Modeling Green DataCenters and Jobs Balancing with Energy Packet Networks and Interrupted Poisson Energy Arrivals
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Abstract
We investigate micro datacenters systems with intermittent energy (due to solar panels or price of energy) which balance their jobs to mitigate these fluctuations. The model is based on energy packet networks recently proposed by E. Gelenbe and his colleagues. These models explicitly represent both energy and data processing in a combined stochastic process. We prove that under some technical conditions on the rates and with a suitable control of the migration rates, the leakage rates and the service rates, the steady state distribution has a product form steadystate distribution.
Keywords
Energy packet network Product form equilibrium distribution Interrupted Poisson process Clouds with harvesting energyIntroduction
Geographical load balancing is an appealing strategy to reduce the energy consumption or cost for datacenters (see for instance [19] or [20]). We consider micro datacenters powered by renewable energy sources like photovoltaic panels [2]. These sources of energy are intermittent and random. Hence we want to ideally adapt the datacenter loads to the energy harvesting rate. Here we develop a model based on Energy Packets networks to prove the equilibrium distributions of a set of microgrids with intermittent energy harvesting and an ideal geographical load balancing mechanism.
Energy packet networks (EPNs), recently introduced by Gelenbe and his colleagues [8, 9, 10, 13], are used to model the flow of intermittent sources of energy like batteries and solar or windbased generators and study their interactions with IT devices consuming energy like sensors, cpu, storage systems and networking elements.
The key idea of EPNs is to represent energy with packets of discrete units called energy packets (EPs). Each EP models a certain number of Joules. Since the EPs are produced by an intermittent source of energy (typically solar, tides and wind), the flow of EPs is associated with some random processes. EPs are consumed by some devices after some random duration to perform requested works or can also be stored in a battery from which they can also leak after a random delay. Note that an independent approach to the EPNs has been presented in the electrical engineering literature under the name “power packet”, see [21]. In this approach, power packets consist in a pulse of current and are associated with a header and a protocol to control the routing using some hardware switching.

Quantity of energy: One EP may be not sufficient to send a DP as in [11]. In [17] the number of EP needed to transfer a DP is a constant K. This energy can even be a discrete random variable like in [15] or associated with a continuous process in [1].

Master: in [18], the DP queue is the initiator of the transfer. The arrival of a DP at the battery triggers the deletion of a an EP and the movement of the data. If a data packet does not find an energy packet, it is lost. Thus we represent losses of DP, while the initial model (i.e., [13]) represents packets delayed due to the lack of energy.

Classes of data packets: in [3], networks with multiple classes of DP are proved to also have a product form steady state distribution. The routing and the energy needed for a DP movement are classdependent.
Some of these EPN models (not all of them) are Gnetworks of queues, introduced by the seminal papers by Gelenbe on networks of queues with positive and negative customers [6], and queues with triggers [7]. As Gnetworks have a product form for the steadystate distribution of jobs in the queues, they allow to optimize the system for some utility function like losses or response time [4, 12, 13, 14, 15]. Note, however, that EPN models are not always related to Gnetworks but may be associated to other stochastic models (for instance, in [1] the authors use a diffusion process).
Each datacenter may be in two modes: ON when it receives energy and OFF where the energy harvesting is impossible (or the cost of energy is too high). During their OFF state, the datacenters are in a sleep mode where the transitions are blocked. The evolution between ON state and OFF state of each datacenter is governed by a global Markov chain called as a modulating phase. We show that the steadystate distribution of jobs in the queues and the energy has a product form provided that a stable solution of a fixed point problem exists. The remainder of the article is organized as follows. In the next section, we describe the model and we prove the main result of the article. In the following section an example is given before we present the conclusions.
Model and Results

\({{{\mathcal {S}}}}_i^{\mathrm{ON}} \cup {{{\mathcal {S}}}}_i^{\mathrm{OFF}} = {{\mathcal {S}}} \),

and \({{{\mathcal {S}}}}_i^{\mathrm{ON}} \cap {{{\mathcal {S}}}}_i^{\mathrm{OFF}} = \emptyset \),

and \(\forall \phi \in {{{\mathcal {S}}}}_i^{\mathrm{OFF}},\quad \gamma _i^{\phi } = 0 \),

while \( \gamma _i^{\phi } = \varGamma _i, \quad \forall \phi \in {{{\mathcal {S}}}}_i^{\mathrm{ON}}\).

The services consist in the consumption of an energy packet (if available) at the EP queue. Once the EP have been consumed, a DP is instantaneously removed from the DP queue (i.e., the jobs is completed). If the DP queue is empty, the energy packet is lost. This model represents two important features of the model: first without energy, the jobs are not served, and second even if there are no jobs in the server, there is still some energy consumption.
The services are exponential with rate \(\mu _i\) when the datacenter is in ON mode while we have:Similarly, the leakages follow a Poisson process with rate \(\varOmega _i\) when the datacenter is in ON mode.$$\begin{aligned} \forall \phi \in {{{\mathcal {S}}}}_i^{\mathrm{OFF}}, \quad \omega _i^{\phi } = 0 \quad \mathrm{and}\quad \mu _i^{\phi } = 0. \end{aligned}$$(4)  The arrivals of jobs are only sent by the scheduler to the datacenter which are in ON mode such that:$$\begin{aligned} \forall \phi \in {{{\mathcal {S}}}}_i^{\mathrm{ON}}, \quad \lambda _i^{\phi } = \varLambda _i \quad \mathrm{and}\quad \forall \phi \in {{{\mathcal {S}}}}_i^{\mathrm{OFF}}, \quad \lambda _i^{\phi } = 0. \end{aligned}$$(5)
Theorem 1
Note that solving the flow equations is a rather simple task. We do not need the complex algorithm presented in [5]. It is only needed for all i to compute \(\beta _i\) with Eq. 6 and then compute \(\rho _i\) with Eq. 7 once \(\beta _i\) is known. Thus checking the stability is an easy task for that model.
A Simple Example
ON–OFF periods for six datacenters
1  2  3  4  

A  ON  ON  OFF  OFF 
B  OFF  ON  ON  OFF 
C  OFF  OFF  ON  ON 
D  ON  OFF  OFF  ON 
E  OFF  OFF  ON  ON 
F  OFF  OFF  ON  ON 
Routing probability during each phase for each datacenter
1  2  3  4  

A  1/2  1/2  0  0 
B  0  1/2  1/2  0 
C  0  0  1/4  1/4 
D  1/2  0  0  1/2 
E  0  0  1/8  1/8 
F  0  0  1/8  1/8 
Concluding Remarks
To the best of our knowledge, this paper is the first attempt to have a non constant arrival rate of energy packets in an EPN model. Note that it is straightforward to modify the proof to consider a single datacenter with energy harvesting where the center stops (or go to sleep mode) when the energy rate goes to zero. We hope that this result will open new research to deal with time varying energy arrivals and load balancing of jobs.
Notes
References
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