The emerging technology of network slicing identified as the main enhancement  of the fifth generation of mobile networks (5G) enables infrastructure providers and mobile network operators (MNOs) to abstract (radio, transport and computational) resources in the form of logically independent “network slices”. Such network slices may be tailored onto customized service types with highly specific requirements, e.g., low-latency, high reliability, high mobility, ect. This breaks down the barriers of an old monolithic business model based only on CAPital EXpenditure (CAPEX) and OPerating EXpenditure (OPEX) savings by allowing MNOs to lease network slices to infrastructure tenants while, at the same time, augmenting economic revenues and maximizing the network resource efficiency.  This emerging business case is known as “Slice as a Service” (SlaaS)  and may significantly benefit vertical segments, such as automotive, e-health or over-the-top applications that do not own network infrastructures to deliver advanced services to their customers.
Differing from conventional cloud environments such as Software-as-a-Service (SaaS)  or Infrastructure-as-a-Service (IaaS) , the entities to be rent in SlaaS, i.e., the network slices, can be highly heterogeneous in resource requirements and dynamic behavior. For instance, massive Machine-Type-Communications (mMTC) slices for Internet-of-Things (IoT) applications require dense wireless accesses but only limited throughput, and are usually created/released at low frequency. In contrast, enhanced Mobile Broadband (eMBB) slices for high-speed transmission need high throughput for medium amount of accesses, and can be created/released very dynamically to match the traffic load. This challenges the network management and orchestration system , calling for intelligent slice brokers  and efficient slice admission control (SAC) , in order to optimize the network resource utilization.
The literature provides an extended overview about network slicing orchestration solutions pursuing network efficiency maximization or revenue maximization. For e.g. the authors in  present a near-optimal solution to dynamically allocate resources amongst slices based on a weighted proportionally fair objective that guarantees desirable fairness and protection across the slices. Another recent work  models the slicing orchestration problem as a congestion game with a distributed solution. Differently, our work proposes to leverage the intrinsic tenant behaviors when a consistent number of network slice requests reach the network. This allows us to propose a practical tenant-based admission control strategy, making it as the first one of its kind.
In particular, a SAC module conditionally declines some tenant requests for network slices to mitigate resource overload making declined requests to be reconsidered for admission after a certain delay. A straightforward solution for delivering such delayed service is queuing system employed as the “First-Come First-Serve” policy. However, when the service demand is dense, slice congestions can occur and most requests will suffer from an over-length waiting time in the queue. In this case, waiting tenants may behave impatiently to avoid potential business deficits. In this paper, we study the business model of tenants in SlaaS environment, to show ) how they shall behave in case of slice-request competition and subsequent congestion and ) which knowledge do they need to make rational (and efficient) decisions.
The remainder of the paper is organized as follows: Section II sets up the SlaaS model with multi-queued SAC and tenant business model. Section III investigates on the rational strategies of impatient 5G network slice tenants in congested queues. Analytical results are then demonstrated and evaluated in Section IV though numerical simulations. We close the paper with our conclusion and outlook in Section V.
Ii System Model
Ii-a Feasibility and Admissibility
A MNO with limited resource pool offers a number of tenants with slices of different predefined types to rent. Every active slice under maintenance occupies a certain resource bundle w.r.t. its type . The active slice set is subjected to the constraint of resource pool, which defines the feasibility region :
where the assigned resources is defined by
Clearly, some slice sets can sufficiently utilize the resource pool so that no admission of new slice is possible, regardless of the slice type. In other cases, the MNO is able to admit at least one new slice with its idle resources, which introduces the admissibility region :
where is the unit slice incremental vector
unit slice incremental vectorof type :
Fig. 1 briefly illustrates the concepts of and with an example where .
Ii-B Slice Admission Control with Heterogeneous Multi-Queue
Generally, a variety of tenants independently and randomly issue service requests to the MNO. As indicated by many existing works [11, 3, 12, 13], the MNO may decline some request instead of admitting them immediately, taking account of both the resource feasibility and prediction of the future load. In this case, a mechanism is needed to allow the MNO to reevaluate the declined requests and deliver a delayed slice admission. In this paper we consider a slice admission control mechanism with queues, where every issued request for slice type will enter and wait in the queue. Only the first awaiting request in every queue can be considered by the MNO for potential admission. Upon admission, the request will be removed from the queue.
When there is more than one queue non-empty, multiple requests of different slice types are simultaneously available for the MNO’s decision. In this case, we consider that the MNO has a certain preference among different slice types, which can be described by a vector :
where is the set of all permutations of . The order of elements in presents the MNO’s preference. More specifically, every change in either the status of queues or triggers the MNO to attempt admitting the first request in its most preferred queue. If this attempt fails due to resource constraint or an empty queue, the MNO continues with its next preferred queue. Once a request is admitted, both and the queues status, and therefore the MNO’s preference as well, are updated, which triggers the aforementioned procedure again. This process continues recursively until no more awaiting request can be accepted by the MNO, thereafter the MNO waits for the next incoming request (that updates the queues status) or slice termination (that updates ).
It is also worth to note that sometimes the MNO may want to reserve its resource for future utilization, despite that it is able to admit another slice. This can be enabled by adding an option for “standby” to the preference vector. In this case, should be extended to the set of all permutations of . A graphical summary of the aforementioned mechanism is illustrated in Fig. 2. We refer to the specific mapping as the admission strategy of the MNO.
As service requests are issued by a number of independent tenants, it is reasonable to consider the request arrival of every certain slice type as an independent Poisson process.
Ii-C Tenant Business Model
Having introduced the SAC mechanism from the MNO’s perspective, now we take the tenant’s point of view and consider the business model of every tenant service instance through its life cycle.
Generally, the motivation of tenants to request network slices is to fulfill the end-user service demands from their own customers. For simplification, here we consider w.l.o.g. that for every certain tenant, every service demand can be supported by one slice of the same type. Once a tenant is granted with the requested network slice, it launches its business session to deliver service to the end-users. The duration of a business session, i.e. the lifetime of the corresponding network slice, is a random variable, that is known by the tenant before issuing the slice request and established in the Service-Level-Agreement (SLA) upon slice admission.
It can take the tenant a one-time cost to issue the service request to the MNO, which is used to issue the request and prepare the end-user service. Besides, this cost may also covers part (or even the whole lump) of the rent for requested network slice, which the MNO may also requires the tenants to prepay as deposit in advance. Additionally, when a request waits in queue, it can consistently generate a periodical waiting cost for the tenant, which is used to keep the tenant standby for launching the business session.
Once a request is accepted, a new network slice will be created and granted to the corresponding tenant to support the desired end-user service. This service is supposed to generate a periodic revenue Rev that we assume as known or well predictable by the tenant. Meanwhile, the tenant has to pay a periodical expenditure Exp that is composed by the operations cost to maintain the service, and the residential rent for the network slice in case that the rent is not completely prepaid to the MNO in the request-issuing phase. Without loss of generality, we can assume that the periodical profit is positive – as the tenant will never issue such a request otherwise.
Ii-D Impatience of Tenants
The longer a request waits in queue, the more waiting cost the tenant pays. In the extreme case, this cost can exceed the total profit that can be generated by the requested slice through its entire lifetime. Therefore, a rational tenant will choose to cancel its request for slice, when it expects to lose by keep waiting. Such impatient behavior commonly exists in various queuing systems, and has been extensively studied in the fields of queuing theory and operations research [14, 15]. Generally, there are two forms of impatient behavior in queues:
A tenant may give up issuing a request, if it considers the current request queue as overlength, which implies a probably long waiting time.
Having issued a request and and waited in queue for a while, a tenant may have a pessimistic expectation of the remaining waiting time, thereby give up waiting and cancel its request without being served.
Different models have been proposed to describe the statistical features of these phenomena, such as the linear balking model , the hyperbolic balking model , the exponential balking model  and the exponential reneging model . However, it remains unclear in the context of SlaaS environment, which model applies the best – or even more fundamentally – if any of them applies. Furthermore, to the best of our knowledge, it has not been clearly stated yet, how each individual tenant request behaves in its queue, or what kind of information is essential for the tenant to make appropriate decision. In the next section of this paper, we attempt to address these problems.
Iii Rational Balking & Reneging Strategies
An arbitrary request can be characterized by a feature vector , where is the slice type, is the expected life time of slice upon acceptance, , and are the business model parameters discussed in Section II-C. Meanwhile, the queue of slice type can be characterized by where is its current queue length, and are the arrival rate and serving rate of type- requests, respectively.
When the business demand arises, i.e. request is generated (not issued yet) by the tenant, the expected total profit that this business session can generate is
Meanwhile, the expected cost of issuing the request and waiting in the queue till acceptance is
where is the time a request must wait in queue until being accepted with other requests ahead of it.
Assume that the tenants can obtain the a priori knowledge about , self-evidently, a rational tenant will propose the request if and only if , which can be described as a binary decision model:
where stands for issuing and for balking.
Iii-a Rational Balking without Reneging
For simplification, we first ignore reneging, so that
Hence, given certain , and , it yields that if the tenant is able to observe before pushing its request into the queue. The balking chance of such a tenant is therefore a function of under any certain distribution of :
Particularly, when and :
which is the linear balking model in . Note that there is an implicit limit for the queue length , even if no such limit is explicitly set by the MNO.
When and (rational distribution):
When and :
which is the hyperbolic balking model in  with the factor of patience . Note that this model assumes every slice remains active for at least an unit time period.
When and :
which is the exponential balking model in  with the exponent of impatience .
Iii-B Rational Reneging
Now we consider the reneging behavior. As we will derive below, the decision of reneging highly relies on the tenant’s knowledge of the queue. So we discuss this problem separately in different cases.
Iii-B1 Full Knowledge
First, we assume that every tenant is not only able to observe the position of its request in the queue in real time, but also informed by the MNO about . In this case, at every step it will rationally choose whether to cancel the request (renege) after waiting in the queue for , based on the observed queue information:
where is the request’s position in the queue, indicates waiting and for reneging.
where is the reneging rate at the queue position for and . Therefore, this case has an equivalent form where the MNO informs the tenant that issues the request in queue about and for all . The values of and converge to stable levels in long term when the business scenario remains consistent, therefore we consider them here as constants that are known by the MNO.
Iii-B2 Knowledge of Current Position and Serving Rate
In this case, we assume that every tenant is informed by the MNO about , and keeps observing the position of its request in the queue, but has no knowledge about . As a tenant generally lacks knowledge of requests issued by other tenants, i.e. the statistics of their parameters
. So no tenant is able to estimate the values ofin this case, which disables the estimation of according to Eq. (19). However, knowing that for all , the tenants can make conservative estimations based on
and therefore Eq. (9) becomes
Iii-B3 Knowledge of Current Position
In this case, every tenant is able to track its request’s current position in queue, but has no a priori knowledge about . Thus, the tenant has to estimate through an online learning process while waiting in queue:
It has to be noted here that the estimation in Eq. (23) is only valid when , and the estimation error decreases as increases. Therefore a threshold should be set whereas is estimated only when . In summary:
Iii-B4 Knowledge of Mean Waiting Time
In this case, all tenants are only informed about the average waiting time since joining the queue till being served, in which the waiting time requests that balk and renege do not count. Meanwhile, the current position in queue is unobservable for a request unless (i.e. when the request gets served).
In this case, a request can only roughly consider the batch of all other requests ahead of it as an integrated entity in queue. As the service to every single request is a Poisson event, we approximately consider the complete service to this batch (of unknown length) as a Poisson event with arriving rate of . Thus, the reneging decision can be made as
Note that Eq. (26) is independent of or so that it always returns the same decision.
If the tenant possesses neither the position of its waiting request in the queue, nor any knowledge about the dynamics of queue, it can only make a blind reneging, where a maximal waiting time is predetermined at the queue entrance. A straightforward solution is to set a maximal cost proportional to the total profit that can be generated by the requested slice upon admission:
where is the factor of risking that indicates the tenant’s intension of waiting in the queue. This yields that
It is worth to note that when and , the blind reneging model becomes the classic reneging model 
where the maximal waiting time is exponentially distributed. Furthermore, whenthe tenants will never renege and therefore become patient.
Iii-C Balking with Renaging
Iv Numerical Evaluation
Iv-a Simulation Setup
In the numerical simulations111Note that all simulations shown in this work are implemented by means of Julia . we define a simplified scenario, where the MNO holds a normalized two-dimensional () resource pool and different slice types are defined and specified with the parameters listed in Table I.
|Slice Type ()|
Under these specifications, the admissibility region is composed of different values of . For the sake of simplicity, we do not consider the option of “standby” in the MNO’s slicing strategy: in this way for every state only two different preference vectors, i.e. and , are available. Therefore, there are in total different slicing strategies applicable for the MNO. We randomly select 1000 from these valid slicing strategies, and with every MNO slicing strategy, we evaluate the rational balking/reneging strategies of impatient tenants with different information available. For every individual evaluation, we simulate the arrivals of tenant requests and the MNO’s operations for 400 periods.
Iv-B Evaluation Results
|Case||Total Profit ()||Mean Profit||Profiting Rate|
|Type 1||Type 2||Type 1||Type 2||Type 1||Type 2|
|Knowledge of current position ()||-97.64||-772.49||-12.23||-19.39||2.17%||0.34%|
|Knowledge of mean waiting time||-55.59||-15.27||-11.14||-14.11||8.28%||26.73%|
|Knowledge of current position and serving rate||4.21||7.78||9.25||8.78||81.64%||87.15%|
During the simulation, we track the end-profit of every issued slice request defined as follows
where is the total waiting time from queue entrance to admission/reneging. Then we evaluate the balking/reneging strategies of tenants with three different metrics:
Total profit: the sum of end-profits obtained by all issued slice requests;
Mean profit: the average end-profit obtained by all issued slice requests;
Profitting rate: the ratio of slice requests that lead to positive end-profits in all issued requests.
The simulation results are listed in Table II.
It can be easily observed from the results that, given the knowledge about position of its request in queue and the queue’s serving rate, a tenant has a high chance to make correct decisions of balking and reneging. Thereby it is able to mitigate most losses caused by excessive waiting in case of request congestion, and thus obtain a positive profit.
The information about reneging rates provides a further improvement in addition, but only to an insignificant degree. One reason of this phenomena could be that, after a rational balking with sufficient knowledge, the reneging rate of requests generally remains limited, and therefore it exhibits a little impact on the waiting time in queue.
In contrast, when provided with only insufficient information, most tenants fail to benefit from their impatience. An impatient tenant knowing only the mean waiting time in queue can reasonably avoid most extreme long waitings by balking, and therefore expected to lose less in average, but its chance to achieve positive profits is not significantly higher than the patient tenants. When knowing only the current position of request in queue, an impatient tenant loses even more than patient ones under request congestions, as it cannot balk but only renege with an inaccurate estimation of the serving rate. The worst decisions are made by the blind tenants that renege after predetermined waiting time, which lead to a negligible chance of making positive profit and a huge total loss. More specifically, we can observe a tendency that the performance of blind reneging increases along with the factor , approaching to the upper bound of patience where .
In summary, network slice tenants need information about queue dynamics from the MNO—at least the minimum information to enable balking—so that they can benefit from impatience in case of slice requests congestion. Otherwise, it is statistically better for the tenants to wait with patience rather than biding.
Iv-C Distribution of Reneging Time
In Section III-A, we have analytically proven the applicability of various classical models of balking statistics in the slice admission control scenario upon different distributions of the slice lifetime . The distribution of reneging time in SAC, however, is relatively challenging to derive in such way.
To evaluate the applicability of existing reneging models, we execute two additional simulations carrying out for each of those times Monte-Carlo test. For both simulations, the environment is configured to the same specifications listed in Table I, and the tenants possess full knowledge of the queuing system. In the first simulation, each test simulates operations periods, where the MNO is always fixed to a static admission strategy that for all . In the second simulation, the MNO is set to a random admission strategy in every Monte-Carlo test. We observe the waiting time of all reneged requests and illustrate the obtained results in Fig. 3. Generally, in both simulations and for both slice types, the maximal waiting time before reneging exhibits a exponential distribution, which supports applying the classical model proposed in  to simplify queue models from the MNO’s perspective.
Iv-D Further Discussion
Certainly, when fed with the knowledge of the current active queues, tenants may be more encouraged to balk or renege from densely congested queues of slice requests, which in turn leads to a decreased number of awaiting slice requests. Nevertheless, it should be noted that the phenomena of balking and reneging are only significant when the queues are considerably long. In this case, the MNO’s resources are already sufficiently utilized, and the utilization rate is hardly impacted by the impatience of tenants. On the other hand, if there is a lack of information about the queues, as demonstrated in Section IV-B, tenants can suffer from high probability of business loss. This will, self-evidently, suppress the tenants’ interest for the MNO’s slice service on long-term windows, leading to a consistent loss of customers from the MNO’s perspective. In summary, we can argue that it is a win-win option for the MNO to share full knowledge of the queues, or at least the request’s current position in queue and the serving rate of queue, to every awaiting tenant.
V Conclusion and Outlook
In this paper we have addressed the network slicing orchestration problem from a different perspective: realistic tenant behaviors to cope with network slice requests congestion. In particular, we have studied the impatient behavior of network slice tenants in the multi-queue slice admission control scenario. We have derived the rational behavior models of tenants in different cases of knowledge, and demonstrated the applicability of some classical statistical balking and reneging models in the discussed environment. We have carried out extended simulations and shown that our results encourage the network operator (MNO) to fully make available information of its current queues to the awaiting tenants to benefit both MNO and tenants in terms of efficiency, waiting time and, in turn, overall revenues.
As an outlook, the statistical balking and reneging models that are proven applicable for the Slice-as-a-Service (SlaaS) paradigm can be taken into account for the design and optimization of advanced MNO’s admission strategies that automatically share the queues status while fostering or discouraging tenants from sending additional slice requests.
This work was supported in part by the European Union Horizon-2020 Project 5G-MoNArch under Grant Agreement 761445, and in part by the Network for the Promotion of Young Scientists (TU-Nachwuchsring), Technische Universität Kaiserslautern with individual funding.
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