Segunda etapa (1971-1985): desafíos en el estudio del clima

Analytical estimations make a simplifying assumption that a node can host any sin- gle EPN on its own and satisfy both 2a and 2b. If this assumption does not hold, then inter-EPN parallelism, similar to intra-operator parallelism used in Gulisano et al. (2010), would be necessary. This is an online model where few of the parame- ters(arrival rates) are dependent on dynamic real time values from the system. More detailed remarks are presented at the end of this section.

Figure 5.1: (a) Internals of an EPN. (b) 1-server, 1-queue model of the EPN

Consider a node hosting the EPNs of some Zx = {EP N1, EP N2, . . . , EP Nk}. Esti-

mations for this scenario are done in two parts as listed below:

1. Modelling a single EP Ni ∈ Zx as a single-queue, 1-server system.

2. Modelling all EPNs of Zx as a M/G/1 multi-class queuing system.

5.2.1

Modelling and Calibrating a Single EPN

Figure 5.1(a) presents the DAG (directed acyclic graph) structure of a simple EPN chosen as an example. There are two input streams with arrival rates AR1 _{and AR}2_.

After they are acted on by distinct operators (Op1 and Op2), they are joined at Op3 which produces an output stream O1 to the environment and an input stream to Op4 which then generates O2. Note that each Op would have its own buffer to store the incoming events which are not shown in Fig 5.1(a).

Irrespective of its DAG structure, the EPN of Fig5.1(a) is modeled as a single server system that receives a single input stream with arrival rate AR = AR1∪ AR2_{, and}

generates a single output stream O = O1∪ O2_{; the incoming event tuples are queued}

and the tuple at the head of the queue is processed by a super operator OP composed of Op1, Op2, Op3, Op4 (see Fig 5.1(b)). The tuples that get past the head of the queue, in the model, are processed by OP as per the logic of the EPN of Fig 5.1(a) and generate O = O1_{∪ O}2_.

The above modelling approach is applied to estimate the average response time of, and the CPU load exerted by, any single EPN. First, a few metrics are defined or recalled, some of which are measured dynamically and the rest established through off-line calibration. Let EP Ni be any EPN in the system.

Event Arrivals denote the arrivals of events of streams in Si and are taken to be

Poisson at the rate of ARi that is supplied to CS at the end of each reporting interval.

Processing Time denotes the total processing time a tuple or a window of tuples needs to undergo to produce an output Oj _{∈ O after a tuple in the window has just gone}

past the head of the queue as in Figure5.1(b). Note that it does not include the time that a tuple spends between its arrival and reaching the head of the queue viz. the queuing delays. Moreover, the processing time depends on the nature of DAGi that

EP Ni implements and also on the particular path that a tuple takes within DAGi

for it to be processed and an appropriate output to be generated.

Given the non-deterministic nature of tuple-flows, the processing time is modeled as a random variable of some unknown distribution. When EP Ni generates several

outputs, the one that takes the maximum processing time will be of interest. For this output, bi is defined as the average processing time and M2,i as the second moment of

the processing times.

The Processing Load imposed by EP Ni on the host node is ρi. Specifically, ρi is the

fraction of time EP Ni uses the node’s CPU. Thus, ρi = ARi× bi and this relation is

used to establish M2,i and bi through calibration as described below.

EP Ni is hosted on its own on a node and subject to small arrival rates ARi (to

eliminate or at least minimize queuing delays); the CPU usage for each rate is observed and the resulting processing times are also computed from the observations. From these times, the most processing intensive output is identified, and M2,i and bi are

established. EPNs typically process input events in groups or windows of, say, w tuples; if so, the arrival rate during calibration should not exceed w events per second.

5.2.2

Modelling Multiple EPNs in a Single Host

Recall that a single node hosts k EPNs of Zx. The collection of EPNs is abstracted

in the same way as the collection of operators of a single EPN; more precisely, the k EPNs of Zx are replaced by a composite EPN and the input events of various EPNs

are placed in a single FIFO queue; when an event or a window of appropriate events gets past the head of the queue, the (virtual) processor of the node processes it by executing the appropriate EP Ni. The model thus becomes a M/G/1 multiple class

Using the well-known M/G/1 results, the expressions are derived for metrics of in- terest, when a node hosts k EPNs of Zx = {EP N1, EP N2, . . . , EP Nk}. Note that

the total arrival rate (AR) at the node is the sum of the arrival rates of the k hosted EPNs (Pk

i=1ARi); similarly, the total load (ρ) on the node is the sum of the load

exerted by every EP Ni ∈ Zx. Thus,

AR = k X i=1 ARi, ρ = k X i=1 ρi = k X i=1 ARi× bi (5.1)

The average response time estimated for any EP Ni ∈ Zx as Wi:

Wi =

(AR × M 2)

2(1 − ρ) + bi, (5.2) where M 2 = _AR1 Pk

i=1M2,i × ARi. Similar to δ m

i defined by expression 7.1. The δei

denote the relative deviation of Wi, with e indicating that δie is based on estimation.

δe_i = Wi− Ti Ti

(5.3) If Wi estimates RTi reasonably accurately, δie ≈ δim, and 2(a) and 2(b) of §5.2 can

be evaluated; more precisely, given that the EPNs are divided into ζ disjoint sets, Z1, Z2, . . . , Zζ, using a unique node to host each Zx, 1 ≤ x ≤ ζ, leads to a viable

configuration, if

∀Zx, 1 ≤ x ≤ ζ : ∀EP Ni ∈ Zx : LW ≤ δei, and (5.4)

∀Zx, 1 ≤ x ≤ ζ : ρ ≤ ρth (5.5)

Remarks. Condition 5.4 states that every EPN in every Zx must have its esti-

mated deviation within the water marks and Condition 5.5 requires that the total load imposed on every Zx must not exceed ρth. They verify 2(a) and 2(b) of Section

4 respectively, provided that each node being used for hosting a set of EPNs is (i) a single CPU or a single core machine, and has a CPU that is as powerful as the CPU of the one used for the calibration of EPNs.

If the CPUs of the host nodes are more powerful, the actual response times would be smaller than the estimates and hence the evaluation of Conditions 5.4 and 5.5 would be pessimistic and would result in more nodes being used than strictly necessary. On the other hand, if the calibration has been done using nodes with more powerful CPUs, response-time targets may be missed frequently.

In inter-EPN parallelism, a given EP Ni is hosted on multiple nodes and each input

stream in Si is temporally split and each split is input to one of the hosting nodes.

For example, let EP Ni be hosted by two nodes as EP Ni1 and EP Ni2; an input stream

si ∈ Si can be split as: first 100 tuples (1 to 100) as s1i, the next 100 tuples (101 to

200) as s2_i, (201 to 300) as s3_i, and so on. Odd splits, (s1_i, s3_i, . . ., s2n−1_i , . . .), are sent to EP N_i1 (in order) and even splits to EP N_i2, thus halving the arrival at each node. The results from EP N_i1 and EP N_i2 would have to be ’reduced’ for the final result.