I Introduction
MissionCritical InternetofThings (MCIoT) will be widely deployed in future wireless networks for remote health monitoring, haptic interaction, and factory automation [2, 3]. Achieving ultrareliable and lowlatency communications (URLLC) (e.g., packet loss probability and ms EndtoEnd (E2E) delay) for MCIoT has been considered as one of the major goals in 5th Generation (5G) cellular networks [4]. Most existing technologies mainly focus on one of the seven layers of the open systems interconnection model, and cannot guarantee the E2E delay [5]. To satisfy the requirements of MCIoT, we need to redesign the physicallayer resource management, the linklayer scheduling policy, and the networklayer user association from a crosslayer perspective.
One of the major differences between MCIoT and enhanced Mobile BroadBand (eMBB) services lies in the sizes of packets. With high data rate, the packet size in eMBB services is relatively large, e.g., thousands of bytes in each packet. However, the packets generated by MCIoT are very small, e.g., or bytes in each packet [6]. To achieve low latency for short packet transmissions, a short frame structure is adopted in 5G New Radio (NR) [7]. When transmitting a short packet in a short frame, the blocklength of channel codes is very limited. As a result, the decoding error probability cannot be ignored when analyzing reliability [8].
On the other hand, the computing ability at each MCIoT device is limited. To reduce processing delay, MCIoT devices will offload some of the packets to the Mobile Edge Computing (MEC) systems for processing [9, 10]. Considering that MCIoT services will coexist with eMBB services, a short packet arriving at the MEC after long packets has to wait in a queue if the packets are processed with a FirstComeFirstServe (FCFS) order. To avoid long queueing delay, other scheduling orders at MEC systems should be considered.
Furthermore, the reliability and delay not only depend on the resource management and scheduling order but also depend on the traffic load. Considering that the radio resources and computing capacity at each Access Point (AP) are limited, the user association and offloading policy should be optimized to balance traffic loads. The problems for optimizing user association and offloading policy are NPhard in general [11]. Lowcomplexity solutions to the NPhard problems are in urgent need for MCIoT since complicated searching algorithms will lead to long computation delay [12].
Ia Related Works
To transmit short packets with low latency, the blocklength of channel codes is short. In the short blocklength regime, Shannon’s capacity is not applicable since it cannot characterize the decoding error probability [8]. Recently, the maximal achievable rate with given decoding error probability in the short blocklength regime was obtained in multiantenna quasistatic channel [13]. How to design transmission schemes and resource allocation in the short blocklength regime has been studied in existing literature [14, 15, 16, 17, 18, 19, 20]. The throughput achieved in cognitive radio channels and relay systems was studied in [14] and [15, 16], respectively. The studies in [17] optimized the scheduling of short packets to maximize energy efficiency. The authors of [18] optimized packet losses caused by decoding errors, queueing delay violation, and packet dropping over deep fading channel subject to the ultrahigh reliability. Considering that the feedback of Channel State Information (CSI) leads to extra delay, the studies in [19] jointly optimized Uplink (UL) and Downlink (DL) resource configurations without CSI at the transmitters. More recently, how to optimize resource allocation among multiple users with different packet arrival processes was studied in [20].
Scheduling policies in computing systems have significant impacts on the QualityofService (QoS) of MCIoT. A nearoptimal policy to minimize the average latency of short packet is the Shortest Remaining Processing Time (SRPT) first scheduler. Such a scheduler is hard to implement in practice since the remaining processing time is not available at the server, and it requires too many priority levels[21]. To reduce the latency of short packets without introducing priority levels, the ProcessorSharing (PS) server is a possible solution, where the total service rate is equally allocated to all the packets in the server[22]. Although the distribution of latency was derived in the large delay regime in the PS server [23], the latency experienced by short packets remains unclear. To derive the delay bound violation probability of URLLC services, martingalesbased analysis, effective capacity, and network calculus were used in [24], [15], and [25], respectively. But all the results were obtained in the FCFS servers. Note that it is very challenging to derive the closedform expression of the distribution of delay, how to formulate the constraints on delay and reliability of MCIoT is still unclear.
Promising network architectures for MCIoT were studied in [26, 27, 28, 29, 30]. A comprehensive overview on MCIoT of industrial scenarios was carried out in [26], where the issues related to architecture design were discussed, such as extensibility, scalability, and modularity. To reduce the routing delay, an adaptive transmission architecture with softwaredefined networks and edge computing was proposed in [27]. Considering that energy consumption of IoT devices is an important issue, an energyaware realtime routing scheme was proposed in [27] to reduce energy consumption and E2E delay in largescale IoT networks. More recently, a fog computing architecture was proposed for 5G tactile IoT [29], where the qualityofexperienceaware model was formulated. By combining stochastic geometry and queueing theory, different QoS requirements were analyzed in ultradense networks [30]. The studies in [26, 27, 28, 29, 30] shed light upon network architecture designs for MCIoT, but decoding errors in the physicallayer were not considered.
Computing offloading has been exhaustively studied in various MEC systems, such as wireless powered MEC [31], ultradense IoT networks [32, 33], and fiberwireless networks [34, 35]. Although these works did not consider MCIoT, they developed useful methodologies for optimizing computing offloading in MEC systems. How to improve the QoS in MEC systems by optimizing task offloading has been addressed in [36, 37, 38, 39]. In [36], the average delay was minimized by optimizing task offloading/scheduling in MEC systems. Considering that the average delay is not suitable for URLLC services, the authors of [37] optimized task offloading and resource allocation under the constraint on a queue length violation probability. The offloading schemes for URLLC in MEC systems were optimized in [38], where a weighted sum of E2E delay and the offloading failure probability was minimized in a singleuser scenario. How to analyze latency in largescale MEC networks was studied in [39], where the average communication and computing latencies were derived.
Most of the existing studies on resource management in MEC systems only analyzed UL transmission and processing delay, and assumed DL transmission can be finished with high transmit power at the APs [36, 37, 39, 38]. Besides, they did not take the decoding errors in the short blocklength regime into account, which is crucial for MCIoT. Although the block error probability was considered in the optimization problem in [38], the radio resource management was not optimized and the packet losses due to the delay bound violation were not considered.
IB Our Contributions
To the best of the authors’ knowledge, there is no resource allocation scheme or offloading scheme that can achieve the target E2E delay and overall packet loss probability for MCIoT in MEC systems. Moreover, how to design scheduling policy and whether the FCFS server is suitable for MCIoT were not addressed. In order to achieve ultrahigh reliability and ultralow E2E delay for MCIoT in MEC systems, the following three issues will be addressed in this work: 1) How to design scheduling and queueing policies in MEC servers and local servers to achieve ultrahigh reliability and ultralow E2E delay for short packets? 2) How to characterize the statistical QoS of short packets when there are both short and long packets in MEC systems? 3) How to improve the fundamental tradeoff between delay and reliability by optimizing user association, packets offloading, and bandwidth allocation in MEC systems? Our major contributions are summarized as follows,

We establish a framework for minimizing overall packet loss probability subject to E2E delay requirement in MEC systems, where processing delay and UL and DL transmission delays are taken into account. PS servers are equipped at MEC systems, where the total service rate is equally allocated to all the packets in each server, and every packet receives service at all times. As such, short packets can bypass long packets and achieve low latency.

We derive a closedform approximation of the Complementary Cumulative Distribution Function (CCDF) of the latency experienced by short packets in the PS server. The approximation is accurate when the number of Central Processing Unit (CPU) cycles needed to process a long packet is much larger than that needed to process a short packet.

We optimize the user association scheme, packet offloading rates, and bandwidth allocation for short packets in MEC systems. We propose an algorithm to solve a mixed integer problem and analyze the convergence and the complexity of it. Our analysis shows that the difference between the obtained solution and the global optimal solution only results from searching numbers of subcarriers in a continuous domain.
Furthermore, our simulation results validate the accuracy of the closedform approximation. Numerical results indicate that when increasing the number of antennas at the AP with a fixed processing capacity, communication is the bottleneck of the reliability when the number of antennas is small, and computing is the bottleneck when the number of antennas is large. Only in a very small region (e.g., from to antennas), the packet loss probability in communications is comparable to the processing delay violation probability. This implies our algorithm converges to a near optimal solution in most of the cases.
The rest of the paper is organized as follows. Section II describes the system model. Section III analyzes delay and reliability. Section IV studies how to optimize the association scheme, packet offloading rates and bandwidth allocation. Section V extends the algorithm into more general scenarios. Section VI provides simulation and numerical results. Section VII concludes the paper.
Ii System Model
Iia The MEC System
As illustrated in Fig. 1, we consider a MEC system with singleantenna devices and multiantenna APs, where the data collected by each MCIoT device and the computation intensive tasks generated by eMBB services can be offloaded to one of the APs for processing. To provide better services in radio access networks and to avoid high backhaul overhead, the partially centralized Control Plane (CP) in [12] is considered. The whole network is decomposed into multiple clusters, each of which includes closely located APs and one CP that optimizes user association scheme, packet offloading rates, and bandwidth allocation for devices in the cluster. To achieve ultralow latency and ultrahigh reliability, strong cochannel interference should be avoided. To this end, orthogonal channels are allocated to different devices in each cluster, and the frequency reuse factor is less than one such that adjacent clusters use different bandwidth. In this work, we focus on one cluster of APs, and our solution is applicable for low mobility scenarios like factory automation and VR/AR applications. For high mobility scenarios, where devices travel across clusters frequently, how to reserve resources in different clusters deserves further study.
IiB Traffic Models
In vehicle networks and factory automation, there are two kinds of packets, i.e., periodic packets with deterministic arrivals and sporadic packets that are driven by some random events [40, 41]. Since analyzing the delay and the reliability of random packet arrival processes is more challenging than deterministic arrivals, we focus on sporadic packets in this work. The experiment in [42] indicates that the packet arrival processes of MCIoT are very bursty, i.e., there is a high traffic state and a low traffic state. For each of the traffic states, the arrival process can be modeled as a Bernoulli process. According to 5G NR, time is discretized into slots with duration . In each slot, a device either has a packet to transmit or stays silent. We assume the traffic state is obtained with the traffic state classification methods in [42]. When a device switches between the high and low traffic states, we only need to change the average arrival rate in our analysis. The aggregation of multiple independent Bernoulli processes at a MEC server can be accurately approximated by a Poisson process [43].
IiB1 Short packets
The data collected by each MCIoT device is contained in short packets for transmission and processing. A packet with the following three features is considered as “short”,

The number of bits in the packet is small. According to [6], the packet size in MCIoT services is around or bytes. In contrast, the packets in eMBB services may include hundreds or thousands of bytes, such as video streaming.

The blocklength of channel codes of the packet is short. For example, if quadrature phaseshift keying is used in modulation, the number of symbols required by a packet with bytes ( bits) is , which is the blocklength of the packet. To achieve lowlatency, the blocklength of channel codes is short in MCIoT [44].

The number of CPU cycles required to process the packet is small. The number of CPU cycles required to process the packet depends on the number of bits in the packet and the processing algorithms. Since a packet in MCIoT services only contains a few bits, the number of required CPU cycles is small.
Let be the number of CPU cycles required to process a short packet. The service rate of the local server at the th device and that of the th MEC server are denoted as and , respectively.
IiB2 Long packets
There are some devices requesting eMBB services in each cluster. The tasks generated by the eMBB services are packetized into long packets. How to optimize resource allocation and computing offloading for eMBB services has been studied in the existing literature, such as [45, 46, 47]. In our work, we focus on MCIoT services, where eMBB services are considered as background services.
The sum of average packet arrival rates of eMBB services at the th AP is denoted as (packets/slot). The number of CPU cycles required to process a long packet is denoted as
, which is a random variable with mean value
. In this work, we do not specify the distribution of . The only assumption on is that , which is reasonable since the packet size of eMBB services is much larger than that of MCIoT services and the algorithm for processing long packets (e.g., high definition pictures) is more complex than that for processing short packets (e.g., the location and velocity of a device). For example, may follow the Pareto distribution with a heavy tail [22], i.e.,(1) 
where , , and is the minimum of . As shown in [22] and the references therein, Pareto distributions have been observed in different application scenarios, such as the service time of UNIX jobs.
IiC User Association and Packet Offloading
IiC1 User association
Each device can associate with one of the APs. We leverage indicators , to represent the user association scheme,
(2) 
The association scheme of the th device is denoted as .
IiC2 Packet offloading
When a short packet is generated by the th device, the device either processes the packet with the local server or uploads the packet to an AP. The average packet rate from the th device to the th MEC server is denoted as (packets/slot), where and . means that the packets are processed at the local server. If , then , where is the average packet arrival rate of the th device. If for all , then .
IiD Queueing Models and Scheduling Policies
As shown in Fig. 2(a), to guarantee the QoS requirements of different devices, packets from different devices are waiting in different queues at a MEC server, and are served according to the FCFS order. In the th MEC server, the service rate allocated to the th device is denoted as . Once the computing resource is allocated to one device, it cannot be shared with the other devices. Such a scheduling scheme is widely used, but is not optimal in terms of minimizing the delay.
The second server in Fig. 2(b) is referred to as a statistical multiplexing FCFS server [22]. Due to statistical multiplexing gain, the average delay in the second server is much shorter than the first server when . Furthermore, as proved in [19], if the sizes of all the packets are identical, to achieve the same delay bound and delay bound violation probability, the required service rate in the statistical multiplexing server is less than the sum of the service rates in the individual server. However, when the distribution of the number of CPU cycles required to process the packets has a heavytail, some short packets arriving at the MEC server after a long packet need to wait for a long time. As a result, the delay requirement of MCIoT services can hardly be satisfied.
The key to low latency is letting short packets bypass queued long packets. One possible solution is the PS server. As shown in Fig. 2(c), every packet in the server receives service at all times. When there are packets in the th MEC server, each packet is processed at rate .
Remark 1.
In practice, a server can be implemented in a timesharing way, i.e., the service time in each slot is equally allocated to all the packets in the server. In this way, the processing delay of packets in the server is the same as that in the ideal PS server [22]. It’s worth noting that there are some other possible scheduling policies if the server is aware of the diverse QoS requirements of different packets. In this work, we do not assume the computing system is aware of the types of packets in the communication systems. We will study more sophisticated scheduling policies for different types of packets in our future work.
We consider the scheduling policies at MEC servers and local servers in Fig. 3, where PS servers are deployed at APs and FCFS servers are deployed at devices. To avoid queueing delay, the UL and DL transmission durations of a short packet equal to one slot.^{1}^{1}1Since the packet arrival rate of each device is less than one packet per slot, there is no queue before UL and DL transmissions. On the other hand, if the required CPU cycles to process different packets are identical, which is the case in the local server of each device, the FCFS server outperforms the PS server [22]. Therefore, FCFS servers are equipped at MCIoT devices.
Iii Analysis of Delay and Reliability
In this section, we study how to characterize E2E delay and overall packet loss probability. We first derive the CCDF of the processing delay of short packets in the PS server. Then, we show how to characterize the transmission delay and decoding error probability of short packets.
Iiia Processing Delay and Delay Violation Probability
A short packet can be processed either at the device or at the AP. The packet arrival process at the local server of the th device is a Bernoulli process with average arrival rate . Denote the service time of a packet at the th local server as
. With the constant service rate at each local server, the queueing model is a Geo/D/1/FCFS model, where “Geo” means that the interarrival time between packets is geometric distributed, and “D” represents deterministic service processes. The CCDF of queueing delay in Geo/D/1/FCFS model has been obtained in
[48]. If , then(3) 
If , the expression of can be found in [48]. Due to the lowlatency requirement, we are interested in the case .
Each AP may serve multiple devices. The aggregation of multiple Bernoulli processes can be modeled as a Poisson process [39]. Thus, the MEC server can be characterized by an M/G/1/PS model, where “M” means the packet arrival process is Poisson process and “G” means that the number of CPU cycles required to process the packets can follow any distributions. To derive a closedform CCDF of the processing delay of short packets in the M/G/1/PS model, we introduce an accurate approximation. Since the short packets are much smaller than the long packets, i.e., , the processing delay of a short packet is much shorter than a long packet. As a result, the number of long packets in the server is nearly constant from the arrival to the departure of a short packet. When a short packet arrives at the server, the number of packets in the server is denoted as . Considering that the value of does not change significantly during the short service time of a short packet, then the service rate allocated to the short packet can be approximated by if . In this case, the processing delay of the short packet is approximated by
(4) 
According to [22], the distribution of can be expressed as follows,
(5) 
which is the distribution of the number of packets in the M/G/1/PS model. The workload of the server is
(6) 
From (4) and (5), we can further obtain that
where . Based on the above expression, the CCDF of the processing delay of short packets in the PS server can be derived as follows,
(7) 
The approximation in (7) is accurate when . We will validate the accuracy of the approximation via simulation.
The processing delay of short packets in the th MEC server can be characterized by a delay bound and a delay bound violation probability, and . From the CCDF in (7), the relationship between and can be expressed as follows,
(8) 
IiiB Transmission Delay and Decoding Error Probability
If a packet is processed at the MEC server, the device first uploads the packet to the AP. After the MEC server finishes the processing, the result is sent back to the device. We introduce a superscript of parameters , where . If , is a parameter in UL transmissions. Otherwise, it is a parameter in DL transmissions. We consider Orthogonal Frequency Division Multiple Access (OFDMA) systems, which will be used to support MCIoT services in 5G NR [7]. The total bandwidth is equally allocated to subcarriers, each with a bandwidth of . Denote the number of subcarriers allocated to the th device for UL and DL transmissions as , , respectively. Since the packet size is small, it is reasonable to assume that is smaller than the coherence bandwidth. As mentioned in the previous section, to avoid queueing delay before UL and DL transmissions, the transmission duration of each packet is one slot, which is smaller than channel coherence time. Thus, each packet is transmitted over a flat fading quasistatic channel. Considering that feedback from receivers to transmitters may cause large overhead and extra delay, CSI is not available at the transmitters. According to [13], the achievable rate in the short blocklength regime over quasistatic flat fading channel can be accurately approximated by
(9) 
where is the largescale channel gain from the th device to the th AP, is the UL or DL smallscale channel fading between the th device and the th AP, is the UL or DL transmit power of one antenna on each subcarrier, is the singleside noise spectral density, is the inverse of Qfunction, is the decoding error probability, and .
Remark 2.
Due to the following two reasons, we only consider the flat fading channel, and do not consider frequencyselective channels. First, the maximal achievable rate over a frequencyselective channel in the short blocklength regime has not been derived in existing studies. Although the upper and lower bounds were obtained in [49], there is no closedform expression. Second, as shown in [19], when the number of antennas is large (e.g., antennas), frequency diversity is not necessary for URLLC. Therefore, we focus on the multiantenna flat fading channel.
Let be the number of bits in a short packet of the th device. When sending a packet of bits within one slot, the decoding error probability can be obtained from (9) by setting
. According to the law of total probability, the packet loss probability due to decoding errors can be expressed as follows
[13],(10) 
where the distribution of smallscale channel gain depends on the number of antennas at each AP, which is denoted as . To compute (10), we need to calculate the integral for a given distribution of . For Rayleigh fading, we can apply the closedform result in [50].
IiiC E2E Delay and Overall Packet Loss Probability
IiiC1 Delay and Reliability at Local Servers
If a packet is processed at the device, then the E2E delay is equal to the sum of the service time and the queueing delay at the local server of the device. Given the E2E delay requirement , the delay violation probability at local servers, , can be obtained by substituting into (3). When ,
(11) 
IiiC2 Delay and Reliability at the MEC server
If a packet is processed at a MEC server, then the UL and DL transmission delays and the processing delay in the server should be considered. The E2E delay can be satisfied under the following constraint,
(12) 
where two slots are occupied by the UL and DL transmissions.
Due to decoding errors and processing delay violation, the overall packet loss probability can be expressed as follows,
(13) 
where the approximation is very accurate since , , and are extremely small in MCIoT. Upon substituting (6) and (12) into (8), we can get the expression of , i.e.,
(14) 
Remark 3.
The factors that lead to packet losses or errors depend on network architectures [51]. For the considered MEC system, reliability only includes queueing delay violations in computing systems and packet losses in radio access networks.
Iv CrossLayer Optimization
In this section, we study how to optimize the association scheme, packet offloading rates, and UL and DL bandwidth allocation to minimize the overall packet loss probability subject to the E2E delay requirement.
Iva Problem Formulation
Note that the packet loss probabilities at the local server and the MEC can be different, the reliability is determined by the worse one. Thus, the packet loss probability of the th device is characterized by
(15) 
The problem that minimizes the maximal packet loss probability experienced by the devices can be formulated as follows,
(16)  
s.t.  (16a)  
(16b)  
(16c)  
(16d)  
(16e) 
where , , is the maximum number of subcarriers that can be allocated to a device without exceeding the coherence bandwidth, and the expressions of , , and can be found in (10), (11), and (14), respectively. Constraint (16a) guarantees that a device can only associate with one AP. If , then all the packets are processed at the local server.
With constraint (16a), each device cannot be served by two or more APs. Constraint (16b) ensures that the packet offloading rate is zero if the th device is not served by the th AP. Constraint (16c) guarantees that the sum of the packet offloading rates at the APs and the packet arrival rate at the local server is equal to the total packet arrival rate of a device. The constraint on the maximal number of subcarriers of the system is given in (16d), where the UL and DL bandwidth allocated to each device does not exceed the coherence bandwidth. When constraint (16e) is satisfied, and are smaller than , and hence the local and MEC servers are stable. By minimizing the objective function, we can check whether constraint (16e) can be satisfied or not. If it cannot be satisfied, the problem is infeasible.
Since CSI is not available at the transmitters, the transmit power on each subcarrier is fixed. In UL transmission, the maximal transmit power of a device, , is equally allocated to subcarriers, . In DL transmission, the maximal transmit power of an AP, , is equally allocated to antennas. Considering that the number of subcarriers for DL transmission can be up to , to satisfy maximal transmit power constraint, the transmit power on each subcarrier is fixed as . Thus, we have .
Problem (16) is a mixed integer optimization problem, which is nonconvex. In typical scenarios, the throughput of eMBB services is much higher than the throughput of MCIoT services, and hence the number of CPU cycles required to process long packets are much larger than that required to process short packets. In the rest part of this section, we first consider the scenario that , and then extend our algorithm into more general scenarios.
IvB Solution in the Typical Scenario
IvB1 Simplified Optimization Problem
According to (16c), . Thus, we have
(17) 
When , the equality in (17) holds, and in (14) is a constant that does not depend on packet offloading rates. Moreover, the packet loss probabilities due to decoding errors in UL and DL transmissions, and , do not change with packet offloading rates. Thus, the second term in does not depend on packet offloading rates. By setting , all the packets are offloaded to the MEC servers. Then, and
(18) 
With (18), problem (16) can be simplified as follows,
(19)  
s.t. 
IvB2 Packet Loss Balance Algorithm
Denote the optimal solution and the minimal packet loss probability of problem (19) as and , respectively. In the following, we propose a binary search algorithm to find the optimal solution. The basic idea of the algorithm is to keep below a threshold , and search the minimal in the regime , where is an initial upper bound of the overall packet loss probability. We refer to the algorithm as the Packet Loss Balance (PLB) Algorithm.
For a given threshold of overall packet loss probability , we search for the optimal association scheme and subcarrier allocation that minimize the total number of subcarriers. If the minimum number of subcarriers exceeds , then . Otherwise, .
The problem that minimizes the total number of subcarriers can be expressed as follows,
(20)  
s.t.  (20a)  
where constraint (16e) is removed since . The above problem can be decoupled into problems since the association schemes and bandwidth allocation of different devices are independent.
Given that the th device is served by the th MEC server, , the required number of subcarriers can be found from the following problem,^{2}^{2}2By solving problem (21) with different , we can obtain the optimal user association scheme and related bandwidth allocation that minimize .
(21)  
s.t.  (21a)  
To solve the inter programming problem, we first relax and as continuous variables, and find the optimal subcarrier allocation. Then, we discretize the number of subcarriers used in UL and DL transmissions. Note that only the discretization step will cause performance loss, which is minor as shown in [52].
Property 1.
The packet loss probabilities and in (10) are convex in .
Proof.
See proof in Appendix A. ∎
Further considering that in (14) does not change with , constraint (21a) is convex. Therefore, problem (21) is a convex problem, and can be solved by techniques such as the interiorpoint method [53]. The algorithm for solving problem (20) is provided in Table I, where is the minimum integer that is equal to or higher than .
Note that problem (20) could be infeasible if is too small. In this case, the minimal overall packet loss probability is higher than . Based on the algorithm in Table I, the PLB algorithm for solving problem (19) is shown in Table II.
IvB3 Convergence of the PLB Algorithm
To prove that the PLB algorithm converges to the minimal packet loss probability of problem (19), we first prove the following proposition,
Proposition 1.
The minimal packet loss probability lies in the region , .
Proof.
See proof in Appendix B. ∎
According to Proposition 1, the minimal packet loss probability lies in the region . After steps of searching, the gap between and the output of the PLB algorithm, , is smaller than . In addition, with the binary search algorithm (i.e. from Line 1 to Line 11 in Table II), the range of decreases according to the following expression, . When is large enough, approaches to .
The above proof holds when the performance loss caused by the discretization step in Line 7 of Table I is negligible. Since the discretization step inevitably causes some performance loss, the related association scheme and bandwidth allocation are near optimal.
IvB4 Complexity of the PLB Algorithm
With the PLB algorithm, we need to solve problem (20) around times. Problem (20) is decoupled into singledevice problem in (21). With the algorithm in Table I, the convex optimization problem in (21) is solved times for devices with possible APs. The complexity of solving the convex optimization problem is denoted as , which is not high. Therefore, the complexity of the PLB algorithm is . Considering that a device will not associate with an AP that is very far from it, will not be very large. For example, a device can only be connected to one of the three or four APs with the highest largescale channel gains. As a result, the complexity of the PLB algorithm increases linearly with the number of devices.
V Solution in General Scenarios
To solve problem (16), we extend the PLB algorithm into general scenarios without the assumption .
Va Extended PLB Algorithm
Although problem (16) cannot be simplified as problem (19), we can still use the algorithm in Table II. The difference between the general scenarios and the scenario with the assumption lies in Line 3 of the algorithm, where problem (20) is obtained from (19). In general scenarios, given the threshold of overall packet loss probability, , the optimization problem that minimizes the total number of subcarriers can be expressed as follows,
(22)  
s.t.  (22a)  
From the expression of in (14), we can see that increases with the packet offloading rate . Besides, the expressions of and in (10) show that the required number of subcarriers increases as decreases. Therefore, to satisfy constraint (22a), the number of subcarriers increases with the packet offloading rate . To minimize the total number of subcarriers, the first step is minimizing packet offloading rates.
Step 1: Optimize offloading rates. To minimize packet offloading rates, we find the maximal packets arrival rate at the local server, denoted as . Note that the delay violation probability at each local server should satisfy , by substituting the expression of in (11) into the constraint, can be obtained via binary search. If , all the packets are processed at the local server, , , and . Otherwise, the packet offloading rate of the th device is . As such, the constraint on the packet offloading rate in (16c) can be expressed as follows,
(23) 
According to (16a) and (16b), each device only offload it’s packets to one AP. Thus, the value of is determined by . If , . Otherwise, .
With the minimal packet offloading rates, problem (22) can be simplified as follows,
(24)  
s.t.  (24a)  
(24a)  
where , and . Different from problem (20), problem (24) cannot be decoupled into subproblems. This is because the workloads of the APs depend on association schemes of all the devices. As a result, changes with . Changing the association scheme of one device will lead to different overall packet loss probabilities of all the other devices.
Based on this fact that the throughput of eMBB services is higher than MCIoT services in most of the scenarios, we optimize the association scheme given the optimal bandwidth allocation obtained from problem (20), and then update bandwidth allocation according to the association scheme and related workloads at MEC servers.
Step 2: Optimize the association scheme . We set and as the values that are obtained under the assumption , and compute , , . The initial workloads of the MEC servers are , . From (8) we can obtain the initial delay bound violation probability . Then, from the st device to the th device, each device selects one AP that can minimize . Denote as the AP that minimizes . The th element of equals to one and , , . After the th device is associated with the th AP, the workload is updated according to .
Step 3: Update bandwidth allocation and . Given , we solve problem (21) for each device, and obtain the bandwidth allocation.
Remark 4.
The packet offloading rates obtained in Step 1 and the bandwidth allocation obtained in Step 3 are optimal with given . If we can obtain the optimal association scheme in Step 2, then we can obtain the optimal solution of problem (22). However, the final workloads of the APs are not exactly the same as the initial values. Thus, obtained in Step 2 is not optimal. However,
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