Real-Time Status: How Often Should One Update?
Age-of-Information for Computation-Intensive Messages in Mobile Edge Computing
Real-time information of computation-intensive messages in mobile edge computing
Abstract
In status update scenario, information age (AoI) is a novel indicator to measure real-time information. Real-time applications need to transmit status update information to destination nodes in time, but some applications’ status information is embedded in packages and will not be displayed until data processing, which is computation-costly and time-consuming.
Consider two options:
- The user’s own local computing
- Unload computing tasks to the edge server
The two methods are unified into a two-node series queue model. The zero-wait rule is adopted, that is, once the previous message leaves the first node, the first node immediately initiates a new message. The second node can be regarded as an M/M/1 queueing model of FCFS
Simulation results show that when the remote computing rate is large enough, remote computing is far superior to local computing, and there is an optimal transmission rate, so remote computing is superior to local computing in the maximum range.
I. introduction
For accurate management, it is important to keep the data fresh. The freshness of the data is defined as the time elapsed since the last update.
It is important that the required updated data not only needs to be transmitted to the controller, but also that the data needs to be preprocessed before useful messages are exposed. But this can be costly and time consuming due to the limited computing power of the local processor. The introduction of edge mobile computing (MEC) solves this problem, not only provides sufficient computing resources, but also shortens the response time.
AoI measures the freshness of data at the destination node. According to existing research, AoI depends on the frequency of packet generation and the delay caused by data transmission and queuing.
This article examines two modes: local computing and remote computing. The calculation process of remote computing follows FCFS rules, and the two computing methods are unified into a two-node series queue model. In remote computing, it is assumed that both the transmission time and the computation time are exponentially distributed and the queue length is infinite. Then, the calculation process can be regarded as an M/M/1 queueing model of FCFS. In the model we take the average AoI of remote and local calculations.
Based on the performance, the region was found to be characterized by remote computing over local computing AoI. The effects of packet size, CPU cycle demand, data rate and edge server computing power on average AoI are studied. According to the results, the amount of CPU cycles required by remote computing decreases, and the computing capacity of edge server increases, so AoI decreases. Considering only the influence of packet size and data rate in remote computing, there exists an optimal packet size and data rate, which minimizes the average AoI.
In local computing, if the number of CPU cycles required by remote computing decreases, AoI decreases. However, packet size and data rate have no effect on AoI. Remote computing is superior to local computing when more CPU cycles are required, or when edge servers have more computing power, with appropriate packet sizes and data rates.
II. Models and formulas
The status monitoring and control system is shown in Figure 1.
A. Local and remote computing models
Local computing (Figure 1A) : Computes computationally intensive data locally and sends the results to the destination node.
Remote computing (Figure 1B) : Compute-intensive packets are sent to edge servers for remote computing.
The zero-wait rule is applied to the local computing process and the remote computing transmission process.
For local computation, a new state packet is generated only if the last packet is fully evaluated by itself. Queue delay is estimated based entirely on the zero-wait principle. Because the processed data is much smaller than the original data, the transfer time is negligible compared to the calculation time.
For remote computing, a new status update packet is generated and transmitted only after the last status packet is sent. The sending queuing delay is zero, which applies the FCFS principle. Therefore, packets need to be queued for processing.
B. Unified model
The two models can be unified into a beam joint series model. Figure 2. For remote computing, C1 is the transmission channel, C2 is the edge server, M1 is the sending queue, M2 is the computation queue waiting for processing. Local computing can be seen as a special case where C2 has infinite service rates, C1 is the local computing server, and M1 is the computing queue (which is empty according to the zero-wait principle). M2 is also empty.
AoI is defined as
T is the current timestamp and u (t) is the time when the data was generated. Under FCFS rules, the change of destination node AoI (△ (t)) is shown in Figure 3.
Real-time Status: How Often Should One Update? Reasoning)
At t=0, the queue is empty △ (0) =△0. The first status update occurs at T1, and so on.
In the absence of any updates, the monitor’s age increases linearly over time and resets to a smaller value when updates are received. The update I is generated at time ti and is received by the endpoint after the computation is completed at time t’i. At time t’i, the end age△ (t’i) is reset to Ti=t’i-ti.
Age Ti is also the system time for updating group I, and is the sum of the time the group spent waiting in the queue and its time spent in service. Thus, the age function △ (t) shows the zigzag pattern shown in Figure 2.
The time-averaged age of status updates is the area under the sawtooth function in Figure 2, normalized by the observed time interval. At an interval (0, τ), the average age is
For simplicity, the length of the observation interval is chosen to be τ= t’n, as shown in FIG. 2. We decompose the region defined by the integral in (1) into the sum of disjoint geometric parts. Starting from t=0, the region can be regarded as the splicing of polygon region Q1, trapezoidal region Qi (I >1), triangle region with length Tn (Tn, t’n), etc.
In the N (T) = Max {N | tn < T = T} said time before arrival times. This decomposition makes
As shown in Figure 2, the area of Qi is an isosceles triangle with sides from Ti -1 to Ti ‘minus an isosceles triangle with sides from Ti to Ti’. define
To generate the elapsed time between I -1 and I, the calculation of Qi is defined as
When the generation of updates can be expressed as a random arrival process, Xi is the time interval for the arrival of update I. When (4) is substituted into (2), the average age of the rearrangement is
Among them
Is the steady state rate at which status update packets are generated. We assume that the limit exists and is finite. As N(T) goes to infinity, the remaining sum of PI (5) is a sample mean that converges to its corresponding random mean. The average status update age is
Where E[.] is the expectation operator, X and T are random variables corresponding to the arrival interval time and system time of update packets respectively.
We note that the average update age in (7) holds under the weak assumption of ergodicity of the service system. In addition, (7) is a general result of a broad category of service systems, where updates are grouped for FCFS processing. For example, when a status update flow shares a service facility with another packet flow, (7) will need to queue. However, assessing age△ can be challenging. In particular, X is a random variable that describes the interval between the generation of update groupings and the generation of previous groupings, while T is the running time of the same groupings in the system. The variables X and T are related. A large arrival interval X allows the queue to be empty, resulting in a small wait time and usually a small system time T. That is, X and T tend to be negatively correlated and this complicates the assessment of E [TX].
Next, we will find the age and server utilization of standard queuing systems that minimize FCFS queues. Let’s start with the M/M/1 system. (M/M/1 queuing model) is a single server queuing model, and the number of arrivals is a Poisson process (the number of events occurring in two mutually exclusive (non-overlapping) intervals is independent random variable). , service time is an exponential distribution (also known as a negative exponential distribution) that describes the probability distribution of time between events in a Poisson process, That is, events occur continuously and independently at a constant average rate), with only one server and an unlimited number of people joining the queue.
IV. M/M/1 FCFS
In this model, the birth rate (i.e. rate of joining the queue) λ is the same in all states, and the death rate (i.e. rate of leaving the queue after completing the service) μ is the same in all states (except state 0, where it is impossible for someone to leave the queue)
We consider that FCFS M/M/1 system has arrival rate λ and service rate μ. That is, the update grouping is generated and submitted to the system as a rate λ Poisson process, so that the state update arrival interval Xi is independent and equally distributed. Poisson distribution E (X) = D (X) = lambda lambda X exponential distribution E (X) = 1 / lambda D (X) = 1 / lambda) index of random variables, including E (X) = 1 / lambda. In addition, the service time is an IID index with an average service time of 1/μ. We will calculate the age of the system and then find the server utilization that minimizes the average age =λ/μ.
For fixed service rate μ, we can minimize the average age △ relative to arrival rate λ, or, equivalently, the supplied load ρ=λ/μ. We obtain the optimal utilization rate ρ satisfying the equation ρ^ 4-2ρ^ 3 +ρ^2-2ρ+ 1 = 0, so ρ = 0.53. Server idle = 47% of time. By choosing a λ to achieve the optimal age, the server biases the server to busy rather than idle. The average number of groups in the system is ρ = (1-ρ*) = 1.13 when the optimal utilization rate is ρ.
Note that if we want to maximize throughput, we want our ρ close to 1, where throughput is the number of packets sent to the monitor per second. If we want to minimize packet delay, that is, the system time of the packet, we want ρ close to 0.
V. M/D/1 FCFS
In some systems, a service facility represents an aggregator of randomly arriving status updates, where the update grouping is of fixed length and the grouping processing time is deterministic. For example, this could describe a medical facility in which a patient’s heart rate updates are collected by a central monitor. Assuming that the generation of this information for patients is independent and equally distributed, the total flow to the medical center can be modeled as a Poisson process of rate λ. Therefore, the system can be abstracted as an M/D/1 system.
When the ith status update packet arrives at M2 in Ti, ti is also the beginning of the processing of the I +1 status update packet by C1 due to the zero wait. C1 provides service at the service rate of U1. T ‘i is taken as the service termination time of the ith packet on C2, and C2 takes U2 as the service rate. The freshness of the destination node increases linearly in C2 without service completion, otherwise it rapidly decreases to a smaller value that updates the processing status of the destination. Assume that the two service times are independent and have the same distribution of exponential distributions.
The average freshness of the processed status packets is the function △ (t) in Figure 3, normalized by time interval. The mean AoI of interval (0, τ) is
VII. The conclusion
In this article, we consider AoI for computation-intensive messages in MECs using two scenarios, one local and one remote. The closed form mean AoI for local and remote computing is derived and the areas where remote computing is superior to local computing are given. Numerical results show that there is an optimal transmission rate, so remote computation is superior to local computation in the maximum range. Remote computing is more likely to outperform local computing at greater remote computing rates. We can see that the adoption of MEC is critical to obtaining the best AoI for computationally intensive data. In future work, it is worthwhile to extend the work to partial remote computing and multi-source-destination pairs.