# Mathematical programming formulation

The scheduler of the shared base station allocates resources by using the optimization model formulated in (1a)?(1h). The proposed techno-economic model runs in real time and controls both the resource allocation and the respective price negotiations in an online manner. Namely, the resource shares of the tenants are dynamically chosen based on their Quality of Service (QoS) expectations (i.e. the achieved rate per user and tenant's time window, Wm), the channel conditions and tenant's market power (i.e. their budget, number of users and traffic mix).

The optimizer dynamically assigns resources to each slice per service type and per tenant to minimize the total gap, i.e., as in (1a), ?_(m?M)??m. By jointly optimizing the resource allocations for all tenants, the scheduler has the flexibility to prioritize the users with the best channel conditions and therefore maximize the utilization of the resources and spectral efficiency. Constraint (1b) sets the gap of tenant m as the difference between its target utility (i.

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e. Uth) and the sum of the achieved utility over its users (i.e. sum of Uk(Rk[n])). Note that within each time window, of length Wm, we evaluate the average by considering the values from the beginning of the time window to the current time slot n, i.e. over am + 1 time slots, where am ? n ? 1 mod Wm. Therefore, the average achieved rate for user k at time slot n is

Rk[n] =1/((1+am)) ?_(i=n-am)^n??xk[i]rk[i]?

Furthermore, we assume that all the users have the same importance to the tenants, thus, U3 = Uth/Km ?k ? Km.

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By selecting the same value of maximum utility, U3, for all the users, the tenants also guarantee neutrality in their provided services. However, depending on the agreements between the service providers and the tenants, as well as in accordance with regulatory constraints, this value can be changed, thus allowing our model to include also non-neutral services. The instantaneous average deviation from the guaranteed resource share, m[n], is given in (1c). Namely, the instantaneous deviation at n for tenant m is given by subtracting the guaranteed resource share Sm from the average assigned resource to the users of m, where the average, as done for the average achieved rate, is evaluated from the beginning of the current time window till time slot n. Constraint (1d) ensures that m[n] is not larger than ?m, which by definition is the tenant-specific maximum allowed deviation. Note that m can either be positive or negative,

?min??(xk[n] )??_(m?M)???m[n]? (1a)

s.t.Uth - ?_(k?Km)??Uk(Rk[n]) ? ?m,?m ? M, (1b) ?

?m[n]= (1/((am + 1) ) ?_(i=n-am)^n??_(k?Km)?xk[i] )- Sm,?m ? M, (1c)

i.e. m ? [??m, ?m]. The former case indicates that the tenant has received - on average and within the current time window - more resources than Sm, while the latter case corresponds to the opposite. Furthermore, constraint (1e) sets the budget constraint per tenant. The first term of the left-hand side scales both CAPEX and OPEX according to Sm, which means that in case of no sharing (when ?m = 0), the tenant will have to pay for the requested resources. The second term, i.e. m[n]Cop, allows tenants to dynamically adjust their total costs according to their resource usage and budget. Namely, if a tenants' actual resource usage is less than the guaranteed resource share (i.e. m[n] < 0), then the tenant will not pay for the OPEX cost of the unused resources. The third term of the budget constraint is a function, fpre(Cpre, ?m), of the pressure cost unit Cpre, defined by the InP, and of the tenant's gap ?m. Namely, the gap considered for the evaluation of the pressure cost is the one obtained at the end of the previous time window (i.e. it varies at every time window, but kept constant within the same time window). The effects of the pressure cost term are evident when, e.g., there is a resource demand that exceeds the available resources. In this case, since the resources are limited, the tenants face non-zero gaps, ?m > 0, which corresponds to an increase of the pressure cost as well as of the total cost of resources. This increase in the cost pushes tenants to increase their ?m and decrease Sm. In the extreme case, tenants opt for full sharing, i.e. ?m = 1, which allows the scheduler to provide the most spectrum efficient and cost efficient allocation. Moreover, the pressure cost allows the infrastructure provider to accumulate additional revenues not directly used for the current infrastructure, but envisioned to support capacity expansion to meet the tenants' quality requirements gap provides an accurate estimation of the capacity needed to satisfy all the tenants.

Constraint (1f) forces the maximum deviation ?m to be at maximum, equal to the resources assigned to the elastic users of tenant m, which implies that tenants are not willing to trade resources used for critical, i.e. non-elastic, services. By setting ?m = 0, tenants indicate that their services are non-elastic and they require the resources they stated by Sm. However, in this case, they also lose the flexibility to adapt to traffic dynamics. Finally, (1g) ensures that the assigned resources do not exceed the total available resources in the system and, similarly, (1h) limits the sum of all Sm to the total amount of resources.

## Two-step approach

The formulation presented in the previous section is able to capture the dynamics of the resource negotiation, considering both the scheduling aspects and the economical constraints (prices and budgets). However, due to its computational complexity, it is not suitable to be used in real-time. Therefore, we decided to split the problem into two, namely the decision on the real time resource allocation and the decision on the negotiations of the sharing parameters (Sm, ?m). In particular, we separate our model into two sub-problems, P1 and P2. The first problem, P1, focuses on the real time resource allocation with the objective of minimizing the total gap and it is solved at every time slot n. During P1, the sharing parameters (Sm, ?m) are assumed to be constant and, therefore, the constraints that regulates the sharing (i.e. (1f) and (1h)) are inactive. The outcome of P1 is then given by the allocated resources and corresponding tenants' gaps. The second problem, P2, is solved at the end of each time window, by the update of the sharing parameters according to the channel conditions of the users at that time and the tenants' targets (i.e. in terms of Uth). In this case, the objective is to find the best sharing parameters so that the total gap of the previous time window is minimized. Namely, P2 receives the achievable rates from the previous time window as input and derives the optimum sharing parameters S opt m and ? opt m by solving (1a)?(1h).

Note that even if both problems, P1 and P2 are derived from the same formulation (1a)?(1h), they are actually separate and different problems since the active variables (and constraints) are different. Formally, P1 and P2 are define as follows:

? P1?=min??( xk[n] )?m[n]

?P2?=min?(xk[n],Sm,?m)????m[n]?

## Update mechanism

As described above, P2 derives the optimum sharing parameters, i.e. S opt m and ? opt m , for all the tenants, in order to achieve the minimum total gap ?_(m?M )??opt m . However, it is important to remember that the optimization problem is solved by using the achievable rates of the previous time window only, meaning that S opt m and ? opt m are optimal only with respect to the previous window. Therefore, to capture the statistic nature of the channel over a longer time span, the sharing parameters are updated with a weighted approach. Namely, the new values for the sharing parameters, S new m and ?new m to be used in the upcoming time window, are derived as:

S new m = ?mSopt m + (1 ? ?m)S old m , (1)

?new m = ?m?opt m + (1 ? ?m)?old m . (2)

where the feature scaling coefficient, ?m, is calculated as:

?m =(?m - ? opt m )/(?m + ? opt m) . (3)

By definition ?m measures the difference between the achievable optimum gap and the actual gap observed by the tenant. For instance, when ?m = ? opt m = 0, the feature scaling coefficient is also 0, which means that the most recently calculated sharing parameters are the optimum values and are therefore also used for the upcoming time window without scaling. In general, with the proposed update mechanism, our framework is able to adapt to the varying channel conditions in a reactive manner. The sharing parameters are automatically updated to provide service quality which is satisfying the tenants' requirements while maintaining proportional fairness among them. A thorough study of the ?m selection and its effects on the model's adaptability has been proposed in [9]. In summary, the following Algorithm 1 is used to solve the dynamic network slicing and resource trading problem introduced in (1a)?(1h).

## Exploiting the channel information

The real-time scheduling problem, P1, exclusively focuses on the optimization of the current time slot n without taking into account the upcoming slots. Thus, it is incapable of fully exploiting the transmission opportunities. As a result, P1 requires a larger amount of resources compared to the one estimated by P2 in order to provide comparable performance. As a matter of fact, P2 derives the minimum values of Sm and ?m, in order to minimize the gap, which are however too restricting for P1. Therefore, to improve the performance of P1, a channel-aware filter is designed to exploit the statistical information of the channel.

Specifically, we design a channel-aware filter to evaluate the rate expectations for the upcoming time slots of each user, while scheduling the resources for the given time slot n. Even though prediction techniques of the channel characteristics are out of scope of this paper,

(a) Varying a1,when a2 = 0. (b)Varying a2,when a1 = 50

### Variation of the sigmoid function for different a1 (left) and a2 (right) values

we assume that the infrastructure provider can learn a statistical profile of the channel behaviors. Therefore, we assume that there is available data on the probable density rate on each available user, k?K, for the infrastructure providers. Based on this data the probability can be evaluated for each specific user within the given time window, whether the user is in the "best" time slot to assign resources, P rk[n] = P(rk[n] ? rk[i] ?i ? W) ? [0, 1], i.e. the slot with best channel conditions compared to the other time slots. In particular, a probability value of 0 indicates that the channel condition at slot n is the worst that can ever be observed, thus, the scheduler should avoid assigning resources, while a value of 1 means that the current channel condition is the best possible and therefore as many resources as possible should be assigned. However, we do not use this probability directly, but we filter it as described below before passing it as input to P1.

We design a two-step filtering function to map the statistical information onto the assignment decisions. As a first step, the statistical information is scaled using a sigmoid function, i.e. f(P rk[n], a1, a2) = 1/(1 +e ?a1(P rk[n]?a2) ), as presented in Fig. 3. The characteristic of the sigmoid function can be controlled by using two parameters, i.e. [a1, a2] (cf. Fig. 3(a) and Fig. 3(b)). The former parameter, a1, controls the slope of the linear region of the sigmoid and indirectly controls the resource efficiency. Namely, assuming that the number of users is low, decreasing the slope of the linear region leads to a situation where unassigned resources exist while the tenants cannot achieve their goals. In contrast, increasing a1 results in assigning resources also with bad channel conditions, thus decreasing the efficiency of the channel utilization. The latter parameter, a2, allows the shift of the sigmoid function (c.f. Fig 3(b)). In this case, choosing large values of a2 gives advantages only to the users with high probabilities. However, when tenants select small time windows, it leads to unassigned resources even in the presence of gaps. In contrast, small values of a2 equalizes all users making the filter ineffective.

The output of the sigmoid function, f(P rk[n], a1, a2), provides an understanding on how good the channel conditions for a specific user are with respect to what the certain user can achieve in the given time window. However, f(P rk[n], a1, a2) does not give information on how good the channel is with respect to the other users in that time slot. Therefore, this first step of the filtering process might not be sufficient to guide the scheduler when there is a significant difference among the distributions of the users' channel.

Consequently, an additional filtering step is introduced to capture these variations among the users' channel conditions. More specifically, taken the output of the sigmoid function, f(P rk[n], a1, a2), the second step outputs f(P rk[n], a1, a2) p , where p is scalar. If the variations in the achievable rates among users are negligible, e.g. the users have similar pathlosses, the p value can be set to 1. In contrast, if the difference is not negligible, a larger value of p should be chosen.

The output of the filter function, referred to as "priority coefficient" and indicated by ?k[n], is then used by the scheduler to give priority to the users with the best channel condition (i.e. ?k[n] = 1) and to discard the users with the worst channel conditions (i.e. ?k[n] = 0). In order to incorporate this information into P1, the constraint (1b) is updated as

Uth -?_(k?Km)???k[n]Uk(Rk[n]) ? ?m,?m ? M. (5)?

X Since the channel information is used to guide the real-time scheduling algorithm, the gap values calculated by P2 are then derived without priority coefficients, as given in (1b).

## CONCLUSION

We have shown that dynamic network slicing offers an ef?cient way of exploiting variable traf?c and channel conditions to share resources among tenants, following different strategies, bearing different characteristics. Our proposed scheme de?nes a new platform where tenants can acquire resources within a short time frame, negotiating through a set of network and economic parameters. Our numerical results demonstrate that the proposed approach provides fairness among both tenants and services and can improve the ef?ciency of resource allocation up to 40% by exploiting simple prediction mechanisms. Despite the tenants share a common infrastructure, results have also demonstrated that it is possible for them to differentiate their services by tuning model parameters. We have also shown that the pricing model can allocate economic resources for capacity expansion and that this is crucial to keep infrastructure sharing convenient for the tenants.

Updated: May 19, 2021