Congestion Probabilities in a Multi-Cluster C-RAN Servicing a Mixture of Tra ﬃ c Sources

: A multi-cluster cloud radio access network (C-RAN) is considered in this paper where the remote radio heads (RRHs) form di ﬀ erent clusters. A cluster includes RRHs that have the same radio resource unit capacity. In addition, all RRHs are separated from the common pool of computational resource units named baseband units. Each RRH accommodates calls whose arrival process can be random, quasi-random, or even bursty. The latter is modeled according to the compound Poisson process where calls arrive in the C-RAN in the form of batches whose size (in calls) is generally distributed. An arriving call requires a radio and a computational resource unit so as to be accepted in the C-RAN. If at least one of these units is not available, the call is blocked. To analyze the proposed multi-cluster C-RAN we model it as a loss system, show that the steady-state probabilities have a product form solution and propose an algorithm for the computation of congestion probabilities. The accuracy of the proposed algorithm is veriﬁed via simulation.


Introduction
The cloud radio access network (C-RAN) architecture consists of a number of base stations where the remote radio heads (RRHs) are separated from the baseband units (BBUs) [1]. The RRHs are grouped in different clusters according to their capacity in terms of radio resource units (RUs). On the other hand, the BBUs form a common pool of computational RUs which can be connected to the RRHs, with a high-capacity fronthaul, via the common public radio interface (CPRI) [2]. In addition, we consider virtualized BBU computational resources (V-BBU) in order to benefit from network function virtualization [1].
In this multi-cluster C-RAN architecture, we concentrate on call-level and study the main performance measure which is call blocking probabilities (CBP). A new call simultaneously requires a radio RU from the RRH that will serve that call and a computational RU from the V-BBU. If at least one of these units is not available, the call is blocked and lost. Otherwise, the new call is accepted in the RRH for an exponentially distributed service time. As far as the call arrival process is concerned, we assume that the C-RAN accommodates random, quasi-random, and bursty traffic. Random traffic refers to calls generated by an infinite number of mobile users (MUs) and is described via the classical Poisson process. It is considered to be the simplest call arrival process in teletraffic modeling due to the fact that it leads to efficient CBP formulas [3,4]. Quasi-random traffic is smoother than random traffic since it refers to calls generated by a finite number of MUs. Finally, bursty traffic, which is  Consider the m-th RRH of class z (z = 1, …, Z and m = 1, …, Mz) and let nz,m be the calls under service in that particular RRH. Then, the steady-state vector where Bc refers to the blocking caused due to insufficient computational RUs while Bz,r expresses the blocking caused solely due to unavailability of radio RUs. According to [14], the case of simultaneous blocking due to both insufficient radio and computational RUs is classified as Bc. In that sense, the sets Ac and Az,r that include the blocking states of Bc and Bz,r, respectively, are mutually exclusive, i.e., . We also keep this distinction in the proposed g-SC-MC model. The values of Bz,tot can be accurately (compared to simulation) calculated either via a BF method (helpful only in small multi-cluster C-RAN examples since it requires enumeration/processing of the system's state space  ) or via a recursive method [14]. Some minor corrections in the CBP formulas of [14] have been presented in [15].
Alternatively, the values of Bz,tot together with the occupancy distribution of the computational RUs can be efficiently determined via the following convolution algorithm [15]: Step 1 For each of the Mz RRHs that belong to class z (z = 1, …, Z and m = 1, …, Mz) determine the occupancy distribution z,m qj () , where j = 1, …, Cz, via Consider the m-th RRH of class z (z = 1, . . . , Z and m = 1, . . . , M z ) and let n z,m be the calls under service in that particular RRH. Then, the steady-state vector n = n 1,1 , . . . , n 1,M 1 , . . . , n z,1 , . . . , n z,M z , . . . , n Z,1 , . . . , n Z,M Z expresses the number of calls serviced in the RRHs of the Z classes while P(n) is the corresponding steady-state probability distribution. The latter has the following PFS [14] α n z,m z /n z,m ! and Ω is the system's state space described as follows Based on (1), we compute the total CBP of calls in a class z RRH, B z,tot , as where B c refers to the blocking caused due to insufficient computational RUs while B z,r expresses the blocking caused solely due to unavailability of radio RUs. According to [14], the case of simultaneous blocking due to both insufficient radio and computational RUs is classified as B c . In that sense, the sets A c and A z,r that include the blocking states of B c and B z,r , respectively, are mutually exclusive, i.e., A c ∩ A z,r = ∅. We also keep this distinction in the proposed g-SC-MC model.
The values of B z,tot can be accurately (compared to simulation) calculated either via a BF method (helpful only in small multi-cluster C-RAN examples since it requires enumeration/processing of the system's state space Ω) or via a recursive method [14]. Some minor corrections in the CBP formulas of [14] have been presented in [15].
Alternatively, the values of B z,tot together with the occupancy distribution of the computational RUs can be efficiently determined via the following convolution algorithm [15]: Step 1 For each of the M z RRHs that belong to class z (z = 1, . . . , Z and m = 1, . . . , M z ) determine the occupancy distribution q z,m ( j), where j = 1, . . . , C z , via The normalization of q z,m ( j) can be achieved via the constant G z,m = C z j = 0 q z,m ( j) and in that case the normalized values become q z,m ( j) = q z,m ( j)/G z,m .
Step 2 Compute the aggregated occupancy distribution Q (−(z,1)) of all RRHs, excluding the (z, 1) RRH, via the formula where the first part of (4) (q 1,1 * . . . * q 1,M 1 ) refers to the M 1 RRHs of the first class, the second part (q z,2 * . . . * q z,M z ) refers to the M z − 1 RRHs of the zth class (the (z, 1) RRH is excluded) and the last part (q Z,1 * . . . * q Z,M Z ) refers to the M Z RRHs of the Zth class while Since the resulting distribution may not be normalized, it is recommended to apply the normalization constant G z,u,w in the results of (5).
Step 3 Compute B z,tot based on the results of the previous step where q z,1 (C z ) expresses the unavailability of radio RUs in the (z, 1) RRH (computed in step 1), Q (−(z,1)) (r) express the normalized values of Q (−(z,1)) (r) and q(T) expresses the un-normalized probability of unavailable computational RUs, obtained via q(T) = T r = 0 Q (−(z,1)) (r)q z,1 (T − r) while G is the (normalization) constant of the operation Q (−(z,1)) * q z,1 calculated in (5). Note that (14) refers to the B z,tot of any RRH that belongs to class z since all RRHs of that class have offered traffic-load a z and capacity C z .
Based on the presented algorithm, we compute the computational RUs' occupancy distribution according to the formulas where G expresses the normalization constant of Q (−(z,1)) * q z,1 .

Description of the Analytical Model
In the proposed g-SC-MC model, we consider again a multi-cluster C-RAN which consists of the V-BBU (T computational RUs) and Z classes of RRHs. Class z (z = 1, . . . , Z) consists of a cluster of RRHs that includes a total number of M z RRHs whose capacity is C z radio RUs. Let the number of M z RRHs be composed of M z,inf RRHs that serve random traffic, M z,fin RRHs that accommodate quasi-random traffic and M z,cP RRHs that accommodate compound Poisson traffic, i.e., M z,inf + M z,fin + M z,cP = M z .
In the case of random traffic, let λ z,P be the rate of Poisson arriving calls in the (z, m) RRH (m = 1, . . . , M z,inf and z = 1, . . . , Z). The corresponding offered traffic-load will be α z,P = λ z,P /µ. In the case of quasi-random traffic, let λ z,m,F = (N z,m − n z,m )v z,m,F be the call arrival rate in the (z, m) RRH (m = M z,inf + 1, . . . , M z,inf + M z,fin and z = 1, . . . , Z) where N z,m denotes the population of MUs that can generate traffic in the (z, m) RRH, n z,m refers to the in-service calls in that RRH and v z,m,F expresses the call arrival rate per idle MU. In the case of the compound Poisson process, batches of calls arrive in the (z, m) RRH (m = M z,inf + M z,fin + 1, . . . , M z and z = 1, . . . , Z) according to a Poisson process, with arrival rate λ z,cP , while the batch size (in number of calls) is generally distributed. Calls that belong to the same arriving batch are treated independently which means that some calls can be accepted in the serving RRH while the rest calls will be blocked, depending on the availability of RUs.
As far as the call admission is concerned, a call is accepted for an exponentially distributed service time with mean µ −1 in a RRH if a radio RU (from that RRH) as well as a computational RU are available when the call arrives in the system. Otherwise, call blocking occurs.
Let the steady-state vector n = n 1,1 , . . . , n 1,M 1 , . . . , n z,1 , . . . , n z,M z , . . . , n Z,1 , . . . , n Z,M Z express the number of in-service calls in the RRHs of the Z classes, where The corresponding steady-state probability distribution is denoted as P gen (n). In order to analyze the g-SC-MC model, we show that P gen (n) can be described via a PFS. To this end, it should be shown that some form of local balance exists between state n and state n + z,m = n 1,1 , . . . , n z,m + 1, . . . , n Z,M Z . More specifically, consider the level L (z,m) n that separates state n + z,m from state n. This level is crossed if one of the following three call-arrival cases occurs: (i) an arriving call that follows a Poisson process requests service from the (z, m) RRH (where m = 1, . . . , M z,inf ), (ii) an arriving call that follows a quasi-random process requests service from the (z, m) RRH (where m = M z,inf + 1, . . . , M z,inf + M z,fin ), (iii) calls of an arriving batch requests service from the (z, m) RRH (where m = M z,inf + M z,fin + 1, . . . , M z ). An additional call-departure case exists when an in-service call departs from the serving RRH after its service is completed. In what follows, we focus on each of the three call-arrival cases and the corresponding call-departure case.
In the first case, we have a Poisson arriving call of rate λ z,P in the (z, m) RRH (m = 1, . . . , M z,inf and z = 1, . . . , Z). We can express the upward probability flow across L The downward probability flow across L (z,m) n takes place when a call departs from the (z, m) RRH and can be expressed as Based on (8) and (9), we have the following local balance equation for L (z,m) n λ z,P P gen (n) = (n z,m + 1)µP gen n + z,m .
In the second case, we have an arriving call, generated from a finite number of MUs, in the (z, m) RRH (m = M z,inf + 1, . . . , M z,inf + M z,fin and z = 1, . . . , Z). We can express the upward probability flow The corresponding downward probability flow across L (z,m) n can be expressed via (9). Based on (11) and (9), we have the following local balance equation for L (z,m) n (N z,m − n z,m )v z,m,F P gen (n) = (n z,m + 1)µP gen n + z,m .
In the third case, we have an arriving batch of calls in the (z, m) RRH (m = M z,inf + M z,fin + 1, . . . , M z and z = 1, . . . , Z). We can express the upward probability flow across L where n −ω z,m = n 1,1 , . . . , n z,m − ω, . . . , n Z,M Z , P gen n −ω z,m is the steady-state probability and S r is the probability that the arriving batch contains r calls.
The corresponding downward probability flow across L (z,m) n can be expressed via (9). Based on (13) and (9), we have the following local balance equation for L (z,m) n n z,m Equations (10), (12), and (14) can be satisfied via the following PFS Having determined the values of P gen (n) we can compute the total time congestion (TC) probabilities in the (z, m) RRH (m = 1, . . . , M z and z = 1, . . . , Z) either according to a BF method or according to a convolution algorithm (presented in Sections 3.2 and 3.3, respectively) and based on the formula where B TC c and B TC z,m,r refer to the unavailable computational RUs in the V-BBU and radio RUs in the (z, m) RRH, respectively.

BF Method for the Computation of Congestion Probabilities
The determination of B TC c can be based on the values of P gen (n) according to the formula where Ω =T = n : On the same hand, the values of B TC z,m,r can be computed (via (15)) via where It is evident that (17) and (18) can be quite complex since it is expected to enumerate and process the state space Ω. Because of this, we adopt the convolution algorithm proposed in the next subsection.

Convolution Algorithm for the Computation of Congestion Probabilities
The PFS of the g-SC-MC model implies that a convolution algorithm can be applied for the accurate and efficient determination of congestion probabilities. To this end, we propose the following three-step convolution algorithm.
Step 1 (a) For each of the M z,inf RRHs that accommodate random traffic and belong to class z (z = 1, . . . , Z and m = 1, . . . , M z,inf ) determine the occupancy distribution q gen,z,m ( j), where j = 1, . . . , C z , via Note that q gen,z,m (0) = 1 while q gen,z,m (x) = 0 for x > C or x < 0. Furthermore, the normalization of q gen,z,m ( j) can be achieved via the constant G gen,z,m = C z j = 0 q gen,z,m ( j) and in that case the normalized values become q gen,z,m ( j) = q gen,z,m ( j)/G gen,z,m .
Step 2 Compute the aggregated occupancy distribution Q gen,(−(z,m)) of all RRHs, excluding the (z, m) RRH, via the formula where q gen,z,u * q gen,z,w = Since the resulting distribution may not be normalized, it is recommended to apply the normalization constant G gen,z,u,w in the results of (23).
Step 3 Compute B TC z,m,tot based on the results of Step 2 where q gen,z,m (C z ) expresses the unavailability of radio RUs in the (z, m) RRH (computed in the first step), while the second summation expresses the un-normalized probability of unavailable computational RUs and G gen is the normalization constant of Q gen,(−(z,m)) * q gen,z,m computed via (23). In addition to TC probabilities (which refer to the proportion of time the multi-cluster C-RAN has no available RUs) we can also compute the call congestion (CC) probabilities (which refer to the proportion of lost calls), B CC z,m , for a new call in an RRH that accommodates quasi-random or compound Poisson traffic. CC and TC probabilities coincide in the case of random traffic. To determine the CC probabilities in the case of quasi-random traffic, we can adopt the convolution algorithm for a system with N z,m −1 sources. On the same hand, to determine the CC probabilities in the case of compound Poisson traffic we can adopt the formula A popular batch size distribution is the geometric distribution which possesses the memoryless property [4,47]. If this distribution is adopted and assuming that β is its parameter, thenŜ = 1/(1 − β).
The determination of n z,m can be based on the formula n z,m = 1 G gen C z j = 1 y gen,z,m ( j)q gen,z,m ( j) T− j r = 0 Q gen,(−(z,m)) (r), (26) where G gen is the normalization constant of the convolution operation Q gen,(−(z,m)) * q gen,z,m determined via (23) and y gen,z,m ( j) expresses the average number of calls that exist in state j of the (z, m) RRH. The determination of y gen,z,m ( j), for j = 1, . . . , C z , can be based on the formula y gen,z,m ( j) = α z,cP q gen,z,m ( j) Based on the convolution algorithm, we can also compute the computational RUs' occupancy distribution according to the formulas q gen (0) = Q gen,(−(z,m)) (0)q gen,z,m (0)/G gen , j = 0 q gen ( j) = j r = 0 Q gen,(−(z,m)) (r)q gen,z,m ( j − r)/G gen , j = 1, . . . , T Considering the computational complexity of (23) is in the order of O(T 2 ) while the corresponding complexity of (28) is in the order of O(MT 2 ), where M = (M 1 , . . . , M Z ).

Evaluation
In this section, we consider an example of a multi-cluster C-RAN and provide both simulation and analytical results for the CC and TC probabilities in the case of the proposed g-SC-MC model and analytical results for the existing SC-MC model. The simulation tool adopted in our example is SIMSCRIPT III [48] while all simulation results are mean values of seven runs. In every run, 200 million calls are generated while the initial 5% of them are not taken into consideration in order to have a warm-up period [49,50]. Regarding reliability ranges, they are less than two order of magnitudes and therefore we do not present them in Figures 2-7.
The C-RAN example presented herein consists of two clusters (Z = 2). The first cluster consists of M 1 = 6 RRHs where the capacity per RRH is C 1 = 10 radio RUs. The second cluster consists of M 2 = 3 RRHs where the capacity per RRH is C 2 = 15 radio RUs. Regarding the computational RUs, we consider that T = 80 RUs. Values of T that are much lower than T = M 1 C 1 + M 2 C 2 = 105 RUs will result in quite high values for the congestion probabilities due to insufficient computational RUs (B TC c ) and therefore are not taken into consideration herein. In the case of the existing SC-MC model, we assume that the offered traffic load (per RRH) in the first cluster is α 1 = 4 erl and in the second cluster α 2 = 7 erl. In the case of the proposed g-SC-MC model, the traffic-load offered in the RRHs of the first cluster is as follows: the first two RRHs accommodate random traffic with α 1,P = 4 erl (per RRH), RRHs numbered 3 to 4 accommodate quasi-random traffic with N 1,3 = N 1,4 = 50 sources and α 1,3,idle = α 1,P /N 1,3 , α 1,4,idle = α 1,P /N 1,4 while RRHs numbered 5 to 6 accommodate compound Poisson traffic with α 1,cP = 4 erl. Regarding the size distribution (in terms of calls) of the arriving batches, we consider the geometric distribution with parameter β = 0.2. Similarly, the traffic-load offered in the RRHs of the second cluster is as follows: the first RRH accommodates random traffic with α 2,P = 7 erl, the second RRH accommodates quasi-random traffic with N 2,2 = 100 sources and α 2,2,idle = α 2,P /N 2,2 while the third RRH accommodates compound Poisson traffic with α 2,cP = 7 erl and a geometrically batch size distribution with β = 0.2.
Electronics 2020, 9, x FOR PEER REVIEW 9 of 19 million calls are generated while the initial 5% of them are not taken into consideration in order to have a warm-up period [49,50]. Regarding reliability ranges, they are less than two order of magnitudes and therefore we do not present them in Figures 2-7.              In the x-axis of Figures 2-7, the values of offered traffic-load increase in steps of 0.2 erl. Therefore, for Figures 2-4 that refer to the first cluster, point 1 in the x-axis is 4 erl while point 11 is 6.0 erl. Similarly, for Figures 5-7 that refer to the second cluster, point 1 in the axis is 7 erl while point 11 is 9.0 erl.
In Figure 2, we present both simulation and analytical results of the TC probabilities (B TC 1,m,r ) for the g-SC-MC model and the corresponding results (B 1,r ) for the SC-MC model, which refer to time congestion due to unavailability of radio RUs. In Figure 3 Similar conclusions are obtained in Figures 5-7 which refer to the congestion probabilities in the RRHs of the second cluster. As a final comment, we mention that the accuracy of the analytical results of the proposed model, compared to simulation, is not affected by an increase in the number of clusters.

Conclusions
We proposed a loss model for the analysis and evaluation of a multi-cluster C-RAN that accommodates random, quasi-random, and bursty traffic. New calls can be accepted in an RRH if their resource requirements (a radio and a computational RU) can be met. If any of these two RUs is not available, then call blocking occurs. We showed that the model has a PFS for the steady-state probabilities and provided an efficient convolution algorithm for the computation of the main performance measures such as congestion probabilities. The accuracy of the proposed algorithm was verified via simulation. As a possible future extension of this work, we intend to study single or multi-cluster C-RAN that accommodates calls: (i) whose RUs may fluctuate between a maximum and a minimum value during this service time, forming the so called "elastic traffic" [51][52][53][54][55]; or (ii) whose requirements in terms of RUs may be different during the call admission phase [56][57][58].

Conflicts of Interest:
The authors declare no conflict of interest.

Appendix A
In this appendix, a tutorial example is presented that shows some intermediate results required for the determination of congestion probabilities in the proposed g-SC-MC model. To this end, we consider a multi-cluster C-RAN of T = 7 computational RUs and Z = 2 classes of RRHs. The first class consists of M 1 = 2 RRHs of C 1 = 3 radio RUs. The second class consists of M 2 = 1 RRH of C 2 = 4 radio RUs. Regarding the first class, arriving calls in the first RRH follow a Poisson process with rate λ 1,P = 1.0, while arriving calls in the second RRH follow a quasi-random process with N 1,2 =10 and v 1,2,F =0.1. Regarding the second class, arriving calls in the RRH follow a compound Poisson process with λ 2,cP = 0.8 and β = 0.5. The service time of all calls is exponentially distributed with mean µ −1 = 1.0. The state space of this example consists of 70 states (whose form is n = (n 1,1 , n 1,2 , n 2,1 )).
Step 3 Based on the above, the TC probabilities due to lack of radio RUs in each RRH are: