1 Bankruptcy Problem in Network Sharing: Fundamentals, Applications and Challenges Angelos Antonopoulos, IEEE Senior Member Telecommunications Technological Centre of Catalonia (CTTC/CERCA) Castelldefels, Spain Email: [email protected] Abstract—Network sharing has been already adopted increase in capital and operational expenditure (CAPEX by mobile network operators as a reliable and effective and OPEX, respectively), threatening the viability of fu- countermeasure to the constantly increasing network ture mobile networks. cost. The introduction of the fifth generation (5G) mobile communications is expected to revolutionize the Network sharing [2] has already become a standard part telecommunications world, however it also brings new of the operating model for mobile operators, while the challenges to the network sharing, as new technologies trend is constantly accelerating. Through sharing both (e.g., network virtualization, cloud architectures, etc.) active and passive equipment, operators have been able and types of resources (e.g., computation and storage) to significantly reduce the total cost of ownership, while come to the fore. Despite their inherent differences, the common characteristic of the emerging resources is that improving network quality. Hitherto, operators have been they are limited and, in many cases, insufficient to cover able to achieve significant savings1, while recent reports the rising traffic demands. In this article, we highlight foresee that these savings will be even more impressive the equivalence between the distribution of a limited (i.e., up to 50%) in 5G, as greenfield deployment is better number of resources and the bankruptcy problem, suited for sharing, since it avoids the cost of network where the demands of different agents over a given commodity exceed its total quantity. In particular, i) consolidation [3]. we review the fundamentals and existing solutions of Nonetheless, despite its strong potential, 5G network the bankruptcy problem, ii) we provide potential appli- sharing is not a clear-cut concept and comes along with cations of this problem in mobile network sharing, and several challenges. More specifically, the evolution of the iii) we list challenges and open issues for the application of the bankruptcy game in the mobile communications mobile networks increases their complexity and, conse- domain. The main goal of our work is to identify new quently, the different types of resources that can be en- research lines that will foster the 5G network sharing. countered. In particular, unlike the traditional sharing Index Terms—Bankruptcy game; Game theory; 5G; schemes that focus on the physical infrastructure and Infrastructure Sharing. the communication resources in basic network parts (e.g., radio access or transport network), the introduction of a series of different paradigms in future networks brings I. Introduction new heterogeneous resources that need to be explicitly The fifth generation (5G) mobile communications is considered. For instance, the cloud radio access network just around the corner, signaling the transformation of (C-RAN) technology generates new network parts (i.e., telecommunication networks from mere communication fronthaul) that did not exist in previous mobile gener- channels into key enablers for a variety of important ations, the adoption of fog and multi-access edge com- vertical industries, such as e-Health, manufacturing and puting (MEC) implies new computational and storage automotive, among others. It is expected that upcoming resources, while there are also hardware developments 5G networks will provide extremely high data rates and with the appearance of unmanned aerial vehicles (UAVs) ultra low latency, thus being able to fulfill the strict that can be part of the network by carrying small base requirements of different traffic types, e.g., augmented stations. On top of this, the embracement of renewable reality (AR) or remote surgery applications. energy sources for the network power supply adds another The high potential of 5G networks, along with the degree of complexity that should be contemplated by the rapid evolution of mobile devices, constitute some of the forthcoming network sharing schemes. main factors for the anticipated data traffic explosion. In Despite their heterogeneity and their inherent differ- particular, according to Cisco forecasts, the global mobile ences, all the aforementioned resources share a common data traffic will increase seven-fold between 2017 and 2022, characteristic, i.e., they are limited and they become even reaching 77.5 exabytes per month by 2022 [1]. Apparently, more valuable as the data traffic grows. Therefore, as to cope with these unprecedented traffic demands, mobile the network expansion and the inclusion of additional network stakeholders (e.g., mobile operators, infrastruc- ture providers, etc.) need to upgrade and expand their net- 1Please see: https://www.gsma.com/futurenetworks/wiki/infrastructure- works, something that intrinsically implies an important sharing-an-overview 2 resources is not always feasible (e.g., due to space lim- are allocated to each agent, respectively. It is worth noting itations or cost), efficient sharing approaches should be that, following this approach, it is possible that one agent established. In this context, this challenging situation of receives an amount higher than the one she claimed. In our sharing limited resources among various interested parties example, the entity would be divided in three equal parts, can be formulated as a bankruptcy problem, which is a i.e., A1 = A2 = A3 = 15. Obviously, the first claimant distribution problem that involves the allocation of a would receive a quantity higher than the requested, while given amount of a single commodity among a group of the requirements of the other two claimants would not be agents, when this amount is insufficient to satisfy all their satisfied. demands [4]. Taking into account the similarities of the limited re- C. Water Filling source sharing with the bankruptcy problem and the fact Water filling solution is another simple method that is that bankruptcy theory has not yet been fully exploited mainly used to avoid that a claimant is awarded with a in mobile network sharing scenarios, the main goal of this portion higher than the requested. In this case, we can article is to identify new research lines in order to pave think the claims as tanks of different height (i.e., propor- the way for the application of the bankruptcy problem tional to each claim) that are gradually filled with part of in network sharing. To that end, our contribution is the entity. Once the lower tank is filled, we continue only mainly threefold. First, we present the fundamentals of the with the rest ones, maintaining always the same “water” bankruptcy problem, along with a list of possible solutions level in the tanks to be filled. However, one important and the key ideas behind these solutions. Then, we focus disadvantage of this approach is that the losses are not on the telecommunications domain and we provide poten- equally shared, as usually the smaller claim is completely tial applications in existing and future mobile networks, satisfied and the losses affect mainly the high claims. Fig. 1 where the resource sharing can be modeled and solved demonstrates the water filling allocation in four different as a bankruptcy problem. Finally, we list some intriguing cases. In our toy example, following this approach, all open research lines and challenges for the application of claimants would receive 10 units (until the demand of the the bankruptcy theory in wireless networks. first claimant is fulfilled) and then the remaining 15 units would be equally split to the other two claimants. II. Bankruptcy Problem: Background The origins of the bankruptcy problem go several cen- turies back, following the fundamental human need for fair E=45 E=210 division of commodities2. In principle, each bankruptcy problem is characterized by an entity E ∈ R that has to be divided among N agents, whose individual claims Ci, i ∈ N, add up to an amount higher than the total entity, PN 15 15 15 70 70 70 i.e., i=1 Ci > E. Apparently, as it is not possible for all C1=100 C2=200 C3=300 C1=100 C2=200 C3=300 claims to be satisfied, different approaches may be followed for the division. In this section, we present a toy example E=340 E=550 along with some of the most popular solutions and the outcome allocations (Ai) they yield, summarized also in Table I. 100 120 120 100 200 250 A. Bankruptcy Problem: Toy Example C1=100 C2=200 C3=300 C1=100 C2=200 C3=300 Without loss of generality, we assume that a total entity Fig. 1: Water filling allocation in the bankruptcy problem of E = 45 units has to be allocated to three claimants, whose claims are C1 = 10, C2 = 20 and C3 = 30, respectively. As the sum of all claims is greater than the P3 D. Loss Sharing total entity, i.e., i=1 Ci = 60 > 45, it is not possible to fulfill all claims. To overcome the unbalanced loss issue, one possible solution would be to observe the problem from a different point of view and focus on the fair distribution of losses B. Equal Sharing instead of sharing the entity. More specifically, as the Equal sharing is the simplest form of sharing a com- sum of the claims is always higher than the entity, we modity among a set of interested agents. Equal sharing may estimate their difference and then split this difference completely neglects the individual claims of the involved equally to the N agents. You may also notice that, in parties and divides the commodity into equal shares that case that there are big discrepancies between the claims, this approach may induce negative values. For instance, if 2The Babylonian Talmud (a record of discussions about Jewish laws and customs) includes various cases with regard to the fair E = 45, C1 = 5 and C2 = 60, the loss sharing method distribution.