I Introduction
The Tactile Internet (TI) is a new service portfolio of the next generation of wireless networks, e.g., the fifthgeneration (5G) wireless networks, where a novel communication paradigm is introduced. For instance, via the TI, touch sensation can be remotely transmitted. One of the most important requirements of TI service is ultra low endtoend (E2E) delay, e.g., E2E delay should be less than one millisecond [1, 2, 3, 4]. These requirements cannot be guaranteed via existing wireless networks such as fourthgeneration (4G) wireless networks [4]. However, 5G platform via its own soft, virtualized, and cloudbased architecture can be leveraged to implement the TI services [1, 2].
For instance, via the concept of cloud radio access network (CRAN) in 5G, spectral efficiency (SE) and energy efficiency (EE) along with cost can be efficiently optimized, where the baseband processing is performed by the baseband units (BBUs) which are connected to remote radio heads (RRHs) via the fronthaul links [5, 6]. Specifically, CRAN reduces energy consumption and cost, and improves throughput in dense environment [7, 8]. Therefore, this RAN architecture is a proper environment for the implementation of the TI services in dense areas.
In 5G, due to the introduction of various services, each to be provided with a high quality of service (QoS) via the virtualization techniques, the concept of slice has been defined for each service in which each slice is a bundle of users with a specific set of QoS requirements [9, 10]. The slice concept adds flexibility to utilize resources which leads to higher SE and EE. However, in this concept, the isolation between slices should be preserved such that the activities of users of one slice do not have harmful effects on QoS of the users of other slices. One of the major issues in the slicing is how to translate the isolation concept to the proper notation for the networks’ procedures. There exists a large body of work for this translation, such as dynamic and static methods [11, 9, 10]. In this paper, we consider the minimum required data rate of each slice as a means of preserving the isolation between slices [11, 12]. Obviously, for this setup due to the complexity of system architecture, diverse transmission parameters such as power, and different QoS requirements, the problem of resource allocation is highly essential which has drawn a lot of attention recently [13, 3, 14, 15, 16]. For instance, in [13], a resource allocation problem for the TI in the LongTerm EvolutionAdvanced (LTEA) is investigated where the average queuing delay and queuing delay violation in one base station (BS) are optimized. Orthogonal frequency division multiple access (OFDMA) and single carrier frequency division multiple access (SCFDMA) are considered for downlink (DL) and uplink (UL), respectively. A crosslayer resource allocation problem for the TI is proposed in [3]
for single BS where the packet error probability, maximum allowable queuing delay violation probability, and packet dropping probability are jointly optimized with the objective to minimize the total transmit power subject to maximum allowable queuing delays. In
[13], queuing delay, packet loss induced by queuing delay violation, packet error, and packet drop caused by channel fading are considered for analyzing the E2E delay of RAN. In [14], the effect of frequency diversity and spatial diversity on the transmission reliability in UL is studied in the TI service where the number of subcarriers, the bandwidth of each subcarrier, and the threshold for each user are optimized for minimizing the total bandwidth to ensure the transmission reliability. In [15], a multicell network based on frequency division multiple access (FDMA) with a fixed delay for backhaul is studied in the TI service. Moreover, queuing delay, delay violation probability, and decoding error probability are considered for analyzing the E2E delay of the TI service [15].In the abovementioned works, a network is considered in which for each user one queue at the BS is assumed. Therefore, by increasing the number of users, a lot of queues are needed at the BS for both UL and DL. However, given that the TI is assumed to be implemented in the 5G framework, it is necessary to consider CRAN architecture. There exists a set of RRHs in the highly dense network which are connected to BBU center via fronthaul links. Furthermore, the results in the above works generally ignore the fronthaul delay. However, due to the importance of delay in the TI, it is crucial to consider queuing delay in fronthaul as well, otherwise, the resulting allocation of the resources may not practically fulfill the requirement of the TI.
To address the mentioned issues, we consider a CRAN architecture serving a set of tactile users. The contributions of this paper are as follows, many of which have been considered for the first time in the TI:

We propose a CRAN scenario in ultra dense environment in 5G platform. This will impose new constraints to the system as far as the number of queues is concerned. For the considered CRAN architecture, we propose a practical queuing model for sequential queues in the TI that can be implemented in realistic networks. Moreover, we consider slicing for the TI service in our work.

Given that TI services are extremely delay sensitive, there is a possibility that due to high channel fading, the delay requirements is not met for some tactile users, i.e., the resource allocation problem is not feasible. To tackle this issue and reach an efficient solution, we propose an admission control (AC) where a set of users who has the worst condition to reach a feasible solution is not admitted.

In contrast to [3, 14, 13] where the fronthaul delay is ignored, we take this delay into consideration. Moreover, we consider dynamic adjustment of the access and fronthaul delays based on channel state information (CSI) for each pair of users instead of fixed maximum delay values per each transmission part of our setup and show that it can significantly reduce the required total transmit power.
The rest of this paper is as follows. In Section II, the system model is described. In Section III, we formulate the optimization problem. Numerical results and simulation are presented in Section IV. Finally, Section V concludes the paper.
Ii System Model
We consider a CRAN network where all RRHs are connected to the BBU via fronthaul links. In this region, there exist several pairs of tactile users where each user aims to send its information to its paired tactile user via the closest RRH through the UL transmission link. Then, RRH sends the received data to the BBU via the fronthaul link. The BBU processes all the received data and then sends the data to the corresponding RRH of its paired tactile user. Finally, this RRH transmits the relevant message to the paired tactile user via the DL transmission link. Assume each RRH has only one queue for UL transmission and all the data of tactile users is stored in this queue. In addition, we consider only one queue in the BBU to store all received data from RRHs. In DL, we assume that each RRH has a queue for each user for sending data to the paired users.
As shown in Fig. 1, in the considered system model, we have RRHs, slices, and pairs of tactile users. Slice contains tactile users and the total number of tactile users in our system model is equal to pairs of users. The terms of access link and fronthaul link often are used to express the RRHuser connection and RRHBBU connection, respectively. In order to reduce the cost of cabling, wireless fronthaul is used instead of fiber fronthaul [17, 5]. We assume that the fronthaul links are provided via wireless channels in an ultradense environment and that there exist two sets of subcarriers and for access and fronthaul links, respectively. Moreover, we define for simplicity. We consider a twophase transmission; in the first phase, all tactile users send their data to the corresponding RRH and simultaneously all RRHs send their buffered data to the BBU via fronthaul links. In the second phase, all RRHs send data to the corresponding tactile users, and simultaneously, BBU sends the buffered data to all RRHs via fronthaul links. These two phases do not perform at the same frequency. Therefore, the proposed system model is based on the frequency division duplex (FDD) transmission mode in which each RRH can transmit and receive simultaneously in different frequencies. In order to isolate slices, a minimum required data rate for each slice must be reserved [18, 11, 19]. By considering the above definitions, we can now proceed to review the system parameters.
Remark 1.
To estimate the CSI for DL transmission, pilot signals are transmitted via RRHs to all users. Then, each user sends the channel estimation to RRHs via feedback channels. To estimate the CSI for UL, pilot signals are transmitted via users to RRHs, and then, the channel estimations are sent to the users. For the CSI estimation, one of the proposed approaches in
[20, 21, 22] can be applied.Iia Access Links Parameters
We introduce a binary variable
which is set to 1 if subcarrier is assigned to user in slice at RRH , i.e.,Since we deploy OFDMA in this setup, each subcarrier can be allocated to at most one user. Therefore, we have the following constraint:
Here, for all , , , , and , the achievable rate for user on subcarrier at RRH can be calculated as [23, 15]
(1) 
where , in which , , and represent the transmit power, channel power gain from RRH to user on subcarrier in slice , and noise power, respectively. Also, is the intercell interference which is equal to . Also , , and represent time unit, the bandwidth of subcarrier , and the inverse of GaussianQ function, respectively. Moreover, is defined as . Moreover, in each time unit (short blocklength regime), the total number of transmitted bits of user at RRH in slice over subcarrier is . From (1), the error probability () can be calculated as follows:
(2) 
Since the reliability is important for the TI services, we consider the following constraint:
where is error probability threshold. Given that the Qfunction does not have a closedform, we deploy approximation as follows [23, 15]:
(3) 
where and .
The total achievable rate in the access links at RRH is as follows:
(4) 
Due to the power limitation of each RRH in DL transmission, we have the following constraint:
Moreover, due to the power limitation of each user, we have
IiB Fronthaul Links Parameters
We introduce a binary variable denoting that subcarrier is assigned to RRH which is defined by
Assuming that OFDMA is also deployed for the fronthaul links, again each subcarrier can be allocated to at most one RRH, and hence, we have the following constraint:
The achievable rate for each RRH on subcarrier is calculated as follows [23, 15]:
(5) 
where is defined as . is the bandwidth of subcarrier and is defined as . In addition, in each time unit, the total number of transmitted bits is . Similar to the previous subsection (IIA) and based on (5), the error probability can be calculated as follows:
(6) 
Given that the Qfunction does not have a closedform, we deploy approximation similar to the previous subsection (IIA).
The total achievable rate in the BBU is obtained as follows:
(7) 
Due to the power limitation of each RRH in UL transmission, we have
Moreover, due to the power limitation of the BBU, we have
IiC Queuing Delay Model
The total delay of this architecture consists of three components: delay resulting from UL queues at RRHs, BBU queue, and DL queues at RRHs, as shown in Fig. 2. Due to delay constraint in the TI service, we have
where , , , and are the delays of UL queues at RRHs, BBU queue, DL queues at RRH, and the total delay, respectively.
IiC1 UL Queuing Delay
The aggregation of receiving bits from several nodes can be modeled as a Poisson process [24, 3]. The effective bandwidth for a Poisson arrival process in RRH is defined as [25, 24, 3]
where is the statistical QoS exponent of the RRH. A larger indicates a more stringent QoS and a smaller implies a looser QoS requirement. is the number of bits arrived at RRH queue defined as . The probability of queuing delay violation for RRH can be approximated as
(8) 
for all where is the RRH delay, is the maximum delay, and is the nonempty buffer probability. Equation (8) can be simplified to
Therefore, we have
IiC2 BBU Queuing Delay
We consider a queue for all RRHs at the BBU for processing data. Therefore, the formulas in the previous section can also be used for this section. The effective bandwidth for each queue in BBU is where is the statistical QoS exponent in the BBU and is the number of bits arrived at the queue in the BBU which is defined as The probability of queuing delay violation at the BBU can be approximated as
(9) 
where is the nonempty buffer probability. Equation (9) can be simplified to
Therefore, we have
IiC3 DL Queuing Delay
The effective bandwidth for each user in RRH is defined as where is the statistical QoS exponent of the user in RRH and is the number of bits arrived at user queue in RRH which is defined as . The probability of queuing delay violation for user can be approximated as
(10) 
where is the user delay in RRH and is the nonempty buffer probability. Equation (10) can simplified to
Therefore, we have
In order to avoid bit dropping, the output rate of queues must be greater than the input rate of queues. Therefore, we have two following constraints:
Iii Optimization Problem Formulation
In this section, our aim is to allocate resources to minimize the overall power consumption in our setup by considering a bounded delay constraint to satisfy the E2E delay requirements. Based on the mentioned constraints C1C14, the optimization problem can be written as
(11)  
The optimization variables in (11) are subcarrier allocation, power allocation, and delay adjustment for different users in the access and fronthaul as well as in both UL and DL where , , , and
are the transmit power, the access subcarrier allocation, fronthaul subcarrier allocation, and delay vector for users, respectively. The rate constraint C15 is used to isolate the network slices.
In problem (11), the rate is a nonconvex function, which leads to the nonconvexity of the problem. In addition, this problem contains both discrete and continuous variables, which makes the problem more challenging. Therefore, we resort to an alternate method to propose an efficient iterative algorithm [26, 27] with three subproblems, namely, subcarrier allocation subproblem, power allocation subproblem, and delay adjustment subproblem which will be explained in the followings.
Iv An Efficient Iterative Algorithm
Due to the complex nature of (11), and specially having C9C15, obtaining feasible initial values for problem (11) is not trivial. Therefore, to find a feasible point for Problem (11), we propose to solve the following optimization problem instead of Problem (11):
(12)  
where is an elastic variable and if the original problem (11) is feasible, the optimal value is . Moreover, is a large coefficient, i.e., . By using this variable, in this section, we propose an AC method to reject users who make the problem infeasible based on a defined criterion and guarantee the QoS of other users. (IV) is also nonconvex and we utilize an iterative algorithm based on the difference of two convex (DC) approximation. To solve (IV), we set all the initial values except to zero and set to a large value which is a feasible point of (IV). After deriving the solution of (IV), we check out if the constraints hold or not. If these constraints hold and , the derived solution of (IV) is an initial value of (11); otherwise, we run the admission control to remove a user and then repeat this procedure.
As mentioned earlier, to solve (IV), we deploy an iterative algorithm that divides the problem into four subproblems and solve them alternately [26, 27].
This procedure is presented in Algorithm.1. Let be the iteration number and , , and be the initial values.
In each iteration, we solve each subproblem by considering the optimization parameters of other subproblems as fixed values derived in the previous steps.
The iteration stops when the error in Step 6 is less than a predetermined threshold, i.e., , or the number of iterations exceeds a predetermined value, i.e., .
The solution of the last iteration is then declared as the solution of (IV). The flowchart of Algorithm.1 is shown in Fig. 3, which is explained following in more detail.
Proposition 1: The presented iterative algorithm which is described in Algorithm 1 converges.
Proof.
See Appendix A. ∎
Iva Subcarrier Allocation SubProblem
With assuming fixed value , and , the subcarrier allocation subproblem is written as follows:
(13) 
While (IVA) has less computational complexity than (IV), it suffers from nonconvexity due to the interference in the rate functions. In addition, this problem contains discrete variables. We apply time sharing method and relax discrete variables as and . To solve this problem, we use DC approximation to transform the problem into a convex form. The subcarrier allocation subproblem is transformed into the following form (See Appendix C):
(14)  
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