Congestion Probabilities in a Multi-Cluster C-RAN Servicing a Mixture of Traffic Sources

Abstract

A multi-cluster cloud radio access network (C-RAN) is considered in this paper where the remote radio heads (RRHs) form different 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.

1. 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 considered to be a dominant part in 5G networks [5,6,7], is more peaked than random traffic and can be well approximated via the compound Poisson process whose applications in loss/queueing systems are numerous [8,9,10,11,12,13]. In the compound Poisson process, batches of calls arrive in the system according to a Poisson process while their size (in number of calls) is generally distributed.

The springboard for the analysis of the proposed system is [14] where a loss model, named single-class-multi-cluster (SC-MC) model has been proposed for the case of random traffic only. The term ‘single-class’ refers to the fact that all new calls have the same requirements in terms of RUs. The authors of [14] consider a multidimensional Markov chain for the description of the SC-MC loss model. This chain is reversible, a fact that leads to the computation of the steady-state probabilities via a product form solution (PFS). The latter leads to an accurate CBP calculation via recursive formulas or via an evaluation method which requires the system’s state space enumeration/processing. Recently, in [15], the SC-MC model has been extended to include the case of quasi-random traffic, only. We name this model finite SC-MC (f-SC-MC) model.

In this paper, we generalize the SC-MC and the f-SC-MC models by assuming that RRHs may serve random, quasi-random and bursty traffic. The proposed model is named generalized SC-MC (g-SC-MC) model while our contribution can be summarized as follows: (1) we initially show that the g-SC-MC model can be analytically described via a continuous time Markov chain and that the steady-state probabilities have a PFS, (2) we determine the congestion probabilities via a brute force (BF) evaluation method, (3) we propose a convolution algorithm for the efficient determination of congestion probabilities in the multi-cluster C-RAN, and (4) we provide a comparison of the analytical results of the g-SC-MC model with those obtained according to the models of [14,15] and verify the accuracy of the proposed convolution algorithm via simulation.

Generally speaking, it is significant in network planning and dimensioning procedures to have convolution algorithms or efficient recursive formulas for the determination of various performance measures including congestion probabilities [16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36]. We focus on convolution algorithms which are adopted in the literature both in PFS [37,38,39] and in non-PFS queueing/loss models [40,41,42]. The advantage of such algorithms is that they can incorporate various resource sharing policies such as the bandwidth reservation policy, the complete sharing policy, as well as threshold-based policies [43,44,45,46].

The organization of this paper is as follows: In Section 2, we present a short review of the SC-MC model and provide a convolution algorithm for the CBP determination. In Section 3, we propose the g-SC-MC model. In Section 3.1, we present a PFS for the determination of the steady-state probabilities while in Section 3.2 and Section 3.3 we propose a BF method and a convolution algorithm for the calculation of congestion probabilities, respectively. In Section 4, we provide analytical and simulation results for the congestion probabilities of the proposed g-SC-MC model and the SC-MC model of [14]. We present our conclusion in Section 5. In Appendix A, we provide a short tutorial example of the necessary congestion probabilities calculations for the proposed g-SC-MC model.

2. The SC-MC Model—A Review

Consider the multi-cluster C-RAN of Figure 1 where the V-BBU and the RRHs are separated. In this network, Z classes of RRHs are considered. Class z (z = 1, …, Z) forms a cluster of RRHs that includes a total number of M_z RRHs. A RRH of class z has a capacity of C_z radio RUs which serve the Poisson arriving calls of the MUs. Similarly, the V-BBU consist of T computational RUs.

Let λ_z be the rate of Poisson arriving calls in a class z RRH. A call is accepted for a generally 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. The corresponding offered traffic-load in a class z RRH is given by ${\alpha }_z={\lambda }_z/\mu$ (in erl).

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 $\mathit{n}=\left(n_{1,1}, \dots ,n_{1,M_1},\dots ,n_{z,1}, \dots ,n_{z,M_z},\dots ,n_{Z,1},\dots ,n_{Z,M_Z}\right)$ expresses the number of calls serviced in the RRHs of the Z classes while $P\left(\mathit{n}\right)$ is the corresponding steady-state probability distribution. The latter has the following PFS [14]

$$P(\mathit{n})=G^{-1}\left({\displaystyle \prod _{z=1}^Z{\displaystyle \prod _{m=1}^{M_z}{\alpha }_z^{n_{z,m}}/n_{z,m}!}}\right),$$

(1)

where G =${\displaystyle \sum _{n\in \mathbf{\Omega }}\left({\displaystyle \prod _{z=1}^Z{\displaystyle \prod _{m=1}^{M_z}{\alpha }_z^{n_{z,m}}/n_{z,m}!}}\right)}$ and $\mathbf{\Omega }$ is the system’s state space described as follows $\mathbf{\Omega }=\left\{\mathit{n}:0\le n_{z,1},\dots ,n_{z,M_z}\le C_z, 0\le {\displaystyle \sum _{z=1}^Z{\displaystyle \sum _{m=1}^{M_z}{n}_{z,m}}}\le T\right\}$.

Based on (1), we compute the total CBP of calls in a class z RRH, B_z,tot, as

$$B_{z,tot}=B_c+B_{z,r},$$

(2)

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., $A_c\cap A_{z,r}=\varnothing$. 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 $\mathbf{\Omega }$) 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

$$q_{z,m}\left(j\right)=\frac{{\alpha }_z^j}{j!} q_{z,m}\left(0\right).$$

(3)

The normalization of $q_{z,m}(j)$ can be achieved via the constant $G_{z,m}={\displaystyle {\sum }_{j=0}^{C_z}{q}_{z,m}\left(j\right)}$ and in that case the normalized values become $q_{z,m}^{\prime }\left(j\right)=q_{z,m}\left(j\right)/G_{z,m}$.

• Step 2

Compute the aggregated occupancy distribution Q_(−(z,1)) of all RRHs, excluding the (z, 1) RRH, via the formula

$$Q_{(-(z,1))}=q_{1,1}^{\prime }*\dots *q_{1,M_1}^{\prime }*\dots *q_{z,2}^{\prime }*\dots *q_{z,M_z}^{\prime }*\dots *q_{Z,1}^{\prime }*\dots *q_{Z,M_Z}^{\prime },$$

(4)

where the first part of (4) ($q_{1,1}^{\prime }*\dots *q_{1,M_1}^{\prime }$) refers to the M₁ RRHs of the first class, the second part ($q_{z,2}^{\prime }*\dots *q_{z,M_z}^{\prime }$) refers to the M_z − 1 RRHs of the zth class (the (z, 1) RRH is excluded) and the last part ($q_{Z,1}^{\prime }*\dots *q_{Z,M_Z}^{\prime }$) refers to the M_Z RRHs of the Zth class while

$$q_{z,u}^{\prime }*q_{z,w}^{\prime }=\left\{\begin{array}{l}{q}_{z,u}^{\prime }\left(0\right)\cdot q_{z,w}^{\prime }\left(0\right), {\displaystyle \sum _{x=0}^1{q}_{z,u}^{\prime }\left(x\right)\cdot q_{z,w}^{\prime }\left(1-x\right)},\\ \dots ,{\displaystyle \sum _{x=0}^T{q}_{z,u}^{\prime }\left(x\right)\cdot q_{z,w}^{\prime }\left(T-x\right)}\end{array}\right\}.$$

(5)

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

$$B_{z,tot}=B_{z,r}+B_c=G^{-1}\left(q_{z,1}^{\prime }\left(C_z\right){\displaystyle \sum _{r=0}^{T-C_z-1}{Q}_{{}_{(-(z,1))}}^{\prime }\left(r\right)}+q\left(T\right)\right),$$

(6)

where $q_{z,1}^{\prime }\left(C_z\right)$ expresses the unavailability of radio RUs in the (z, 1) RRH (computed in step 1), $Q_{{}_{(-(z,1))}}^{\prime }\left(r\right)$ express the normalized values of $Q_{{}_{(-(z,1))}}^{}\left(r\right)$ and $q\left(T\right)$ expresses the un-normalized probability of unavailable computational RUs, obtained via $q\left(T\right)={\displaystyle \sum _{r=0}^T{Q}_{(-(z,1))}^{\prime }\left(r\right)q_{z,1}^{\prime }\left(T-r\right)}$ while G is the (normalization) constant of the operation $Q_{(-(z,1))}^{\prime }*q_{z,1}^{\prime }$ 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

$$\begin{array}{l}{q}^{\prime }\left(0\right)=G^{-1}\left(Q_{{}_{(-(z,1))}}^{}\left(0\right)q_{z,1}^{\prime }\left(0\right)\right), j=0\\ q^{\prime }\left(j\right)=G^{-1}\left({\displaystyle \sum _{r=0}^j{Q}_{{}_{(-(z,1))}}^{}\left(r\right)q_{z,1}^{\prime }\left(j-r\right)}\right), j=1,\dots ,T\end{array}$$

(7)

where G expresses the normalization constant of $Q_{(-(z,1))}^{}*q_{z,1}^{\prime }$.

3. Proposed g-SC-MC Model

3.1. 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 ${\alpha }_{z,\mathrm{P}}={\lambda }_{z,\mathrm{P}}/\mu$. In the case of quasi-random traffic, let ${\lambda }_{z,m,\mathrm{F}}=\left(N_{z,m}-n_{z,m}\right)v_{z,m,\mathrm{F}}$ 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 $\mathit{n}=\left(n_{1,1}, \dots ,n_{1,M_1},\dots ,n_{z,1}, \dots ,n_{z,M_z},\dots ,n_{Z,1},\dots ,n_{Z,M_Z}\right)$ express the number of in-service calls in the RRHs of the Z classes, where M₁ = M_1,inf + M_1,fin + M_1,cP, M_z = M_z,inf + M_z,fin + M_z,cP and M_Z = M_Z,inf + M_Z,fin + M_Z,cP. The corresponding steady-state probability distribution is denoted as $P_{\mathrm{gen}}\left(\mathit{n}\right)$. In order to analyze the g-SC-MC model, we show that $P_{\mathrm{gen}}\left(\mathit{n}\right)$ can be described via a PFS. To this end, it should be shown that some form of local balance exists between state $\mathit{n}$ and state ${\mathit{n}}_{z,m}^{+}=\left(n_{1,1}, \dots ,n_{z,m}+1,\dots ,n_{Z,M_Z}\right)$. More specifically, consider the level $L_{\mathit{n}}^{(z,m)}$ that separates state ${\mathit{n}}_{z,m}^{+}$ from state $\mathit{n}$. 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 $L_{\mathit{n}}^{(z,m)}$ as

$$f^{(up)}\left(L_{\mathit{n}}^{(z,m)}\right)={\lambda }_{z,\mathrm{P}}{P}_{\mathrm{gen}}\left(\mathit{n}\right).$$

(8)

The downward probability flow across $L_{\mathit{n}}^{(z,m)}$ takes place when a call departs from the (z, m) RRH and can be expressed as

$$f^{(down)}\left(L_{\mathit{n}}^{(z,m)}\right)=\left(n_{z,m}+1\right)\mu P_{\mathrm{gen}}\left({\mathit{n}}_{z,m}^{+}\right).$$

(9)

Based on (8) and (9), we have the following local balance equation for $L_{\mathit{n}}^{(z,m)}$

$${\lambda }_{z,\mathrm{P}}{P}_{\mathrm{gen}}\left(\mathit{n}\right)=\left(n_{z,m}+1\right)\mu P_{\mathrm{gen}}\left({\mathit{n}}_{z,m}^{+}\right).$$

(10)

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 across $L_{\mathit{n}}^{(z,m)}$ as

$$f^{(up)}\left(L_{\mathit{n}}^{(z,m)}\right)={\lambda }_{z,m,\mathrm{F}}{P}_{\mathrm{gen}}(\mathit{n})=\left(N_{z,m}-n_{z,m}\right)v_{z,m,\mathrm{F}}{P}_{\mathrm{gen}}\left(\mathit{n}\right).$$

(11)

The corresponding downward probability flow across $L_{\mathit{n}}^{(z,m)}$ can be expressed via (9). Based on (11) and (9), we have the following local balance equation for $L_{\mathit{n}}^{(z,m)}$

$$\left(N_{z,m}-n_{z,m}\right)v_{z,m,F}{P}_{\mathrm{gen}}(\mathit{n})=\left(n_{z,m}+1\right)\mu P_{\mathrm{gen}}\left({\mathit{n}}_{z,m}^{+}\right).$$

(12)

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 $L_{\mathit{n}}^{(z,m)}$ as

$$f^{(up)}\left(L_{\mathit{n}}^{(z,m)}\right)={\displaystyle \sum _{\omega =0}^{n_{z,m}}{P}_{\mathrm{gen}}\left({\mathit{n}}_{z,m}^{-\omega }\right)}{\lambda }_{z,\mathrm{cP}}{\displaystyle \sum _{r=\omega +1}^{\infty }{S}_r},$$

(13)

where ${\mathit{n}}_{z,m}^{-\omega }=\left(n_{1,1}, \dots ,n_{z,m}-\omega ,\dots ,n_{Z,M_Z}\right)$, $P_{\mathrm{gen}}\left({\mathit{n}}_{z,m}^{-\omega }\right)$ is the steady-state probability and S_r is the probability that the arriving batch contains r calls.

The corresponding downward probability flow across $L_{\mathit{n}}^{(z,m)}$ can be expressed via (9). Based on (13) and (9), we have the following local balance equation for $L_{\mathit{n}}^{(z,m)}$

$${\displaystyle \sum _{\omega =0}^{n_{z,m}}{P}_{\mathrm{gen}}\left({\mathit{n}}_{z,m}^{-\omega }\right)}{\lambda }_{z,\mathrm{cP}}{\displaystyle \sum _{r=\omega +1}^{\infty }{S}_r}=\left(n_{z,m}+1\right)\mu P_{\mathrm{gen}}\left({\mathit{n}}_{z,m}^{+}\right).$$

(14)

Equations (10), (12), and (14) can be satisfied via the following PFS

$$P_{\mathrm{gen}}\left(\mathit{n}\right)=G^{-1}\left({\displaystyle \prod _{z=1}^Z{\displaystyle \prod _{m=1}^{M_{z,\inf }}\frac{{\alpha }_{z,\mathrm{P}}^{n_{z,m}}}{n_{z,m}!} }{\displaystyle \prod _{m=M_{z,\inf }+1}^{M_{z,\inf }+M_{z,\mathrm{fin}}}\left(\begin{array}{l}{N}_{z,m}\\ n_{z,m}\end{array}\right) }{\alpha }_{z,m, \mathrm{idle}}^{n_{z,m}}{\displaystyle \prod _{m=M_{z,\inf }+M_{z,\mathrm{fin}}+1}^{M_z}{P}_{n_{z,m}}^{(z,m)}}}\right),$$

(15)

where $G={\displaystyle \sum _{\mathit{n}\in \mathbf{\Omega }}\left({\displaystyle \prod _{z=1}^Z{\displaystyle \prod _{m=1}^{M_{z,\inf }}\frac{{\alpha }_{z,\mathrm{P}}^{n_{z,m}}}{n_{z,m}!} }{\displaystyle \prod _{m=M_{z,\inf }+1}^{M_{z,\inf }+M_{z,\mathrm{fin}}}\left(\begin{array}{l}{N}_{z,m}\\ n_{z,m}\end{array}\right) }{\alpha }_{z,m, \mathrm{idle}}^{n_{z,m}}{\displaystyle \prod _{m=M_{z,\inf }+M_{z,\mathrm{fin}}+1}^{M_z}{P}_{n_{z,m}}^{(z,m)}}}\right)}$, $\mathbf{\Omega }$ is described as $\mathbf{\Omega }=\left\{\mathit{n}:0\le n_{z,1},\dots ,n_{z,M_z}\le C_z, 0\le {\displaystyle \sum _{z=1}^Z{\displaystyle \sum _{m=1}^{M_z}{n}_{z,m}}}\le T\right\}$, ${\alpha }_{z,m,\mathrm{idle}}=v_{z,m,\mathrm{F}}/\mu$ expresses the offered traffic-load per idle MU, $P_{n_{z,m}}^{(z,m)}={\displaystyle \sum _{\omega =1}^{n_{z,m}}\frac{{\alpha }_{z,\mathrm{cP}}}{n_{z,m}}}{P}_{n_{z,m}-\omega }^{(z,m)}{\widehat{S}}_{\omega -1}$, ${\alpha }_{z,\mathrm{cP}}={\lambda }_{z,\mathrm{cP}}/\mu$ and ${\widehat{S}}_{\omega }={\displaystyle \sum _{r=\omega +1}^{\infty }{S}_r}$ is the complementary batch size distribution.

Having determined the values of $P_{\mathrm{gen}}\left(\mathit{n}\right)$ 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 Section 3.2 and Section 3.3, respectively) and based on the formula

$$B_{z,m,tot}^{TC}=B_c^{TC}+B_{z,m,r}^{TC},$$

(16)

where $B_c^{TC}$ and $B_{z,m,r}^{TC}$ refer to the unavailable computational RUs in the V-BBU and radio RUs in the (z, m) RRH, respectively.

3.2. BF Method for the Computation of Congestion Probabilities

The determination of $B_c^{TC}$ can be based on the values of $P_{\mathrm{gen}}\left(\mathit{n}\right)$ according to the formula

$$B_c^{TC}={\displaystyle \sum _{n\in {\mathbf{\Omega }}_{={\rm T}}}^{}{P}_{\mathrm{gen}}\left(\mathit{n}\right)},$$

(17)

where ${\mathbf{\Omega }}_{={\rm T}}=\left\{\mathit{n}:{\displaystyle \sum _{z=1}^Z{\displaystyle \sum _{m=1}^{M_z}{n}_{z,m}}}=T\right\}$.

On the same hand, the values of $B_{z,m,r}^{TC}$ can be computed (via (15)) via

$$B_{z,m,r}^{TC}={\displaystyle \sum _{n\in {\mathbf{\Omega }}_{{}_{<{\rm T}}}^{z,m,C_z}}{P}_{\mathrm{gen}}\left(\mathit{n}\right)},$$

(18)

where ${\mathbf{\Omega }}_{<{\rm T}}^{z,m,C_z}=\left\{{\mathbf{\Omega }}_{}^{z,m,C_z}\cap {\mathbf{\Omega }}_{<{\rm T}}^{}\right\}, {\mathbf{\Omega }}_{}^{z,m,C_z}=\left\{\mathit{n}:n_{z,m}=C_z\right\}, {\mathbf{\Omega }}_{<{\rm T}}^{}=\left\{\mathit{n}:{\displaystyle \sum _{z=1}^Z{\displaystyle \sum _{m=1}^{M_z}{n}_{z,m}}}<T\right\}$.

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.

3.3. 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 $q_{\mathrm{gen},z,m}(j)$, where j = 1, …, C_z, via

$$q_{\mathrm{gen},z,m}\left(j\right)=\frac{{\alpha }_{z,\mathrm{P}}^j}{j!} q_{\mathrm{gen},z,m}\left(0\right).$$

(19)

(b) For each of the M_z,fin RRHs that accommodate quasi-random traffic and belong to class z (z = 1, …, Z and m = M_z,inf + 1, …, M_z,inf + M_z,fin) determine the occupancy distribution $q_{\mathrm{gen},z,m}(j)$, where j = 1, …, C_z, via

$$q_{\mathrm{gen},z,m}\left(j\right)=\left(\begin{array}{l}{N}_{z,m}\\ j\end{array}\right) {\alpha }_{z,m, \mathrm{idle}}^j{q}_{\mathrm{gen},z,m}\left(0\right).$$

(20)

(c) For each of the M_z,cP RRHs that accommodate compound Poisson traffic and belong to class z (z = 1, …, Z and m = M_z,inf + M_z,fin + 1, …, M_z) determine the occupancy distribution $q_{\mathrm{gen},z,m}(j)$, where j = 1, …, C_z, according to the formula

$$q_{\mathrm{gen},z,m}\left(j\right)={\displaystyle \sum _{l=1}^j\frac{{\alpha }_{z,\mathrm{cP}}}{j}{q}_{\mathrm{gen},z,m}\left(j-l\right)}{\widehat{S}}_{l-1}.$$

(21)

Note that $q_{\mathrm{gen},z,m}\left(0\right)=1$ while $q_{\mathrm{gen},z,m}\left(x\right)=0$ for x > C or x < 0. Furthermore, the normalization of $q_{\mathrm{gen},z,m}(j)$ can be achieved via the constant $G_{\mathrm{gen},z,m}={\displaystyle {\sum }_{j=0}^{C_z}{q}_{\mathrm{gen},z,m}\left(j\right)}$ and in that case the normalized values become $q_{\mathrm{gen},z,m}^{\prime }\left(j\right)=q_{\mathrm{gen},z,m}\left(j\right)/G_{\mathrm{gen},z,m}$.

• Step 2

Compute the aggregated occupancy distribution Q_{gen,(−(z,m))} of all RRHs, excluding the (z, m) RRH, via the formula

$$Q_{\mathrm{gen},(-(z,m))}=q_{\mathrm{gen},1,1}^{\prime }*\dots *q_{\mathrm{gen},1,M_1}^{\prime }*\dots *q_{\mathrm{gen},z,1}^{\prime }*\dots *q_{\mathrm{gen},z,m-1}^{\prime }*q_{\mathrm{gen},z,m+1}^{\prime }*\dots *q_{\mathrm{gen},z,M_z}^{\prime }*\dots *q_{\mathrm{gen},Z,1}^{\prime }*\dots *q_{\mathrm{gen},Z,M_Z}^{\prime },$$

(22)

where

$$q_{\mathrm{gen},z,u}^{\prime }*q_{\mathrm{gen},z,w}^{\prime }=\left\{\begin{array}{l}{q}_{\mathrm{gen},z,u}^{\prime }\left(0\right)\cdot q_{\mathrm{gen},z,w}^{\prime }\left(0\right), {\displaystyle \sum _{x=0}^1{q}_{gen,z,u}^{\prime }\left(x\right)\cdot q_{gen,z,w}^{\prime }\left(1-x\right)},\\ \dots ,{\displaystyle \sum _{x=0}^T{q}_{gen,z,u}^{\prime }\left(x\right)\cdot q_{gen,z,w}^{\prime }\left(T-x\right)}\end{array}\right\}.$$

(23)

Since the resulting distribution may not be normalized, it is recommended to apply the normalization constant $G_{\mathrm{gen},z,u,w}$ in the results of (23).

• Step 3

Compute $B_{z,m,tot}^{TC}$ based on the results of Step 2

$$B_{z,m,tot}^{TC}=B_{z,m,r}^{TC}+B_c^{TC}=G_{\mathrm{gen}}^{-1}\left(q_{\mathrm{gen},z,m}^{\prime }\left(C_z\right){\displaystyle \sum _{r=0}^{T-C_z-1}{Q}_{\mathrm{gen},(-(z,m))}^{}\left(r\right)}+{\displaystyle \sum _{x=0}^T{Q}_{\mathrm{gen},(-(z,m))}^{}\left(x\right)q_{\mathrm{gen},z,m}^{\prime }\left(T-x\right)}\right),$$

(24)

where $q_{\mathrm{gen},z,m}^{\prime }\left(C_z\right)$ 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 $Q_{\mathrm{gen},(-(z,m))}^{}*q_{\mathrm{gen},z,m}^{\prime }$ 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), $B_{z,m}^{CC}$, 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

$$B_{z,m}^{CC}=\frac{{\alpha }_{z,\mathrm{cP}}\widehat{S}-{\overline{n}}_{z,m}}{{\alpha }_{z,\mathrm{cP}}\widehat{S}},$$

(25)

where $\widehat{S}={\displaystyle \sum _{l=1}^{\infty }l{\widehat{S}}_l}$ is the average batch size (in number of calls) and ${\overline{n}}_{z,m}$ refers to the average number of calls accommodated in the (z, m) RRH.

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, then $\widehat{S}=1/(1-\beta )$.

The determination of ${\overline{n}}_{z,m}$ can be based on the formula

$${\overline{n}}_{z,m}=\frac{1}{G_{\mathrm{gen}}}{\displaystyle \sum _{j=1}^{C_z}{y}_{\mathrm{gen},z,m}(j)q_{\mathrm{gen},z,m}^{\prime }(j)}{\displaystyle \sum _{r=0}^{T-j}{Q}_{\mathrm{gen},(-(z,m))}^{}\left(r\right)},$$

(26)

where G_gen is the normalization constant of the convolution operation $Q_{\mathrm{gen},(-(z,m))}^{}*q_{\mathrm{gen},z,m}^{\prime }$ determined via (23) and $y_{\mathrm{gen},z,m}(j)$ expresses the average number of calls that exist in state j of the (z, m) RRH.

The determination of $y_{\mathrm{gen},z,m}(j)$, for j = 1, …, C_z, can be based on the formula

$$y_{\mathrm{gen},z,m}(j)=\frac{{\alpha }_{z,\mathrm{cP}}}{q_{\mathrm{gen},z,m}^{\prime }(j)}{\displaystyle \sum _{l=1}^j{q}_{\mathrm{gen},z,m}^{\prime }(j-l)}{\widehat{S}}_{l-1},$$

(27)

Based on the convolution algorithm, we can also compute the computational RUs’ occupancy distribution according to the formulas

$$\begin{array}{l}{q}_{\mathrm{gen}}^{\prime }\left(0\right)=Q_{{}_{gen,(-(z,m))}}^{}\left(0\right)q_{\mathrm{gen},z,m}^{\prime }\left(0\right)/G_{\mathrm{gen}}, j=0\\ q_{\mathrm{gen}}^{\prime }\left(j\right)={\displaystyle \sum _{r=0}^j{Q}_{{}_{{}_{gen,(-(z,m))}}}^{}\left(r\right)q_{gen,z,m}^{\prime }\left(j-r\right)}/G_{\mathrm{gen}}, j=1,\dots ,T\end{array}$$

(28)

Considering the computational complexity of (23) is in the order of O(T²) while the corresponding complexity of (28) is in the order of O(MT²), where M = (M₁, …, M_Z).

4. 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 Figure 2, Figure 3, Figure 4, Figure 5, Figure 6 and Figure 7.

The C-RAN example presented herein consists of two clusters (Z = 2). The first cluster consists of M₁ = 6 RRHs where the capacity per RRH is C₁ = 10 radio RUs. The second cluster consists of M₂ = 3 RRHs where the capacity per RRH is C₂ = 15 radio RUs. Regarding the computational RUs, we consider that T = 80 RUs. Values of T that are much lower than T = M₁C₁ + M₂C₂ = 105 RUs will result in quite high values for the congestion probabilities due to insufficient computational RUs ($B_c^{TC}$) 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 ${\alpha }_1^{}$ = 4 erl and in the second cluster ${\alpha }_2^{}$ = 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 ${\alpha }_{1,\mathrm{P}}^{}$ = 4 erl (per RRH), RRHs numbered 3 to 4 accommodate quasi-random traffic with $N_{1,3}=N_{1,4}=$ 50 sources and ${\alpha }_{1,3,\mathrm{idle}}={\alpha }_{1,{P}}/N_{1,3}$, ${\alpha }_{1,4,\mathrm{idle}}={\alpha }_{1,\mathrm{P}}/N_{1,4}$ while RRHs numbered 5 to 6 accommodate compound Poisson traffic with ${\alpha }_{1,\mathrm{cP}}$ = 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 ${\alpha }_{2,\mathrm{P}}^{}$ = 7 erl, the second RRH accommodates quasi-random traffic with $N_{2,2}=$ 100 sources and ${\alpha }_{2,2,\mathrm{idle}}={\alpha }_{2,\mathrm{P}}/N_{2,2}$ while the third RRH accommodates compound Poisson traffic with ${\alpha }_{2,\mathrm{cP}}$ = 7 erl and a geometrically batch size distribution with β = 0.2.

In the x-axis of Figure 2, Figure 3, Figure 4, Figure 5, Figure 6 and Figure 7, the values of offered traffic-load increase in steps of 0.2 erl. Therefore, for Figure 2, Figure 3 and Figure 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 Figure 5, Figure 6 and Figure 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 ($B_{1,m,r}^{TC}$) 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, we present simulation and analytical results of the TC probabilities ($B_c^{TC}$) for the g-SC-MC model and the corresponding results ($B_c^{}$) for the SC-MC model, which refer to time congestion due to unavailability of computational RUs. Finally, in Figure 4, we present simulation and analytical results of the CC probabilities ($B_{1,m}^{CC}$) for the g-SC-MC model and the corresponding results ($B_{1,tot}\equiv B_1^{CC}$) for the SC-MC model. According to Figure 2, Figure 3 and Figure 4, we observe that: (i) simulation and analytical results are quite close in the proposed g-SC-MC model, (ii) all congestion probabilities increase as the offered traffic-load increases, (iii) the choice of T = 80 RUs results in an increase of $B_c^{TC}$ and $B_c^{}$ especially for higher values of the offered traffic-load (in the case of T = M₁C₁ + M₂C₂ = 105 RUs, both $B_c^{TC}$ and $B_c^{}$ can be considered negligible), and (iv) the existing SC-MC model cannot capture the behavior of the proposed g-SC-MC model since the former accommodates only random traffic and not bursty or quasi-random traffic. This failure is depicted even for a small value of β. Higher values of β will increase the difference between the results obtained via the two models.

Similar conclusions are obtained in Figure 5, Figure 6 and Figure 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.

5. 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].