Newsvendor Model Of Capacity Sharing R. Berry, M. Honig, T. Nguyen, V. Subramanian, H. Zhou EECS Department Northwestern University Evanston, IL 60208 {rberry, mh, thanh, vjsubram, hang.zhou}@eecs.northwestern.edu R. Vohra CMS-EMS, Kellogg School of Management Northwestern University Evanston, IL 60208 r-vohra@kellogg.northwestern.edu 1. INTRODUCTION Capacity sharing in the form of roaming agreements have long been a fixture of cellular service. Historically, the main driver behind roaming agreements is to extend the cover- age of a wireless carrier’s network into regions where that carrier has no infrastructure, thus making the service more attractive to customers who “roam” into a new region. Here we focus on a different form of capacity sharing, namely ac- quiring “overflow” capacity from another provider during periods of high demand. Such sharing may provide carriers with an attractive means to better meet their rapidly in- creasing bandwidth demands. On the other hand, the pres- ence of such a sharing agreement could encourage providers to under-invest in their networks, resulting in poorer perfor- mance. We consider a stylized model of such a situation to gain insight into these trade-offs. Specifically, we adapt the newsvendor model from operation management [8] to this situation. The newsvendor model applies to a single firm that is determining how much inventory to stock in the face of uncertain demand. Here, we consider two wireless car- riers, who are determining how much capacity to invest in, also in the face of uncertain demand. Without any capac- ity sharing, this reduces to the standard newsvendor model for each carrier. However, with sharing, the carriers’s in- vestment decisions become coupled since the revenue one carrier earns form sharing capacity with another depends on the other’s capacity investment. We model this interac- tion as a game and give conditions for when the game has a unique pure strategy Nash equilibrium. Our analysis applies for two different sharing models one in which a carrier first serves its own customers before allocating an excess capacity to the other provider’s customers and one in which carriers do not discriminate between their own traffic and the other carrier’s. Numerical comparisons of the resulting equilibria are also given for different demand distributions. In terms of related work, there has been a line of literature studying how carriers set prices in roaming agreements (see e.g. [4]). Much of this work focuses on a known demand and Copyright is held by author/owner(s). seeks to understand if firms pricing decision lead to collusive behavior. An extension of this line of work that also con- siders investment can be found in [9]. Here we assume that prices for roaming are given exogenously and focus instead on the carriers investment decisions in the face of uncertain demand. There has also been increasing interest in sharing ”raw spectrum” between different providers (e.g. [1]). This differs from the capacity sharing model considered here in that if a carrier receives raw spectrum from another carrier, it would still have to use its own infrastructure to utilize this spectrum. Finally, we note that the type of capacity sharing model considered here may be applicable to other settings as well, such as sharing between providers of renewable energy in real-time electricity markets [6]. 2. NON-SHARING MODEL We consider a situation with two carriers, 1 and 2, who are faced with unknown demands in each of their markets. They each procure capacity/spectrum as in the newsven- dor model [8]. The initial scenario assumes that there is no interaction between the two carriers. We denote the re- ward per unit demand of the carriers by pi and the cost per unit capacity by ci for i =1, 2. The demands are denoted by a non-negative stochastic quantity Di with probability density function fi (·) and cumulative distribution function Fi (·) (complementary cumulative distribution function be- ing F C i (·)). Initially we will consider independent demands. The carriers chose to buy capacity qi at the beginning. Then the expected profit of each carrier ˜ πi is given E[˜ πi ]= pi E[min(qi ,Di )] - ci qi = pi qi F C i (qi )+ pi  q i 0 xfi (x)dx - ci qi Since min(qi ,x) =: qi ∧ x is a concave function of qi for any x ∈ R, it follows that pi E[min(qi ,Di )] is also a concave function of qi . If, in addition, a density exists for Fi (·), then by taking derivatives, it is easy to see that the expected profit is a strictly concave function of qi and the optimal amount of capacity purchased is given by q * i = F -1 i  1 - ci pi 