Optimal trunk reservation for an overloaded link

Operations Research Letters 38 (2010) 499–501

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Optimal trunk reservation for an overloaded link John A. Morrison Bell Laboratories, Alcatel-Lucent, Room 2C-378, 600 Mountain Avenue, Murray Hill, NJ 07974, USA

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Article history: Received 12 July 2010 Accepted 25 July 2010 Available online 25 August 2010 Keywords: Asymptotics Average reward Overloaded link Trunk reservation

abstract We consider a single overloaded link with a large number of circuits which is offered two kinds of calls. The traffic intensity of the primary calls is assumed to be strictly less than 1, and R of the circuits are reserved for these. Rewards are generated when calls are accepted, and it is assumed that the reward for primary calls is greater than that for secondary calls. We determine the trunk reservation parameter R which asymptotically maximizes the long run average reward. © 2010 Elsevier B.V. All rights reserved.

1. Introduction In this paper we consider a single link with a large number of circuits. There are two kinds of calls, which arrive as Poisson processes with different rates, and have exponentially distributed holding times with the same mean. The traffic intensity of the primary calls is assumed to be strictly less than 1, but the total traffic intensity is assumed to be strictly greater than 1, corresponding to an overloaded link. R of the circuits are reserved for primary calls, so if fewer than R + 1 circuits are idle when a secondary call arrives it is not accepted. If all the circuits are busy when a primary call arrives it is not accepted. Rewards are generated when calls are accepted, and it is assumed that the reward for primary calls is greater than that for secondary calls. The problem is to choose the trunk reservation parameter R so as to maximize the long run average reward. Trunk reservation is of fundamental importance in circuitswitched networks. State-dependent routing on symmetric loss networks with trunk reservation has been investigated by Mitra et al. [9,10] and Mitra and Gibbens [8]. Basic to the investigations is the analysis of a single link. The network is then analyzed by means of fixed point approximations (see [2,3]), which are based on the assumption that each link acts independently. The optimization problem for a single link, when calls have differing mean holding times, as well as arrival rates, capacity requirements and reward rates, was investigated by Hunt and Laws [1]. The blocking probabilities, but not the optimization problem, for a single link with two kinds of calls with differing mean holding times, have been investigated for various asymptotic scalings by Morrison [11], Knessl and Morrison [5,6], and Morrison and Knessl [12,13].

E-mail address: [email protected]. 0167-6377/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.orl.2010.08.001

Loss networks were also investigated by Key [4]. Importantly, for a single link with two kinds of calls, he gives a condition which implies that the long run average reward will increase if the trunk reservation parameter R is increased by 1. This condition was used by Reiman [14] to determine the asymptotically optimal trunk reservation parameter for a critically loaded link, in which the total traffic intensity is close to the large number of circuits. In this paper we determine the asymptotically optimal trunk reservation parameter for an overloaded link. Although Key [4] briefly mentions an overloaded link, he states only that the optimal trunk reservation parameter is of the order of the logarithm of the number of circuits, whereas our result is more explicit. We establish the matching of our result with that in [14] as critical loading is approached. The remainder of the paper is organized as follows. In Section 2 we formulate the problem and state the main result for the asymptotically optimal trunk reservation parameter, and establish the matching with the result for a critically loaded link. In Section 3 we state results, established in the Appendix, which lead to the main result. 2. Formulation and summary We consider a single link with a large number C of circuits. There are two kinds of calls which arrive as Poisson processes with rates σ C and (τ − σ )C , for the primary and secondary calls respectively, where 0 < σ < 1 < τ . The calls are assumed to have exponentially distributed holding times with mean unity. There is no facility for calls to be queued, and preemption of calls in progress is not allowed. R of the circuits are reserved for the primary calls, so if more than C − R − 1 circuits are busy an arriving secondary call is blocked. If all C circuits are busy then an arriving primary call is blocked. There are rewards w1 and w2 generated when primary and secondary calls, respectively, are accepted, and it is assumed that

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J.A. Morrison / Operations Research Letters 38 (2010) 499–501

w1 > w2 . Let B1 and B2 denote the stationary blocking probabilities

3. Asymptotic analysis

of the primary and secondary calls, respectively. The problem is to choose the trunk reservation parameter R so as to maximize the long run average reward

We proceed to establish Theorem 1. The following results are established in the Appendix.

W = C [w1 σ (1 − B1 ) + w2 (τ − σ )(1 − B2 )].

(2.1)

According to Key [4], W will increase when R is changed to R + 1 if w1 /w2 > Ψ (R) where, in our notation,

Ψ (R) =

τ (C − R − 1)!(σ C )R+1 σ   σ × Erl(τ C , C − R − 1) + − 1 τ R − C! × , ( C − l )!(σ C )l l =0 C!



N − λi i=0

Erl(τ C , C − R − 1)

+

=

1



τ

(2.2)

C!

 −1 (2.3)

i!

R −

C!

l =0

(C − l)!(σ C )l [

=

1

Theorem 1. If τ > 1 is fixed, then asymptotically for C ≫ 1, the optimal trunk reservation parameter is given by

χ (R, σ ) =

R∗ =  

 − 1 + o(1),  

− log σ



Rσ

√

Corollary 2. If τ = 1 + β/ C , where 1 ≪ β ≪ asymptotically

R∗ =  

log

√

C (w1 − w2 )(1 − σ )β/w2

(2.5)

R∗ =  

log

√

C (w1 −w2 )(1−σ ) w2 [h(β)−β]

If σ = O R

(2.6)

2 h(β) = e−β /2 /

β

 − 1 + o(1),  

(2.7)

∞

e

−z 2 /2

2 e−z /2 dz .

dz = e

−β 2 /2

[

1

(2.8)

−

β

β

1

β

3

 +O

1

β

5

]

.

(2.9)

It follows from (2.8) that h(β) − β =

1

β

 +O

1

β

3



2C

 ] 1

+o

C

.

(3.2)

+

1 C

χ (R, σ ) + o

− (R − 1) − 2

 ] 1

C

,

(3.3)

σ (1 − σ R ) . (1 − σ )2

(3.4)

 [ ( 1 − σ R +1 ) R + 1 +

1

]

(τ − 1) ]  R (3.5) − (1 − σ ) (R + 1) + χ (R, σ ) + o(1) .

1 C

then, from (3.4) and (3.5), we find that

[

1 C (1 − σ )σ R+1

] (τ − σ ) + o(1) . (1 − σ )(τ − 1)

(3.6)

Acknowledgements The author is grateful to Qiong Wang for bringing this problem to his attention, for checking some of the analysis and for comments on the first draft of this paper, and to Marty Reiman for some clarifying discussions. He is also grateful to the reviewer and the associate editor for some helpful comments. Appendix We first establish Result 1, making use of the representation [8]

Integration by parts shows that, for β ≫ 1,

∫

R +1

C (1 − σ )σ R+1

,

which establishes the matching of (2.7) with (2.6).

(2.10)

∞

∫

1 ∞

R(R + 1)

Theorem 1 follows from (2.4) and (3.6).

where

∫

(3.1)

[

Ψ (R) − 1 =

 − 1 + o(1).  



− log σ



.

2

C , then

We next show that this matches with the result in Reiman [14] for a critically loaded link. √ In [14], in √ our notation, we replace N by C , α1 N + β1 N by σ C , α2 N + β2 N by (τ − σ )C , ri by wi√ , i = 1, 2, and k∗ (N ) by R∗ . Then, asymptotically, for τ = 1 + β/ C and C ≫ 1, it follows from (4.6) in [14] that



σ

1−

R +1

1

Ψ (R) − 1 =

√



− log σ



[

1

R

(1 − σ )

where ⌈x⌉ denotes the smallest integer greater than or equal to x.



1

C

It follows from (2.2) and Results 1–3 that



w2 (τ −σ )

=

1−σ

(1 − σ )σ R

where

C (w1 −w2 )(1−σ )2 (τ −1)

(τ − 1)

  +o

Result 3. If C ≫ 1 and 0 ≤ R ≤ O(log C ), then asymptotically

the region of interest, which will be the case. We now summarize our result for the case of an overloaded link.



C

]

1

R+1+

(C − R − 1)!(σ C )

Ψ (R∗ ) ≥ w1 /w2 > Ψ (R∗ − 1), (2.4) provided that Ψ (R) is a monotonically increasing function of R in

log

[

R +1

is Erlang’s loss formula. Hence, as in Reiman [14], the optimal trunk reservation parameter R∗ satisfies



1

τ −1+

Result 2. If C ≫ 1 and 0 ≤ R ≤ O(log C ), then asymptotically

and

λN Erl(λ, N ) = N!

Result 1. If τ > 1 is fixed, C ≫ 1 and 0 ≤ R ≤ O(log C ), then asymptotically

Erl(τ C , C − L)



e− u 1 +

= 0

u C −L

τC

du,

(A.1)

which follows by using the binomial theorem, and the integral representation for i! in (2.3). We assume that τ > 1 is fixed and 0 ≤ L ≤ O(log C ). Then, log

[

1+

] u C −L

τC  [ 3 ] u u2 u = (C − L) − 2 2 +O . τC 2τ C τC

(A.2)

Since the integrand in (A.1) is exponentially small for u ≫ 1, it follows that

J.A. Morrison / Operations Research Letters 38 (2010) 499–501

[ ] (τ − 1) 1 exp − = u Erl(τ C , C − L) τ 0 [  ]   1 u2 1 × 1− Lu + du + o Cτ 2τ C [ ]   τ τ 1 1 = − L + + o . 2 (τ − 1) C (τ − 1) (τ − 1) C ∫

∞

and R − l(l − 1) l =0

(A.3)

If we let L = R + 1, and take the reciprocal of (A.3), we obtain Result 1. Next we establish Results 2 and 3, and consider 0 ≤ l ≤ O(log C ). Now [7], log m! = log Γ (m + 1) = m(log m − 1) +

+



1

+O

12m

1



m2

,

1 2

log(2π m)

m ≫ 1.

(A.4)

Hence, for C ≫ 1,

[ log

C!

]

(C − l)!C l       1 l log C =− C −l+ log 1 − −l+O 2 2

=−

l(l − 1) 2C

C

[ +O

(log C )

3

]

C2

C

.

(A.5)

It follows that C!

(C − l)!C l

=1−

l(l − 1) 2C

[ +O

(log C )4 C2

]

.

(A.6)

If we let l = R + 1 we obtain Result 2. To establish Result 3 we use the elementary results R − 1 ( 1 − σ R +1 ) = , l σ (1 − σ )σ R l=0

501

(A.7)

=

σl R(R − 1)

(1 − σ )σ R

−

2R

(1 − σ )2 σ R−1

+

2(1 − σ R )

(1 − σ )3 σ R−1

.

(A.8)

Result 3 follows from (A.6)–(A.8). References [1] P.J. Hunt, C.N. Laws, Optimization via trunk reservation in single resource loss systems under heavy traffic, Ann. Appl. Probab. 7 (1997) 1058–1079. [2] R.P. Kelly, Blocking probabilities in large circuit switched networks, Adv. Appl. Probab. 18 (1986) 473–505. [3] F.P. Kelly, Loss networks, Ann. Appl. Probab. 1 (1991) 319–378. [4] P.B. Key, Optimal control and trunk reservation in loss networks, Probab. Engrg. Inform. Sci. 4 (1990) 203–242. [5] C. Knessl, J.A. Morrison, Blocking probabilities for an underloaded and an overloaded link with trunk reservation, SIAM J. Appl. Math. 66 (2005) 82–97. [6] C. Knessl, J.A. Morrison, A two-dimensional diffusion approximation for a loss model with trunk reservation, SIAM J. Appl. Math. 69 (2009) 1457–1476. [7] W. Magnus, F. Oberhettinger, R.P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics, Springer, New York, 1966. [8] D. Mitra, R.J. Gibbens, State-dependent routing on symmetric loss networks with trunk reservations, II: asymptotics, optimal design, Ann. Oper. Res. 35 (1992) 3–30. [9] D. Mitra, R.J. Gibbens, B.D. Huang, Analysis and optimal design of aggregatedleast-busy-alternative routing on symmetric loss networks with trunk reservations, in: A. Jensen, V.B. Iversen (Eds.), Teletraffic and Datatraffic in a Period of Change, vol. ITC-13, Elsevier, Amsterdam, 1991, pp. 477–482. [10] D. Mitra, R.J. Gibbens, B.D. Huang, State-dependent routing on symmetric loss networks with trunk reservations, I, IEEE Trans. Commun. 41 (1993) 400–411. [11] J.A. Morrison, Blocking probabilities for a single link with trunk reservation, J. Math. Anal. Appl. 203 (1996) 401–434. [12] J.A. Morrison, C. Knessl, Asymptotic analysis of a loss model with trunk reservation, I: trunks reserved for fast traffic, J. Appl. Math. Stoch. Anal. (2008) 1–34. [13] J.A. Morrison, C. Knessl, Asymptotic analysis of a loss model with trunk reservation, II: trunks reserved for slow traffic, Stud. Appl. Math. 122 (2009) 153–193. [14] M. Reiman, Optimal trunk reservation for a critically loaded link, in: A. Jensen, V.B. Iversen (Eds.), Teletraffic and Datatraffic in a Period of Change, vol. ITC-13, Elsevier, Amsterdam, 1991, pp. 247–252.