PORA: Predictive Offloading and Resource Allocation in Dynamic Fog Computing Systems

Within each time slot $t$, in addition to the current arrivals in the arrival queue, EFN $i$ can also forward future arrivals in the prediction queues. We define $\mu_{i,w}(t)$ as the amount of output workload from $A_{i,w}(t)$, for $w\in\{-1,0,...,W_{i}-1\}$. Such workload should be distributed to the local processing queue and offloading queue. We denote the amounts of workloads to be distributed to the local processing queue and offloading queue as $b_{i}^{(e,l)}(t)$ and $b_{i}^{(e,o)}(t)$, respectively, such that

where each $b^{(e,\beta)}_{i,\text{max}}$ is a positive constant. As a result, we have

Next, we consider the queueing dynamics for different types of queues in EFN, respectively.

Regarding $A_{i,w}(t)$, it is updated whenever pre-service is finished and the lookahead window moves one slot ahead at the end of each time slot. Therefore, we have

1. (i)

   If $w=W_{i}-1$, then

   $$A_{i,W_{i}-1}\left(t+1\right)=A_{i}\left(t+W_{i}\right).$$

   (3)

2. (ii)

   If $0\leq w\leq W_{i}-2$, then

   $$A_{i,w}(t+1)=[A_{i,w+1}(t)-\mu_{i,w+1}(t)]^{+},$$

   (4)

where $[x]^{+}\triangleq\max\{x,0\}$ for $x\in\mathbb{R}$. In time slot $(t+1)$, the amount of workload that will arrive after $(W_{i}-1)$ time slots is $A_{i}(t+W_{i})$ and it remains unknown until time slot $(t+1)$.

Regarding the arrival queue $A_{i,-1}(t)$, it records the actual backlog of EFN $i$ with the update equation as follows:

Note that $\mu_{i,-1}(t)$ denotes the amount of distributed workload that have already being in $A_{i,-1}(t)$.

Next, we introduce an integrate queue with a backlog size as the sum of all prediction queues and the arrival queue on EFN $i$, denoted by $Q_{i}^{\left(e,a\right)}\left(t\right)\triangleq\sum_{w=-1}^{W_{i}-1}A_{i,w}\left(t\right)$ . Under fully-efficient [16] service policy, $Q_{i}^{\left(e,a\right)}\left(t\right)$ is updated as

The input of integrate queue $Q_{i}^{(e,a)}(t)$ consists of the predicted workload that will arrive at EFN $i$ in time slot $(t+W_{i})$, while its output consists of workloads being forwarded to the local processing queue and the offloading queue. Note that $b^{(e,l)}_{i}(t)+b^{(e,o)}_{i}(t)$ is the output capacity of integrate queue $Q_{i}^{(e,a)}(t)$ in time slot $t$. If the capacity is larger than the queue backlog size, the true output amount will be smaller than $b^{(e,l)}_{i}(t)+b^{(e,o)}_{i}(t)$.

In time slot $t$, workload in queue $Q^{(e,o)}_{i}(t)$ will be offloaded to CFNs in set $\mathcal{M}_{i}$. The transmission capacities are determined by the transmit power decisions $(p_{i,j}(t))_{j\in\mathcal{M}_{i}}$, where $p_{i,j}(t)$ is the transmit power from EFN $i$ to CFN $j$. The transmit power is nonnegative and the total transmit power of each EFN is upper bounded, i.e.,

According to Shannon’s capacity formula [17], the transmission capacity from EFN $i$ to CFN $j$ is

where $\tau_{0}$ is the length of each time slot, $B$ is the channel bandwidth, $H_{i,j}(t)$ is the wireless channel gain between EFN $i$ and CFN $j$, and $N_{0}$ is the system power spectral density of the additive white Gaussian noise. Note that $H_{i,j}(t)$ is an uncontrollable environment state with positive upper bound $H_{\text{max}}$. We do not consider the interferences among fog nodes and tiers. By adjusting the transmit power $p_{i,j}(t)$, we can offload different amounts of workload from EFN $i$ to CFN $j$ in time slot $t$. Accordingly, the update equation of offloading queue $Q^{(e,o)}_{i}(t)$ is

where $\sum_{j\in\mathcal{M}_{i}}R_{i,j}(t)$ is the total transmission capacity to EFN $i$ in time slot $t$. The inequality here means that the actual arrival of $Q_{i}^{(e,o)}(t)$ may be less than $b_{i}^{(e,o)}(t)$, because $b^{(e,o)}_{i}(t)$ is the transmission capacity from integrate queue $Q_{i}^{(e,a)}(t)$ to offloading queue $Q_{i}^{(e,o)}(t)$ instead of the amount of truly transmitted workload. Recall that we assume the transmission latency from EFT to CFT is negligible compared to the length of each time slot, the workload transmission in each time slot can be accomplished by the end of that time slot.

The arrivals on CFN $j$ consist of workloads offloaded from EFNs in the set $\mathcal{N}_{j}$. We denote the amounts of workloads distributed to the local processing queue and offloading queue in time slot $t$ as $b_{j}^{(c,l)}(t)$ and $b_{j}^{(c,o)}(t)$, respectively, such that

where each $b^{(c,\beta)}_{j,\text{max}}$ is a positive constant. Accordingly, $Q_{j}^{(c,a)}(t)$ is updated as follows:

For each CFN $j\in\mathcal{M}$, its offloading queue $Q^{(c,o)}_{j}(t)$ stores the workload to be offloaded to the cloud. We define $D_{j}(t)$ as the transmission capacity of the wired link from CFN $j$ to the cloud during time slot $t$, which depends on the network state and is upper bounded by some constant $D_{\text{max}}$ for all $j$ and $t$. Then we have the following update function for $Q^{(c,o)}_{j}(t)$:

Note that the amount of actually offloaded workload to the cloud is $\min\{Q_{j}^{(c,o)}(t),D_{j}(t)\}$.

In each time slot $t$, under PORA, each fog node $k\in\mathcal{N}\cup\mathcal{M}$ decides the amounts of workload scheduled to the offloading queue and the local processing queue, denoted by $b_{k}^{(\alpha,l)}(t)$ and $b_{k}^{(\alpha,o)}(t)$, respectively. Such decisions are obtained by solving the following problem:

where $\beta\in\{l,o\}$. Accordingly, the optimal solution to (20) is

From (21), we see that, to determine the optimal solutions $b_{i}^{(e,l)}(t)$ and $b_{i}^{(e,o)}(t)$, each EFN $i$ would compare its integrate queue backlog size $Q_{i}^{(e,a)}(t)$ with its local processing queue backlog size $Q_{i}^{(e,l)}(t)$ and offloading queue backlog size $Q_{i}^{(e,o)}(t)$, respectively. Particularly, if there is too much workload in its integrate queue compared to its local queue ($Q_{i}^{(e,l)}\left(t\right)<Q_{i}^{(e,a)}(t)$), then it will offload as much workload (up to $b^{(e,l)}_{i,\text{max}}$) as possible to its local queue. Likewise, if its integrate queue is loaded with more workload than its offloading queue ($Q_{i}^{(e,o)}\left(t\right)<Q_{i}^{(e,a)}(t)$), it will offload up to an amount of $b^{(e,o)}_{i,\text{max}}$ workload to its offloading queue.

Notably, if the backlog size of the EFN $i$’s integrate queue is larger than both its local queue and offloading queue, then the EFN will transmit the workload one by one unit (e.g. packets); each unit of workload is either sent to the EFN $i$’s local queue or its offloading queue, such that the amounts of workload distributed to such two queues are no greater than $b_{i,\text{max}}^{(e,l)}$ and $b_{i,\text{max}}^{(e,o)}$, respectively. In practice, the workload distributing strategy is left as a degree of freedom to be specified in the implementation of PORA. In our simulation, we adopt the following distributing strategy. When an EFN $i$’s integrate queue backlog size is greater than both its local queue and its offloading backlog size, then it will transmit workload to its local queue until the amount of transmitted workload reaches $b_{i,\text{max}}^{(e,l)}$. Then the rest workload in the integrate queue is transmitted to the offloading queue until the amount of distributed workload reaches $b_{i,\text{max}}^{(e,o)}$. Such a process terminates whenever the integrate queue becomes empty.

The decision making process is similar for CFNs. Specifically, each CFN $j$ determines $b_{j}^{(c,l)}(t)$ and $b_{j}^{(c,o)}(t)$ by comparing its arrival queue backlog size $Q_{j}^{(c,a)}(t)$ with its local processing queue backlog size $Q_{j}^{(c,l)}(t)$ and offloading queue backlog size $Q_{j}^{(c,o)}(t)$, respectively.

Remark: For each EFN, we can view the difference between the backlog sizes of its integrate queue and its local processing/offloading queue as its willingness of workload transmission. If such willingness is positive, then the EFN will transmit as much workload as possible from its integrate queue; otherwise, the EFN will leave the workload not distributed in the current time slot. In such a way, PORA always endeavors to balance the integrate queue backlog and the local/offloading queue backlog. Likewise, under PORA, each CFN determines its offloading decisions upon the difference between the backlog sizes of its arrival queue and its local processing/offloading queue to ensure the queue stability.

Under PORA, in each time slot $t$, each fog node $k\in\mathcal{N}\cup\mathcal{M}$ sets its local CPU frequency $f_{k}^{(\alpha)}(t)$ by solving the following subproblem:

By setting the second derivative of the objective function in (22) to zero, we can obtain the optimal CPU frequency $f_{k}^{(\alpha)}(t)$ to be set by fog node $k$ as

We prove the optimality of (23) in Appendix B.

Remark: When $f^{(\alpha)}_{k}(t)<f_{k,\text{max}}^{(\alpha)}$, the allocated CPU frequency $f_{k}^{(\alpha)}(t)$ is proportional to the square root of the backlog size of local processing queue $Q_{k}^{(\alpha,l)}(t)$ and the inverse of the value of parameter $V$. This shows that, on the one hand, PORA would allocate as much CPU frequency as possible to process the workload in the queues. On the other hand, the value of parameter $V$ determines the tradeoff between power consumption and the backlog sizes of queues: a small value of $V$ will encourage the fog node to allocate more CPU frequency to process the workload and hence a small queue backlog size; in contrast, a large value of $V$ will make the fog node more conservative to allocate resources, leading to less power consumptions but a large queue backlog size as well. In practice, the choice of the value of $V$ is dependent on the system design objective.

In each time slot $t$, under PORA, each EFN $i\in\mathcal{N}$ determines its allocated transmit power $\boldsymbol{p}_{i}(t)$ by solving the following optimization problem.

where $m_{i,j}(t)\triangleq(Q_{i}^{(e,o)}(t)-Q_{j}^{(c,a)}(t))B$ and $l_{i,j}(t)\triangleq\frac{H_{i,j}(t)}{N_{0}B}$. By applying water-filling algorithm[22], we obtain the optimal solution to problem (24) as

where $\lambda^{*}$ is the optimal Lagrangian variable that satisfies

The optimality of such solutions is proven in Appendix C. We adopt bisection method (line 15-25 in Algorithm 1) to obtain the value of $\lambda^{*}$ with its lower and upper bounds as $\lambda_{\text{min}}$ and $\lambda_{\text{max}}$, respectively. Note that the value of $\lambda^{*}$ converges asymptotically to the optimum $\lambda^{\text{opt}}$ as the tolerance parameter $\varepsilon$ approaches zero, such that $|\lambda^{*}-\lambda^{\text{opt}}|\leq\varepsilon/2$.

Remark: PORA tends to allocate more transmit power to the CFN with smaller arrival queue backlog size $Q_{j}^{(c,a)}(t)$ for load balancing. When $Q_{j}^{(c,a)}(t)\geq Q_{i}^{(e,o)}(t)$, we have $m_{i,j}(t)\leq 0$ and $p_{i,j}(t)=0$, i.e., EFN $i$ allocates no transmit power to CFN $j$ unless the backlog size of the arrival queue on CFN $j$ is greater than that of the offloading queue on EFN $i$. By increasing the value of $V$, transmit power consumption will be reduced but the backlog size will increase as well.

Let $P^{*}$ be the achievable minimum of $\bar{P}$ over all feasible non-predictive polices. We have the following theorem.

The proof is quite standard and hence omitted here.

Remark: By Little’s theorem[23], the average queue backlog size is proportional to the average queueing latency. Therefore, Theorem 1 implies that by adjusting parameter $V$, PORA can achieve an $[O(1/V),O(V)]$ power-latency tradeoff in the non-predictive case. Furthermore, the average power consumption $\bar{P}$ approaches the optimum $P^{*}$ asymptotically as the value of $V$ increases to infinity.

We analyze the latency reduction induced by PORA under perfect prediction compared to the non-predictive case. In particular, we denote the prediction window vector $(W_{i})_{i\in\mathcal{N}}$ by $\boldsymbol{W}$ and the corresponding delay reduction by $\eta(\boldsymbol{W})$. For each unit of workload on EFN $i$, let $\pi_{i,w}$ denote the steady-state probability that it experiences a latency of $w$ time slots in $A_{i,-1}(t)$. Without prediction, the average latency on its arrival queues is $d=\sum_{i\in\mathcal{N}}\lambda_{i}\sum_{w\geq 1}w\pi_{i,w}/\sum_{i\in\mathcal{N}}\lambda_{i}$. Then we have the following theorem.

We relegate the proof of Theorem 2 to Appendix D.

Remark: Theorem 2 implies that predictive offloading conduces to a shorter workload latency; in other words, with predicted information, PORA can break the barrier of $[O(1/V),O(V)]$ power-latency tradeoff. Furthermore, the latency reduction induced by PORA is proportional to the inverse of the prediction window size, and approaches zero as prediction window sizes go to infinity. In our simulations, we see that PORA can effectively shorten the average arrival queue latency with only mild-value of future information.