## Network sharing and the probability of ruin

There is now increasing interest in network sharing to support wireless communications, where the main focus is on how operators should get access to the scarce spectrum (at least in the radio bands). A more general question is how different parties can provide different components of the network. Although spectrum is one component, the physical transmitting devices and the backhaul are also key components.

In a previous post, I discussed some different ways that infrastructure can be owned by different parties, especially in the setting where users are in a facility (e.g., mine, power plant or large residential area). I briefly mentioned the need for new evaluation metrics in order to optimize network sharing agreements and it is this aspect I want to explore further in this post. In particular, I will introduce the notion of the probability of ruin.

Suppose that you own a component of a wireless communication network. This might be spectrum, it might be a number of transmitting devices (e.g., small-cells), or you might even be an internet service provider (ISP) with backhaul infrastructure to connect small-cell networks. Irrespective of your particular component, the key question is whether or not providing it is profitable.

The basic idea of profitability is that what you pay is less than what you earn; your income is higher than your expenses. This simple idea pervades economic analysis of many systems. However, we need to account for the fact that the income and expenditures vary over time. If we are interested in a business, how do we decide that the business is profitable after 6 months, or a year, or indefinitely?

In the context of the economic analysis of wireless network sharing arrangements, the common approach is to consider the expected profit; that is, the income and expenditures are averaged. For example, see:

• Duan, L., Huang, J. and Shou, B., “Duopoly competition in dynamic spectrum leasing and pricing,” IEEE Transactions on Mobile Computing, vol. 11, no. 11, pp. 1706-1719, 2012.
• Yang, Y. and Quek, T., “Optimal subsidies for shared small cell networks–a social network perspective,” IEEE Journal on Selected Topics in Signal Processing, vol. 8, no. 4, pp. 690-702, 2014.
• Park, J., Kim, S. and Zander, J., “Asymptotic behavior of ultra-dense cellular networks and its economic impact,” in Proc. IEEE Global Communications Conference (GLOBECOM), 2014.
• Berry, R., Honig, M., Nguyen, T., Subramanian, V., Zhou, H. and Vohra, R., “On the nature of revenue-sharing contracts to incentivize spectrum-sharing,” in Proc. IEEE INFOCOM, 2013.
• Cano, L., Capone, A., Carello, G., Cesana, M. and Passacantando, M., “Cooperative infrastructure and spectrum sharing in heterogeneous mobile networks,” IEEE Journal of Selected Areas in Communications, vol. 34, no. 10, pp. 2617-2629, 2016.

To make the notion of expected profit more precise and to explore its limitations, it is helpful to introduce the basic revenue surplus process:

Definition (Basic Revenue Surplus Process): Let $(V_i, i \in \mathbb{N})$ be the income stochastic process and $(C_i, i \in \mathbb{N})$ be the expenditure process. Then, the revenue surplus process is $S = (S_i, i \in \mathbb{N})$, where the random variable $S_i = \sum_{j=1}^i V_j - C_j$.

The basic revenue surplus process provides insight into the financial resources of the business at each time $t$, without accounting for the interest (i.e., the interest rate is zero). As assuming an interest rate of zero does not obscure any of the main ideas, in the remainder of this post, I will focus on this case.

Assuming taking expectations is meaningful, the expected profit at time $t$ is then $\mathbb{E}[S_t]$. This may seem like a natural thing to do, but it is worthwhile exploring the implications. For example, it means that the main criterion for profitability is that $\mathbb{E}[S_t] \geq \rho$; i.e., a business is unprofitable if its expected profit drops below a threshold $\rho$.

But in practice, if a businesses profit drops below zero at any point in time before the target time $t$, then the business must borrow money which may not be desirable. If instead, we require that the revenue surplus $S_i > 0$ for all $i \leq t$, then it is clear that the expected profit does not do the job.

To properly account how the revenue surplus varies over time, we need the more refined notion of the probability of ruin. This idea has been widely considered in mathematical insurance and risk theory; for example, see

• De Vylder, F. and Goovaerts, M., “Recursive calculation of finite time ruin probabilities,” Insurance: Mathematics and Economics, vol. 7, pp. 1-7, 1988.
• Asmussen, S., (2000), Ruin Probabilities. Singapore: World Scientific Publishing Co.

The key question in ruin theory is: what is the probability that the revenue surplus drops below zero at a time before a given time $t$? To answer this question, we need to define the time of ruin:

Definition (Time of Ruin): The time of ruin $L_R$ is defined as $L_R := \inf \{n: S_n < 0\}$.

The probability of ruin is then just the probability that the time of ruin is before time $t$. (Readers might notice the similarities between the probability of ruin and the probability of buffer overflow in queuing theory.)

In a future post, I will explore a concrete calculation of the probability of ruin in the context of wireless network sharing, accounting for non-zero interest rates. For now, I just want to point out some implications of using the probability of ruin rather than the expected profit.

One implication is that if the income and expenditures are linked to usage, then the dynamics of the demand can influence the probability of ruin in ways that do not affect the expected profit. This is due to the fact that the probability of ruin does not just consider the first moment. As such, the variance (and higher order moments) also play a role, which suggests that if user demand has large fluctuations then the expected profit may not properly capture these affects.

Another implication is that financial parameters such as the interest rate become coupled to the revenue and expenditure processes. If these processes are dependent on the infrastructure design, then there can be unexpected dependence of the ruin probability on the resource allocation in the wireless network.

Recently, in

• Ballesteros, L., et al., “Effect of network performance on smartphone user behavior,” in the Workshop on Perceptual Quality of Systems (PQS), 2016

the authors showed that the network performance (i.e., available resources to serve users) influences the way users behave; e.g., which apps they use.

If the ability to use certain apps leads to a higher revenue, then the probability of ruin will be jointly dependent on financial parameters (such as the interest rate) and also physical layer parameters  (such as how time-frequency resources are allocated). To cope with this kind of complicated coupling, it may be necessary to introduce specialized contracts for network sharing. Perhaps there will even be some interest in contract theory; e.g.,

• Tirole, J., The Theory of Corporate Finance. Princeton University Press, 2006.