A SHORT NOTE ON THE MONOTONICITY OF THE ERLANG C FORMULA IN THE HALFIN-WHITT REGIME. Bernardo D Auria 1

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1 Working Pper (31) Sttistics nd Econometrics Series December, 2011 Deprtmento de Estdístic Universidd Crlos III de Mdrid Clle Mdrid, Getfe (Spin) Fx (34) A SHORT NOTE ON THE MONOTONICITY OF THE ERLANG C FORMULA IN THE HALFIN-WHITT REGIME Bernrdo D Auri 1 Abstrct We prove monotonicity condition stisfied by the Erlng C formul when computed in the Hlfin-Whitt regime. This property ws recently conjectured in Jnssen et l. [2011] Keywords: Hlfin-Whitt regime, Erlng C formul. Acknowledgements: This reserch is prtilly supported by the Spnish Ministry of Eduction nd Science Grnts MTM , SEJ nd RYC Deprtmento de Estdístic, Universidd Crlos III de Mdrid, Avd. Universidd 30, Legnes (Mdrid), Spin; emil: bernrdo.duri@uc3m.es

2 A SHORT NOTE ON THE MONOTONICITY OF THE ERLANG C FORMULA IN THE HALFIN-WHITT REGIME BERNARDO D AURIA Abstrct. We prove monotonicity condition stisfied by the Erlng C formul when computed in the Hlfin-Whitt regime. This property ws recently conjectured in Jnssen et l. [2011]. Recently, there hs been renewed interest in the Erlng C formul ( n n 1 ) 1 k (1) C(n, ) = n!(1 ρ) k! + n, n!(1 ρ) k=0 tht gives the probbility of witing for n rriving customers to n M/M/n system whose trffic intensity is ρ = /n. The vlue is the offered lod, tht is the rtio between the rrivl rte λ nd the service rte µ. The Erlng C formul founds ppliction in the dimensioning of lrge cll servers, i.e. when the trffic intensity pproches the instbility region (ρ 1) while the system keeps high its efficiency by ccordingly incresing the number of servers (n ). If the scling is done in n pproprite wy, the limiting system still shows non degenerte behvior. This limit is known s the Hlfin-Whitt regime, from Hlfin nd Whitt [1981], nd it requires tht the scling is done vi the squre-root stffing principle, see lso Borst et l. [2004]. The ide is to let the number of servers increse more thn linerly with respect to the offered lod by dding term proportionl to the squre root of, i.e. (2) n() = + β. In the scling procedure the number of servers is llowed to tke not integer vlues, hence it is useful to consider the extended version the Erlng C formul to the positive rel numbers, see Jgers nd vn Doorn [1986], ( ) (3) C(s, ) = (1 + t) s t t e t dt, 0 Dte: December 19, Mthemtics Subject Clssifiction. 60K25. Key words nd phrses. Hlfin-Whitt regime, Erlng C formul. This reserch hs been prtilly supported by the Spnish Ministry of Eduction nd Science Grnts MTM , SEJ nd RYC

3 MONOTONICITY OF C( + β, ) 3 tht is vlid for 0 < < s. In Hlfin nd Whitt [1981], it is shown tht ( (4) C (β) = lim C( + β, ) = 1 + β Φ(β) ) 1 φ(β) with Φ nd φ being respectively the distribution nd the density functions of stndrd Norml rndom vrible. As observed in Jnssen et l. [2011], the vlue C (β) in (4) cn be used s first pproximtion of C( + β, ) for lrge, nd eventully s dimensioning tool for lrge cll centers s shown in Borst et l. [2004], nd supported by Jnssen et l. [2009]. In prticulr in Jnssen et l. [2009], it hs been conjectured tht for ny vlue of β the function C(+β, ) is decresing, tht is the limit in (4) is pproched from bove nd in prticulr the vlue of C (β) is performnce lower bound for ny system with offered lod < 1. The purpose of this note is to prove tht indeed this conjecture holds true. The min tool of the proof is in relizing tht the function 1/C( + β, ) cn be written in term of the moments of some specil rndom vribles tht re proved to be stochsticlly ordered. It is interesting to note tht even if the role of these rndom vribles is crucil in the proof we were not ble to give physicl interprettion to them. Probbly if this could be done it would dd dditionl insights to the structure of the function C( + β, ) nd in generl to the symptotic queueing system in the Hlfin-Whitt regime. Theorem 1. The function C( + β, ) is strictly decresing in for ny fixed β > 0. In prticulr, for ny > 0, C( + β, ) > C (β). Proof. Consider the function g(t, ) = t e t (1 + t) 1, with, t > 0. It is non negtive nd 0 g(t, ) dt = 1 so we cn look t it s density function of non negtive rndom vrible, sy X. Let Y = (1 + X ), we cn express the Erlng C function, see (3), in terms of the inverse of the β moments of the rndom vribles Y, indeed C( + β, ) = 1/E[(1 + X ) β ] = 1/E[Y β ], nd we need to prove tht the β moments, with β > 0 fixed, re strictly incresing in. Since the power function ( ) β is incresing, it is enough to prove tht the fmily of rndom vribles {Y } >0 is incresingly stochsticlly ordered. First we compute their density functions. Let y(x) = (1 + x), we hve tht the inverse x(y) = y 1 1, with x > 0 nd y > 1 nd x (y) = 1 y 1 1, tht is positive for ny y > 1. It follows tht the density function f(t, ) of Y is given by f(y, ) = g(x(y), ) x (y) = y 1 (y 1 1)e (y 1 1),

4 4 B. D AURIA while its til distribution hs the following expression (5) F (y, ) = Pr{Y > y} = y f(t, ) dt = y e (y 1 1) for y > 1. An esy check tht Y, with > 0, is indeed rndom vrible cn be done by differentiting eqution (5) with respect to y nd getting f(y, ). Hving tht f(y, ) is non negtive for y 1 it follows tht F (y, ) is decresing function in y, nd the check is complete by noticing tht F (y, ) 0 s y nd F (1, ) = 1. To prove tht the stochstic order holds it is sufficient to show tht the function F (y, ) is incresing in > 0. We cn rewrite the til distribution in the following wy (6) F (y, ) = exp{(log y) 2 log y (log y)2 ( log y )2 (e log y 1)} = exp{(log y) 2 h( log y )} with h(x) = x + x 2 (1 e 1/x ). Since the exponentil nd the squre root functions re incresing functions in their respective domins, we only need to show tht h(x) is n incresing function for x > 0. Representing the exponentil function by its Tylor series we hve (7) h(x) = x + x 2( 1 n=0 x n ) n! = x n=1 x 2 n n! = n=0 x n (n + 2)! nd the result follows becuse it is sum of incresing functions. As finl remrk, we mention tht the Hlfin-Whitt regime cn be obtined in nother wy s well. Tht is, insted of writing the number of servers s function of the offered lod, like in (2), the ltter is written s function of the former in the following wy (8) (n) = n β n where in generl β > 0, nd the reltion is vlid for n > β 2. A nturl question is to sk if the monotonicity property stisfied by C(s(), ), > 0 is lso vlid for C(s, (s)), with s > β 2. However the behviour of the C(s, (s)) looks more complicted s the following grphs show for two vlues of the prmeter, β = 1/10 nd β = 3.

5 MONOTONICITY OF C( + β, ) C(s, s 1 10 s) C(s, s 3 s) References S. Borst, A. Mndelbum, nd M. I. Reimn. Dimensioning lrge cll centers. Oper. Res., 52(1):17 34, S. Hlfin nd W. Whitt. Hevy-trffic limits for queues with mny exponentil servers. Oper. Res., 29(3): , A.A. Jgers nd E.A. vn Doorn. On the continued Erlng loss function. Oper. Res. Lett., 5(1):43 46, A.J.E.M. Jnssen, J.S.H. vn Leeuwrden, nd B. Zwrt. Refining squre root sfety stffing by expnding Erlng C. Oper. Res., To pper. A.J.E.M. Jnssen, J.S.H. vn Leeuwrden, nd B. Zwrt. A lower bound for the Erlng C formul in the Hlfin Whitt regime. Queueing Syst. Theory Appl., 68: , Mdrid University Crlos III, Sttistic Deprtment Avd. Universidd, 30, Legnés (Mdird), Spin E-mil ddress: bernrdo.duri.uc3m.es