Bertrand Games between Multi-class Queues

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1 Bertrand Games between Multi-class Queues Parijat Dube and Rahul Jain Abstract We develop a framework to study differentiated services when there are competing network providers We adopt a multi-class queueing model, where providers post prices for various service classes Traffic is elastic and users are Quality of Service (QoS)-sensitive, and choose a queue and a class with one of the providers We model the relationship between capacity, QoS and prices offered by service providers in a competitive network services market We establish sufficient conditions for existence of Nash equilibrium in the multi-class queueing game I INTRODUCTION The network neutrality debate [18] has put fresh spotlight on service differentiation in networks While the network service providers want to introduce service differentiation through price differentiation, the content providers (who can be regarded as major users on the Internet) are opposed to such an arrangement Over time, such service and price differentiation may be offered for all users Of course, the market for network services is a competitive marketplace with multiple competing providers Thus, it is in this context that we must seek to understand the introduction and pricing of differentiated services on the Internet Our objective is to understand what kind of resource allocation outcomes we may expect, and whether they are optimal in any sense We consider a simple queueing game model to study differentiated services with multiple providers There are N providers Each player operates service classes as GI/GI/1 type queue Users arrive according to a renewal process (which can be Poisson) Upon arrival, each user then has to decide which queue to join If it joins a particular queue, it has to pay a service price to the operator Each user is also delay sensitive and his service utility is a non-increasing function of the delay it suffers Thus, he joins a queue that offers the maximum net utility, ie, service utility minus delay cost and price We will assume that the queues are partially observable in the sense that the expected delay metric is available to the user upon his arrival but not the instantaneous queue lengths Each provider s objective on the other hand is to maximize his total revenue The players are competitive and strategic The question at hand is what kind of outcomes may one expect from the competition between the providers? And are these optimal in some sense Of course, each queue can be operated with multiple priority classes And a different price can be charged for each class by a provider In that case, each user chooses not only a provider but also the class of the queue he wants Parijat Dube is with the IBM TJ Watson Research, Hawthorne, NY pdube@usibmcom Rahul Jain is with the University of Southern California, Los Angeles, CA rahuljain@uscedu to join The objective of a provider then is to pick a price vector that maximizes his total payoff taking into account the competition provided by the other providers Pricing as a means to queueing stability was first considered in [19] The effect of tolls on queueing behavior was studied with Poisson arrivals with an exponential server queue It was shown that if balking is allowed, the socially optimal price is greater than the revenue maximizing price The implication is that the revenue maximizing expected delay is larger than the socially maximizing one Without balking, it was shown in [3] that the two prices are the same In [4], a scheduling policy was derived for a multiclass queue that maximizes the expected discounted net value (service value minus delay cost) Mendelson and Whang [15] introduced a stylized model for a single queuing service provider that has become accepted as the model of choice for analysis for such problems There are multiple priority classes and each user has private information about job parameters (such as delay cost and expected service time) They introduced an incentive-compatible priority pricing rule for the M/M/1 queue which is efficient as well (maximizes the social welfare) A variant of this model where the users are impatient and may not wait for service to finish is considered in [5] In [21], a general equilibrium model with congestion externality (ie, queueing delays) is considered They conclude that the competitive price is social welfare maximizing A dynamic pricing model is considered in [14] with adaptive learning by the users Various models of learning of congestion delays are considered: rational expectations, Markovian expectations and exponentially smoothed expectations and conditions of network stability are established The case of incomplete information about job parameters (service utility and delay cost) was considered in [16] It establishes the incentive compatibility of the generalized cµ scheduling In [8], a cost sharing perspective is taken on sharing the total delay cost, and the Aumann-Shapley mechanism is used to determine individual user payments Most of the work discussed above is summarized in the book [6] A dynamic price adaptation scheme that modulates the flow to keep performance in various differentiated queued classes close to announced profiles is introduced in [2] The scheme is analyzed using the ode approach to stochastic approximation algorithms While single server models have been much analyzed, the analysis of multiple queueing service provider models is not similarly rich The earliest work on this is [13] It was shown that the equilibrium prices are always different from the social welfare maximizing ones A variation of the Mendelson- Whang model for two identical servers (with unobservable

2 queues) was considered in [12] It was established that the total arrival rate under the Cournot equilibrium is smaller than the total arrival rate under Bertrand equilibrium, which is smaller than the total arrival rate under social optimization In [11], a multi-server extension of the Mendelson-Whang model is considered However, a competitive setting is considered where there are enough servers in the market, and the influence of each is negligible, ie, the servers act as price takers Thus, the existence of competitive equilibrium is established More recently, [7] considered the Bertrand game between multiple network service providers each offering a single service class They considered linear affine delay functions (a very rough approximation for delay in any queueing model) and showed the game to have a price of anarchy of 3125 This result was sharpened by [17] to 15 General convex delay functions were considered in [20], and a price of anarchy of 15 was established which was shown to be tight In all the recent works [7], [20], [17], it was conjectured that a Nash equilibrium may not exist in general We gave sufficient conditions for its existence in [9] In this paper, we consider the model with multiclass queues Traffic is elastic and arriving users cannot balk Though our focus is on delay functions which are strictly increasing and convex, we also consider other models Provider post prices for various classes and arriving users know expected delay for each class in each queue However, they cannot observe the current state of the queue They then decide which queue and class to join We give sufficient conditions for existence of Nash equilibrium in such a multiclass queueing game II PRELIMINARIES We consider N providers each of whom offers a queued service to customers Each provider operates an GI/GI/1 queue and provides (non guaranteed) best-effort service However, he offers priority classes of service The service discipline could be either non-preemptive scheduling or processor-sharing (or any other as long as it satisfies Assumption 1(ii)) Let x ik be the traffic rate at the ith queue in the kth class Provider i has service capacity y i, service quality metric (eg delay) d ik (x ik ; x i, k ) for class k as a function of flows x i (x,,x i ), and prices p i (p,,p i ) per unit flow A higher value of d ik implies a degradation in service quality If the delay suffered by a unit flow is d, then the cost per unit is θd Note that x i, k denotes (x,,x i,k 1,x i,k1,,x i ) while flow vector x (x 1,,x N ) and price vector p (p 1,,p N ) We assume that each arriving user is infinitesimal and non-strategic (ie, price-taker) and their preferences can be aggregated so that the total utility derived by all the users is V ( ik x ik) Note that for given prices p ik and delays d ik, the total demand or flow can be given by z (V ) 1 (min i,k {p ik θd ik }) The p ik θd ik term shall be called the full price of queue i and class k We will take the net social welfare to be given by S(x) :V ( i,k x ik ) θ i,k x ik d ik (x ik ; x i, k ) We will call a traffic vector x to be socially efficient if it maximizes the social welfare maximization problem max x 0 S(x), where x 0 implies x ik 0, i, k A user sends the marginal traffic (eg a packet in the communication network) to queue Q i in class k if (i, k) arg min j,l p jl θd jl (x jl ; x j, l ) We assume that each customer type sends Poisson traffic, the rate x being determined by the prices and the delay cost it faces Thus, a natural outcome of such decision-making by each user is the following Definition 1 (Wardrop equilibrium): For a given price vector p, a flow vector x (p) is said to be a Wardrop equilibrium if and only if x (p) is a solution of xjl max V ( x jl ) ((p jl θd jl (0))x jl θ d jl (x)dx) x 0 j,l j,l 0 We will assume that instantaneous queue lengths are not observable but the expected delays in each class for each queue is available to all arriving users We will assume that the queues have infinite buffers for every class For provider i, if total traffic is x i with prices p, then his total revenue is Π i (p i,p i ) k p ik x ik (p ik,p i, k ) Thus, given prices of other providers, he must pick a price vector p i that maximizes his total revenue In our model, users are indifferent between different priority classes as long as they pay the same full price However, with users that have different traffic profiles and delay requirements (eg, voice versus web traffic), this need not be the case In that case, we must consider models with multiple user types, and hence various utility functions V i corresponding to various types However, in this paper we will address the problem of only a single user type A natural question is does there exist some kind of equilibrium where providers announce expected delay guarantees, and users then choose traffic rates and the system equilibrates through some prices and traffic rates (such that the full price is the same for all queues in all classes), delays guarantees offered are met, and all providers are satisfied at these prices? When players are non-strategic, then the existence of competitive equilibrium was established in [21] When players are strategic however, then they would try to manipulate the market by changing prices In that case, we must look at Nash equilibrium Definition 2 (Nash equilibrium): A price vector p is a Nash equilibrium if (i) the corresponding Wardrop equilibrium is x (p ), and (ii) for each i, given p i, p i arg max p ik x p i ik(p i,p i) k We will make the following assumptions: Assumption 1: (i) V (x) is strictly increasing and concave (ii) d ik (x ik ; x i, k ) are strictly increasing and convex in x ik

3 for given x i, k (iii) V C 3, and d ik C 2, ie, V and d ik are continuous (iv) V ( i µ i) θd ik (µ i ), i, k A justification for these assumptions is provided in [9] III MULTI-CLASS BERTRAND QUEUEING GAME When the providers are strategic, they try to anticipate the actions of the other players (providers) and it may not be possible to achieve socially optimal allocations The providers strategize by picking prices Capacity available to them will be assumed fixed We study equilibria in such a pricing game (also called a Bertrand game) between queueing service providers In particular, we establish the existence of Nash equilibrium in such games Each queue will be assumed to have multiple classes and service is firstcome, first-served For given prices p, we first establish conditions for a flow x to be a Wardrop equilibrium, ie, For all i, k, we have V ( jl x jl ) θd ik (x ik ; x i, k ) p ik ) 0, (1) x ik V ( x jl ) θd ik (x ik ; x i, k ) p ik ) 0 (2) jl Furthermore, the prices p i which maximize the aggregate revenue of provider i from all the classes, Π i k1 p ikx ik, satisfy, x ik k1 p ik x ik p ik 0, k (3) Note that we also must establish joint concavity of payoff Π i in p i (p,,p i ) Let (x,p ) satisfy (1), (2) and (3) (along with concavity) Then, p is a Nash equilibrium We denote by Ui k p ik x ik, the revenue of provider i at p Lemma 1: If p is a Nash equilibrium, then there exists a Nash equilibrium p with equality in (1) Proof: If x ik > 0, define p ik p ik If x ik 0and there is strict inequality in (1), define p ik V ( j,k x jk) θd ik (x ik) Note that this is always non-negative by assumptions 1(i)- (iv) With price p, the allocation is still x, implying that the corresponding revenue Ũi k p ikx ik is optimal and equal to Ui for all i Further (x, p) also solves (1) with equality Thus, it is a Nash equilibrium The implication of this lemma is that if a Nash equilibrium exists, then we can look for an equilibrium of type p for which we have equality for every i in (1) We now obtain sufficient conditions for the existence of such an equilibrium Let {(x ik,p ik )} satisfy Wardrop equilibrium condition in (1) with equality For any variable z, let ż ik ( z ik ) denote the first-order (second-order) partial derivative wrt p ik From (1) we have, V p jl θd jl (x jl ; x j, l ), j, l (4) At equilibrium, taking partial derivative of (4) wrt p ik, V N ẋ ik mp θ d jl x jq ẋ ik jq I(j i, k 1) (5) where I( ) is the indicator function Denote by x i the vector of allocations to firm i, x i (x ik,k 1,,) and by x (x i,i 1,,N) the vector of rate allocation to all the firms We can write (5) in matrix form Aẋ ik B ik for all i, k, where A V I 1 θd with I 1 a square matrix of size M, M N, with all elements equal to 1 and D is block diagonal matrix, D diag(d 1,,D N ), where D i,i 1,,N is a matrix with (r, s)th element defined as Dr,s j djr x js, for r, s 1,, B ik is a column vector of size M with Br ik 1and Bs ik 0,s r, s 1,,M and r (i 1)k ẋ ik is a column vector of size M given by {ẋ ik jl,j 1,,N; l 1,,} Thus for any (i, k), the existence and uniqueness of ẋ ik is established if A is invertible To establish that A is invertible we first state a result on the inverse of the sum of matrices We will set θ 1for simplification for the rest of this paper Lemma 2 ([10]): Let B be a nonsingular matrix and u and v be column and row vectors respectively, then (B buv ) 1 B 1 b 1bv B 1 u B 1 uv B 1 Thus, we can write A 1 {V I 1 D} 1 as D 1 V 1 V N D 1 I 1 D 1 (6) j1 r,s1 Dj r,s Thus, we need to show the existence of D 1 as V 0 Proposition 1: D is non-singular for any strict priority (priority queueing with premptive) service discipline among classes at each provider Proof: Consider strict priority at provider i with classes labeled in decreasing order of priority, ie, class p has higher priority than class q, if p<qfor p, q 1,, This implies that the delay at a higher priority class is independent of the load at a lower priority class, ie, d ip x iq 0, for p < q The corresponding matrix D i is a lower triangular matrix and its determinant is simply the product of its diagonal elements which is Π k1 d ik x ik Since d ik x ik > 0 by our assumption, D i is non singular and hence its inverse exists Thus if each firm i maintains some strict priority among its classes (D i ) 1 exists for all i 1,,N which in turn implies the existence of D 1 which is given by D 1 diag D 1 1,,D N 1 Corollary 1: For any i, k, ẋ ik exists and is unique Proposition 1 implies A is invertible establishing the existence and uniqueness of ẋ ik since ẋ ik A 1 B ik i, k We now establish the existence of Nash equilibrium Theorem 1: If Π i is jointly concave in p i, and under Assumption 1, there exists a Nash equilibrium p in the N- player multi-class Bertrand queueing game

4 Proof: Writing (3) in matrix form, we have, for j 1,,N, X j ẊjP j 0, where X j and P j are each a column vector of size with Xj T {x jl,l 1,,} and Pj T {p jl,l 1,,} Ẋ j is a matrix, Ẋ j (ẋ j1t,, ẋ jt ) T (B j1t A 1T,,B jt A 1T ) T where ẋ jit (ẋ ji j1, ẋji j2,, ẋji j j ) Further since elements of P j also satisfies (4) at equality, we have, P j (V d jl,l 1,,) T (7) We now have N fixed point equations of the form (with j 1,,N): X j g j (X j,j 1,,M): ẊjP j B j1t A 1T V d jl (8) B jt A 1T V d j with P j given by (7) Since we have existence of A 1 from Proposition 1 and continuity of V and d jl by our assumption, the existence of fixed point, (x ik,i 1,,N; k 1,,), of (8) follows from Brouwer s fixed point theorem The corresponding prices p (p ik,i 1,,N; k 1,,) are defined by ẊjP j X j, j 1,,N A Concavity of the payoff function in price We now establish the concavity of Π i in prices p i To establish concavity we need to show that the Hessian of Π i is negative semi-definite The Hessian is given by: p 2 p p 2 Π i p p i H (9) p p i p p i p 2 i Thus we need to evaluate second order and cross derivatives of Π i wrt prices Differentiating Π i wrt p ik twice, we get p 2 ik 2 x ik p ik l1 2 x il p il p 2 (10) ik This requires evaluating first and second order derivatives of flows wrt prices We first present a simple observation Lemma 3: For any arbitrary m n matrix B and a n m unitary matrix U nm, BU nm B is a n m matrix with (BU nm B) i,j S i r(b)s j c(b), i [1:n],j [1:m], (11) where S i r(b) is the sum of the elements in the ith row and S j c(b) is the sum of the elements in the jth column of B From (5) we have: (V U D)ẋ ik B ik (12) where U ij 1, i, j and B ik is a column vector of size M as defined earlier From (12) we have ẋ ik (D V U) 1 B ik (13) Finding the inverse in (13) using (6) we have: (D V U) (D i 1 ) 11 (D i 1 ) 1 diag V S, (14) (D i 1 ) 1 (D i 1 ) with 1 V N j1 r,s1 (Dj 1 ) r,s and where elements of S corresponding to provider i are given by Sr 1 (D 1 1 )Sc 1 (D 1 1 ) Sr 1 (D 1 1 )Sc (D 1 1 ) Sr (D 1 1 )Sc 1 (D 1 1 ) Sr (D 1 1 )Sc (D 1 1 ) From (13) and (14), we have, ẋ ik i (D i 1 ) 1k V S1 r (D i 1)S k c (D i 1 ) (D i 1 ) k V S r (D i 1)S c k (D i 1 ) (15) Having obtained the first derivative of flow we next find second derivatives of flows wrt prices Differentiating (5) again wrt p ik, we get in matrix form N (V U D)ẍ ik V ( ẋ ik mp) 2 u M b ik (16) where u M [1,,1] is a unit vector of size M, ẍ ik [ẍ ik 11,,ẍ ik 1, ẍik 21,,ẍ ik N, ] and b ik is defined as b ik [ d 11,x 2 1q (ẋ ik 1q) 2,, d N,x 2 Nq (ẋ ik Nq) 2 ] Defining b ik V ( N m1 p1 ẋik mp) 2 u M b ik, we get from (16), We next find expression for ẍ ik (D V U) 1 bik (17) p ik p ir p ik p ir x ik p ik x ir p ir ẍ ik,ir mp using 2 x ik,ir iq p iq (18) p ik p ir This requires expressions for cross derivative of flows wrt prices Differentiating again (5) wrt price p ir for r k, we get N N N V V This can be written in matrix form as ẋ ik mp ẋ ir mp d jl,x 2 jq ẋ ik jqẋ ir jq d jl,xjq ẍ ik,ir jq, j, k Ẍ ik,ir (D V U) 1 dik,ir, (19) where d N ik,ir V ẋik mp ( N ẋ ir mp)u M d ik,ir, with d ik,ir [ d il,x 2 iqẋik ẋir iq,l 1,,]

5 Proposition 2: If V 0and delays of all classes of provider i are linear, satisfying: (D i 1 ) kk V Sk r (D i 1 )Sc k (D i 1 ) (D i 1 ) ll V Sl r(d i 1 )Sc(D l i 1 ) 0 (20) k, l then Π i is concave in prices p i for any number of classes, 1 Proof: To show that Π i is concave in price vector p i (p ik,k 1,,) we need to show the Hessian of Π i is negative semi-definite Observe that with linear delays and V 0, b ik ik,ir M d M 0, k, r Thus the second derivatives of flows are all identically zero, ie, Ẍi ik Ẍik,ir i 0and the Hessian of Π i is given by 2 ẋ ẋi ẋ 2 ẋi H ẋi i ẋ ẋi i 2ẋi i (21) To show that H is negative semidefinite, we need to look at the signs of all the principal minors By definition, H is negative semi-definite if and only if all the tth order principal minors are 0 if t is odd and 0 if t is even For 1, we have only one first order principal minor, M from (15) as (D i 1 ) 11 V S1 r (D i 1)S c 1 (D i 1 ) 0 For 2, we have two first order principal minors, M and M and one second order principal minor given by M ( ) 2 ( ) 2 Thus M iff ẋ which implies from (15) (D i 1 ) 11 V S1 r (D i 1 )Sc 1 (D i 1 ) (D i 1 ) 22 V S2 r (D i 1 )Sc 2 (D i 1 ), (22) which is true under the conditions given Thus Π i is concave in p i for 2 Observe that (20) implies that ẋk ik, k Thus H has all entries same and equal to With these for any we have all the first order principal minors 0 and all the second and higher order principal minors identically equal to 0 Hence Π i is concave in p i for any 1 Corollary 2: If V is linear and delay functions of different classes of provider i linear with equal first derivative, d ik,xik d iq,xiq 0, q, k, then Π i is concave for any, 1 Proof: Since the delays are linear and V is linear, to show concavity of Π i for all >1 we just need to check if the delay functions satisfy (20) For linear V, we have V 0, hence from (20) we need d,x d,x 0 If delays are of the form d iq a i x iq f(x ik,k 1,,,k q),a i 0, ie, (i) separable function of its own flow and the flow of other classes, (ii) linear in its own flow with coefficients of x iq independent of class, and (iii) any arbitrary bounded function of flows from other classes, then Π i concave in prices p i B Special Case: 2with strict priority When 2, with strict priority, as defined in Proposition 1, we have D i 1 1 d,x 0,x 0 d,x d,x,x d,x From this and (15), we have, with 1 V ( i d 1,x d,x ),,x 1 V d 1,x (1 d,x ),x [1 V (1 d 1,x d,x ) ],x [ d,x V (1 d 1,x d,x ) (1,x d,x )] V d 1,x (23) We first find conditions for the first derivatives of flows ẋ il ik 0, k, l 1, 2 to be monotone in prices Define Γ min d,x max{d,x,d,x } and Γ d max,x min{d,x,d,x } Observe that Γ max Γ min From definition of first derivatives in (23) we have the following two Lemmas, whose proofs are straightforward Lemma 4: If delays d iq, q 1, 2 are convex in flows x and x then, (a) or (ii) V / 0 and Γ min 1, 0 and ẋ 0 when (i) V / 0 and Γ max 1, (b) 0 when (i) V / 0, d,x1 V (1 Γ min)(1 Γ max ) and Γ max 1 or Γ min 1, or (ii) V / 0, Γ max 1, Γ min 1, and d,x1 V (1 Γ max)(1 Γ min ), (c) 0 when V / 0 From Lemma 4 and Proposition 2 we have Proposition 3: If delays are linear such that d ik,xik d iq,xiq 0, q, k and, either (i) V 0, V 0, > 0, Γ max 1, or (ii) V 0, V 0, < 0, Γ min 1, or (iii) V 0, V 0, > 0, Γ min 1, or (iv) V 0, V 0, < 0, Γ max 1, then Π i is concave with strict priority classes Proof: The conditions on V, and Γ max or Γ min guarantee that (D i 1 ) kk V Sk r (D i 1)S c k (D i 1 ) 0, k 1, 2 and hence the first order prinicipal minors of the Hessian are negative For delays to also satisfy the equality in (20) we need to show,x 1 V d 1,x (1 1 V (,x ) d 1 d,x ) (1,x d,x ) 1 V (d 1,x ) 0,

6 which holds when d,x d,x We next look at cases when V 0 For 2we get from (17), with σ ik N 2 ẋik mp Observe that, from (23), we can write: Ẍi ẍ ẍ ẋ (24) where, we can write b 1 b 2 b j V ( σ ) 2 2 d ij,x 2 iq (ẋ iq )2 Similarly, Ẍ i ẍ ẍ ẋ b 1 b 2, (25) where b j V ( σ ) 2 2 d ij,x 2 (ẋ iq iq )2 The cross derivative from (19) can also be written as ẋ,(26) Ẍ, i ẍ, ẍ, d, 1 d, 2, with d j V σ σ 2 d ij,x 2 iqẋ iqẋ Observe that: p 2 p 2 2 p ( b 1 b 2 ) p ( b 1 b 2 ) (27) 2 p ( b 1 b 2 ) p ( b 1 b 2 ) (28), p ( d 1, d 2 p p ), p ( d 1, d 2 ) (29) Proposition 4: If delays d iq,q 1, 2 are convex in flows x and x, V / 0, Γ max 1 and d,x V (1 Γ max )(1 Γ min ) then 2 Π i 0, k 1, 2, ie, the profit p 2 ik function is marginally concave in prices Proof: With convex delays and V 0, we have b j, b j 0, j 1, 2 Further with V / 0, Γ max 1 and d,x V (1 Γ max)(1 Γ min ) we have ẋ ik ij, j, k 1, 2 0 The proof then follows from (27) and (28) [7] A Hayrapteyan, E Tardos and T Wexler, A network pricing game for selfish traffic, ACM SIGACTS-SIGOPS Symposium on Principles of Distributed Computing (PODC), 2005 [8] M Haviv, The Aumann-Shapley price mechanism for allocating congestion costs, Operations Research Letters 29: , 2001 [9] R Jain and P Dube, N-player Bertrand-Cournot queueing games: Existence of equilibrium, Proc Allerton Conference, 2008 [10] enneth S Miller, On the Inverse of the Sum of Matrices, Mathematics Magazine, Vol 54, No 2 (Mar 1981), pp [11] P Lederer and L Li, Pricing, Production, Scheduling, and Deliverytime Shceduling, Operations Research 45(3): , 1997 [12] C Loch, Pricing in markets senstitive to delay, PhD Dissertation, Stanford University, 1991 [13] I Luski, On partial equilibrium in a queuing system with two servers, Review of Economic Studies 43(3): , 1976 [14] Y Masuda and S Whang, Dynamic pricing for network service: equilibrium and stability, Management Science, 45(6): , 1999 [15] H Mendelson and S Whang, Optimal incentive-compatible priority pricing for the M/M/1 queue, Operations Research, 38(5): , 1990 [16] J Van Mieghem, Price and service discrimination in queueing systems: incentive compatibility of Gcµ scheduling, Management Science, 46(9): , 2000 [17] J Musacchio and S Wu, The price of anarchy in a network pricing game, Proc of 45th Annual Allerton Conference, September 2007 [18] J Musacchio, G Schwartz and J Walrand, Network neutrality and provider investment incentives, submitted to Rev of Network Economics, 2007 [19] P Naor, The regulation of queue size by levying tolls, Econometrica, 37(1):15-24, 1969 [20] A Ozdagler, Price competition with elastic traffic, Networks, 2006 [21] D Stahl and A Whinston, A general economic equilibrium model of distributed computing, in New directions in computational economics, eds W Cooper and A Whinston, luwer Acad Pub, pp , 1994 [22] G Weintraub, R Johari and B Van Roy, Investment and Market Structure in Industries with Congestion, preprint, December 2007 Remark: We note that the conditions in Lemma 2, Propositions 3 and 4 involve endogeneous variables but can be expressed in terms of exogeneous variables such as upper and lower bounds on derivates of delay and utility function REFERENCES [1] C Border, Fixed Point Theorems with Applications to Economics and Game Theory, Cambridge University Press, 1985 [2] V Borkar and D Manjunath, Charge-based control of Diffserv-like queues, Automatica, 40(12), 2004 [3] N Edelson and D Hildebrand, Congestion tolls for Poission queueing processes, Econometrica, 43(1):81-92, 1975 [4] JM Harrison, Dynamic scheduling of a multi-class queue: Discount optimality, Operations Research 23(2): , 1975 [5] R Hassin and M Haviv, Equilibrium strategies for queues with impatient customers, Operations Research Letters 17:41-45, 1995 [6] R Hassin and M Haviv, To Queue or Not to Queue, luwer Academic Publishers, 2003