171 1 Abstract: W e consider an M / M / ∞ service system in which an a rriving customer is served by the fir st idle ser ver in an infinite sequence S 1 , S 2 , . . . of s ervers. W e determine the fir st tw o ter ms in the asymptotic expansions of the moments of L a s λ → ∞ , where L is the index of the server S L serving a newly arriv ing customer in eq uilibr ium, and λ is the ratio of the a rriv al r ate to the service rate. The lea ding ter ms of the moments show that L/λ tends to a unifor m distr ibution on [0 , 1]. Keyw ords: Queueing theory , a symptotic expansio ns . Sub ject Classi fication: 60K26, 90B22 1. In tro duction W e cons ider a stream o f cus tomers, w ith indep endent exp onentially distr ibuted interarriv al times, a r- riving at rate λ to an infinite se quence S 1 , S 2 , . . . of servers. Each arriving customer engages the ser ver S l having the low est index among currently idle servers, and renders that ser ver busy for an indep endent exp onentially distr ibuted service time w ith mean 1 . This sto chastic se rvice system, which is conv ent ionally denoted M / M / ∞ , has been extensiv ely studied in the limit λ → ∞ ; see Newell [N]. W e shall b e interested in a question mentioned only tangentially by Newell: wha t is the distribution of the r andom v a riable L defined as the index o f the server S L serving a newly a r riving customer when the system is in equilibrium? New ell [N, p. 9] states that L “is a pproximately uniformly distributed ov e r the interv al” [1 , λ ], ba sing this a ssertion on the approximation Pr[ L > l ] ≈ ( 1 − l λ , if l < λ , 0 , if l > λ . (1 . 1) But no er ror b ounds are given for this or other a pproximations stated by Newell, and no t even the fact that the first moment has the as y mptotic b ehavior Ex[ L ] ∼ λ 2 (1 . 2) that it would have under the unifor m distribution is esta blis hed r igorously . Our g oal in this pa per is to give a rigoro us version of (1.1) that will suffice to establish not only (1.2), but a lso the next term, Ex[ L ] = λ 2 + 1 2 log λ + O (1) , (1 . 3) and mo re generally Ex[ L m ] = λ m m + 1 + m λ m − 1 log λ 2 + O  λ m − 1  (1 . 4) for m ≥ 1. In par ticular, we have V ar[ L ] = Ex[ L 2 ] − Ex[ L ] 2 = λ 2 12 + λ log λ 2 + O ( λ ) . Since the interv al [0 , 1 ] is b ounded, formula (1.4) shows that the m -th moment of L/ λ tends to 1 / ( m + 1) as λ → ∞ for all m ≥ 1, and th us suffices to show that the dis tr ibution o f L/λ tends to the uniform distribution on the interv al [0 , 1]. W e note that a problem that is in a sense dual to our s (finding the la r gest index of a busy ser ver, rather than the smallest index of an idle server) has b een treated by Coffman, Kado ta and Shepp [C]. The key to our results is the pr obability Pr[ L > l ], which is s imply the pr obability that the firs t l servers S 1 , . . . , S l are a ll busy . It is well known that this pr obability is given by the Erla ng loss for m ula 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 (1 . 5) 1 (see for example Newell [N, p. 3]). The sum D l can be expressed as an integral, D l = Z ∞ 0  1 + x λ  l e − x dx (see for ex ample Newell [N, p. 7]), and mo st of Newell’s analysis is based on such a representation. B ut we shall work directly with the ex pression o f D l as the sum in (1.5). W e shall divide the r ange of summation in (1.5) into tw o parts. The fir st, which we shall c a ll the “ bo dy” of the distr ibutio n, will be 0 ≤ k ≤ l 0 = λ − s , wher e s = √ λ . The second, which we sha ll call the “ tail”, will b e l > l 0 . In Section 2, we shall derive an estimate for Pr[ L > l ] in the b o dy , a nd in Sectio n 3, we shall derive an estimate for the ta il. In Section 4, we shall combine thes e estima tes to establish (1.4). 2. The Bo dy In this section we shall establish the estimate Pr[ L > l ] = (1 − l /λ ) + 1 λ (1 − l/ λ ) + O  1 λ  + O  1 λ 2 (1 − l/ λ ) 3  (2 . 1) for l ≤ l 0 = λ − s , where s = √ λ . W e b egin by using the principle of inclusio n-exclusion to derive b ounds on the denominator D l . W e b egin w ith a low er 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 hav e X 0 ≤ k ≤ l  l λ  k = 1 + O  ( l/ λ ) l  1 − l /λ . W e note that the loga rithm of ( l /λ ) l has a non- negative seco nd deriv a tive for l ≥ 1. Thus ( l /λ ) l assumes its max im um in the interv al 0 ≤ l ≤ l 0 for l = 0, l = 1 or l = l 0 . Its v alues there are 0, 1 /λ and (1 − s/λ ) λ − s = (1 − 1 / √ λ ) λ − √ λ ≤ e − √ λ +1 , res pectively . As λ → ∞ , the larg est o f these v a lue s is 1 /λ , so we hav e O  ( l/ λ ) l  = O (1 /λ ) for 0 ≤ l ≤ l 0 . Thus the first s um is X 0 ≤ k ≤ l  l λ  k = 1 + O (1 / λ ) 1 − l /λ . F or the s econd s um we hav e X 0 ≤ k ≤ l  k 2   l λ  k − 1 = 1 + O  l 2 ( l/ λ ) l  (1 − l/ λ ) 3 . 2 The loga rithm of l 2 ( l/ λ ) l has a non-negative s e cond deriv ative for l ≥ 3, so a n arg umen t s imilar to that used for the first sum shows that O  l 2 ( l/ λ ) l  = O (1 /λ ) for 0 ≤ l ≤ l 0 . Thus we hav e X 0 ≤ k ≤ l  k 2   l λ  k − 1 = 1 + O (1 / λ ) (1 − l /λ ) 3 and the low er b ound D l ≥ 1 + O (1 / λ ) 1 − l /λ − 1 + O (1 / λ ) λ (1 − l /λ ) 3 . (2 . 2) F or a n upp er b ound, we have 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 cal 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 (2.1). 3 3. The T ail In this section we shall establish the estimate Pr[ L > l ] = O ( e − λ λ l /l !) (3 . 1) for l ≥ λ − s , where s = √ λ . T o obtain a n upp er b ound on Pr[ L > l ], we o btain 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 . 2) bec ause l − ⌊ λ − s ⌋ ≥ l − ( λ − s ) ≥ 0 by ass umption a nd ⌊ λ − 2 s ⌋ ≥ 0 for all s ufficien tly lar ge λ . The r e are ⌊ λ − 2 s ⌋ − ⌊ λ − 2 s ⌋ + 1 ≥ s terms in the s um (3 .2 ). F urthermore, the smallest o f these terms is the last, b ecaus e its denominator contains factors of λ where the pr eceding terms co n tain fa c tors smaller 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 denomina to r of this b ound, we shall use the estimate n ! ≤ e √ n e − n n n , which holds for a ll n ≥ 1 (b ecause the trap ezoidal rule under estimates the integral R n 1 log x dx of the conc ave function log x ). This e s timate yields 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 . 3) 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 (3.3) yields D l ≥ l ! e λ e 7 λ l . T ak ing the recipro cal of this b ound yields (3.1 ). 4 4. The Moments In this section we shall use (2.1 ) and (3.1) to prov e (1.4). W e write ∆ m ( l ) = l m − ( l − 1) m = m l m − 1 + O ( l m − 2 ) for the backw ard differences o f the m -th p owers o f l . Then partial summation yields Ex[ L m ] = X l ≥ 0 l m Pr[ L = l ] = X l ≥ 0 ∆ m ( l ) Pr[ L > l ] = X l ≥ 0 m l m − 1 Pr[ L > l ] + O   X l ≥ 0 l m − 2 Pr[ L > l ]   (4 . 1) This formula shows that we sho uld ev a lua te sums of the form T n = X l ≥ 0 l n Pr[ L > l ] . (4 . 2) W e sha ll show tha t T n = λ n +1 ( n + 1)( n + 2) + λ n log λ 2 + O ( λ n ) . (4 . 3) Substitution of this formula into (4.1 ) will then yield (1.4). W e sha ll break the range of summation in (4 .2) at l 0 = λ − s , where s = √ λ , using (2.1) fo r 0 ≤ l ≤ l 0 and (3 .1) for l > l 0 . Summing the firs t term in (2.1), we hav e 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 ) . Summing the second term in (2.1), we have 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 ) , 5 where we hav e used P 1 ≤ k ≤ n 1 /k = log n + O (1). Summing the third ter m in (2.1) o f cour s e yields O ( λ n ). Summing the last term in (2.1 ), we have 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 ). Combining these estima tes, we obta in 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 ) . (4 . 4) Finally , summing (3.1) 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 summatio n on the rig ht -hand side is the n -th moment o f a Poisson rando m v ariable with mean λ , which is a p olynomial of degr ee n in λ . Thus X l>l 0 l n Pr[ L > l ] = O ( λ n ) . Combining this estimate with (4 .4 ) yields (4.3 ) and completes the pro of of (1.4). 5. Conclusio n W e have obtained the first tw o terms in the asymptotic expa ns ions of the moments o f L as λ → ∞ . An obvious op en question is whether one can obtain a c o mplete asymptotic ex pansion, o r even just the c onstant term in (1.3) and the cor r esp onding terms in (1 .4). While our es tima tes for the contributions to the O (1) term in (1.3 ) could b e improv ed (for example, by a b etter choice of the parameter s ), it is clear that new techn iques will b e needed to obtain an error term tending to zero. 6. Ac knowledgment The r esearch repor ted here was suppor ted by Gra nt CCF 0917 026 from the National Science F oundation. 7. References [C] E. G. Coffman, Jr., T . T. Kadota and L. A. Shepp, “A Sto chastic Mo del of F r agmentation in Dynamic Storage Allo cation”, SIAM J. Comput. , 14 :2 (198 5) 416 –425. [N] G. F. Newell, The M / M / ∞ S ervic e System with Ranke d Servers in H e avy T r affic , Springer-V erlag, Berlin, 1984 . 6