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Hardware support for aggressive parallel discrete event simulations

Hardware Support for Aggressive Parallel Discrete Event Simulation

Sudhir Srinivasan

Paul F. Reynolds, Jr.

Computer Science Report No. TR-93-07

January 11, 1993

Hardware Support For Aggressive Parallel

Discrete Event Simulation

Every Parallel Discrete Event Simulation (PDES) protocol can bene?t signi?cantly from the av ailability of global information: non-aggressiv e protocols require deadlock detection and recovery; aggressive protocols require Global Virtual Time (GVT), and global and local status information is critical to adaptiv e protocols. To rapidly compute and disseminate global information, Reynolds proposed a framework for all PDES’s. As proposed, the framework could clearly operate with a non-aggressive protocol. To demonstrate its utility to aggressive protocols, we describe an algorithm to compute GVT using the framework in a Time Warp system, and we establish its correctness. Aggressiveness, used in moderation, (i.e. adaptive behavior) will most likely lead to the most ef?cient protocols. The algorithms described in this paper can be used as a foundation for designing adaptiv e protocols which utilize the framework to obtain status information. Finally, our approach to computing GVT is nov el. Simulation studies show that v ery accurate GVT can be made av ailable at no cost to the simulation itself.

Categories and Subject Descriptors: D.1.3 [Programming Techniques]: Concurrent Programming -parallel programming; D.4.1 [Operating Systems]: Process Management -deadlocks; D.4.1 [Operating Systems]: Process Management -synchronization; D.4.4 [Operating Systems]: Communications Management -message sending; D.4.7 [Operating Systems] Organization and Design -distributed systems; D.4.8 [Operating Systems] Performance -simulation; I.6.8 [Simulation and Modeling] Types of Simulation -parallel and discrete event

General Terms: Algorithms, Performance

Additional Key Words and Phrases: Aggressiv e protocols, GVT computation, reduction network, synchronization algorithm, Time Warp

List of ?gures

Figure 1 - Global Computation Model

Figure 2 - Framework Algorithm

Figure 3 - Hardware Con?guration

Figure 4 - A Parallel Reduction Network

Figure 5 - Aggressive Framework Algorithm

Figure 6 - GVT Computation Model for AFA

Figure 7 - AFA2P: Host Processor Algorithm

Figure 8 - GVT Computation Model for AFA2P

Figure 9 - Timestamp pro?le of LPi

Figure 10 - Possible pro?les

Figure 11 - Pro?le of the root of the rollback chain

Figure 12 - Pro?le of the root of a rollback chain with a valley

Notation

-T -value used to track the logical clock value of LP i -smallest next event time of LP i -smallest unreceived message time of LP i -T -value used for primary acknowledgment by LP i -T -value used for secondary acknowledgment by LP i

-minimum timestamp of AP i =σi ηi υi ρi τi μi MIN σi υi ,()

1Introduction

1.1Overview

Parallelizing Discrete Event Simulation (DES) is a challenging problem in parallel computing. Since simulated events have causal relations among them, they must be simulated in the order de?ned by these causal relations. A dependence graph can be constructed using these causal relations, which de?nes a partial order on all of the simulated events. Events which are not related under this partial order can be executed concurrently. It appears that typical DES’s exhibit substantial concurrency in their dependence graphs [9, 15, 19] which makes DES a candidate for parallelization. Ironically though, this concurrency is dif?cult to extract in practice. The well-investigated approach of extracting concurrency from sequential programs fails with DES due primarily to two reasons: (i) There is a single (main) data structure, the events list, which is modi?ed very frequently (ii) The DES algorithm is basically a single loop in which there are several inter-iteration dependencies (because execution of events may schedule more events in the logical future). Consequently, a different method for parallelizing DES has been adopted [4, 19]. The essence of this method is to divide the DES into several logical processes (LP’s), each responsible for simulating a part of the physical system. These LP’s are sequential DES’s by themselves and they synchronize with each other when required using timestamped messages, in order to guarantee accurate simulation (i.e, freedom from causality violations). The method used to synchronize the LP’s of a Parallel Discrete Event Simulation (PDES) is generally referred to as a protocol. We present a modi?ed version of a previously proposed synchronization algorithm [22], modi?ed to operate with a class of protocols which we call aggressive protocols. This class includes the so-called optimistic protocols as well as an interesting and as yet relatively unexplored subclass:adaptive protocols in which the aggressiveness of the LP’s is controlled. Also, we prove the correctness of our algorithm. The algorithm of [22] as well as our algorithm operate on special-purpose hardware designed to desseminate global information rapidly. Thus, the primary contribution of this work is the establishment of a correct method for rapidly disseminating critical synchronization information in support of aggressive (and therefore adaptive) protocols at essentially no cost to the PDES itself. Our algorithm will serve as a foundation for the design of adaptive protocols which use the hardware to gather status information at very low cost.

1.2Background

In the past, researchers have proposed several protocols [9], most of which belong to one of two categories:non-aggressive, accurate and without risk (commonly known as conservative) and aggressive,accurate and with risk(commonly optimistic) with some exceptions. In a non-aggressive protocol, an LP generally executes its next scheduled event only after it con?rms that no causality constraints can be violated in the future due to this execution. Such protocols tend not to exploit much of the concurrency in the simulation. In an aggressive protocol, an LP executes events without the guarantee of freedom from causality errors. As a result, causality errors may occur. One of the ways of dealing with causality errors is to ignore them, but we do not consider this further here. For the simulation to be accurate, some error recovery must be performed; commonly,rolling back to an earlier, correct, saved state is the method of choice. Consequently, aggressive protocols suffer the penalty of saving state periodically and rolling back on errors. The

costs of these actions can become prohibitively high in implementations. Another problem aggressive protocols face is that LP’s periodically require a global value called global virtual time (GVT) [10]. Computing GVT has thus far been an expensive operation. Note that the general class of aggressive protocols includes optimistic protocols.

Researchers have reported success with both non-aggressive and aggressive protocols [9]. Studies have shown that there are applications for which aggressive protocols will outperform non-aggressive ones [12, 14] and vice-versa [16]. As a result, there is no optimal protocol. In [21], Reynolds points out that there are several other categories of protocols, based on a set of design variables used to characterize protocols. An interesting category of protocols that has not received much attention is the class of adaptive protocols in which the LP’s dynamically adapt some aspect or aspects of their processing. An interesting and promising subclass of adaptive protocols are those in which the LP’s dynamically control their aggressiveness. Recent research [1, 6, 7, 16, 17] suggests that controlled aggressiveness will most likely be a feature of a universally ef?cient protocol. In the rest of this paper, we will use the term adaptive to refer to this subclass of adaptive protocols which is characterized by controlled aggressiveness.

Most PDES protocols require global information of some sort [24]. For example, Time Warp LP’s require the value of GVT for several reasons. Correspondingly, non-aggressive (or blocking) protocols, which have the potential to deadlock, need to be able to identify the LP with the event that must be executed next, to break a deadlock. Iterative protocols which restrict executable events to those inside a computation window require the LP’s to collectively determine the lower and upper bounds of the window. In adaptive protocols as well, there is need for global information, since the amount of aggressiveness of any LP will be controlled by its state relative to the state of others in the system.

The need for global synchronization information motivated a universal framework for all parallel simulations, proposed by Reynolds [22]. This framework has the capability of rapidly providing the global information required in any PDES. For instance, it can be used to compute GVT, the minimum lookahead in the system, the ?oor and ceiling of a computation window or to break deadlocks. We believe that adaptive protocols will bene?t most from this framework because global information required to make adaptive decisions dynamically, which has been very expensive (in time) to compute thus far, can be made available very rapidly using the framework. The framework consists of three parts: (i) A small set of global values required by any PDES protocol (ii) H igh speed special purpose hardware to rapidly compute and disseminate these global values and (iii) A synchronization algorithm executed by each LP to correctly maintain the global values. (i) and (iii) are described in [23] while (ii) is described in [25]. It must be noted that the synchronization algorithm is not a protocol, but a foundation for PDES protocols. It is intuitively clear that the synchronization algorithm can operate with non-aggressive protocols, in which the logical clocks of the LP’s never move backwards. We present an aggressive version of the synchronization algorithm and prove its correctness, thus demonstrating that the framework can be used to support a protocol which allows aggressive processing. In the future, we intend to use our algorithm as a basis for designing ef?cient adaptive protocols which use the framework hardware to collect global information. In the rest of the paper we refer to the hardware desgined

for the framework as the framework hardware and to the synchronization algorithm as the framework algorithm .

In the next section, we brie?y describe the hardware and synchronization algorithm of the framework to set the context for our work. We present our algorithm in Section 3 and in Section 4,we show its correctness. Sections 5 and 6 present and establish the correctness of a modi?ed version of our algorithm, which is required due to the organization of the hardware described in Section 2. Section 7 describes three properties of the value of GVT computed by our algorithm. In Section 8, we present the ?ndings of preliminary studies of our algorithm and compare this with previous work in this area. Section 9 concludes the paper.

2 A Framework for PDES

We begin with a brief review of the Reynolds framework. Details can be found in [23]. We ?rst describe the framework algorithm and then the hardware support for the algorithm.

2.1Framework algorithm

The main purpose of the algorithm is to correctly maintain a set of global values required by the protocol being used in the PDES. At each LP , the algorithm maintains the local counterpart of each global value. For instance, the algorithm could maintain the event processing rate at each LP so that the corresponding global value could identify the fastest LP’s. In [22], Reynolds identi?ed two such global values: , the smallest of the timestamps in the future events list of all the LP’s and , the smallest of the timestamps of messages that are outstanding (as yet unreceived). Since these values are logical timestamps, we call them T-values . As we shall see,these T -values are functions of real time. The upper-case “T ” in “T -values” signi?es logical time,which is the range of these functions, as against real time, which is their domain. To compute these two global T -values, each LP maintains two local T -values, one corresponding to each global T -value. We de?ne and as follows:

where is the timestamp of the ?rst event in LP i ’s events list (which is assumed to be sorted in non-descending order) and is the smallest of the timestamps of all the messages that LP i has sent out which have not yet been received by their intended receivers. We call the next event time and accordingly, the minimum next event time . Similarly, is the smallest unreceived message time and is the minimum unreceived message time . We emphasize here that the responsibility of the algorithm is only to correctly maintain and (or other such global T -values). The values are actually used by the PDES protocol working in conjunction with the framework. For completeness, the algorithm presented here uses the two global T -values to identify the next safe event (or events) in the system, where an event is safe if its execution is guaranteed not to violate any causality constraints in the future.

η′υ′η′υ′η′MIN LP i

ηi ()=υ′MIN LP i

υi ()=ηi υi ηi η′υi υ′η′υ′

this, the protocol uses two phases: in the ?rst phase, the sender of the message being acknowledged determines that the receiver of the message is acknowledging the message and in the second phase, the receiver determines that its acknowledgment has been observed by the sender.

To acknowledge a message sent to it by LP s , LP r initiates the ?rst phase by setting its to the timestamp and message identi?er of the received message. We refer to some having a value other than as a primary acknowledgment (or simply acknowledgment ) and the act of setting as submitting a primary acknowledgment . The timestamp of a primary acknowledgment is the same as that of the message which it is acknowledging. For now, assume that the timestamp of is the smallest among all other acknowldegments submitted simultaneously . Thus, will equal . When this is observed by LP s (and it is guaranteed to ultimately do so, since LP r does not change ),it knows that its message to LP r is being acknowledged and the ?rst phase is complete. LP s initiates the second phase by setting its to . As with , we refer to some having a value other than as a secondary acknowledgment and the act of setting as submitting a secondary acknowledgment . Since an LP submits a secondary acknowledgment only after observing a primary acknowledgment, every secondary acknowledgment corresponds to a primary acknowledgment and the timestamp of a secondary acknowledgment is equal to that of its primary acknowledgment. After some time, will equal and therefore . This is ultimately observed by LP r (because LP s does not change ) which then knows that LP s has seen its acknowledgment. LP r then removes its acknowledgment by changing . After some delay,this change is re?ected in the output of the reduction network as a change in . When LP s observes this change, it knows that LP r has removed its acknowledgment and so it relinquishes the secondary acknowledgment by setting to and the second phase is complete.

At any time during the two phases described above, a new acknowledgment with a timestamp smaller than may be submitted. As a result, will equal this new acknowledgment. This change in may occur either before or after LP s observes the old . In the ?rst case, the ?rst phase of the protocol is preempted; this situation is the same as having several acknowledgments with the new acknowledgment having the smallest timestamp. When the new acknowledgment completes, again becomes the acknowledgment with the smallest timestamp and the ?rst phase is restarted. In the second case, LP s has observed the primary acknowledgment but the second phase is preempted. LP s observes that its message is no longer being acknowledged and consequently relinquishes the second acknowledgment by setting to . After the new acknowledgment completes, reverts to . When LP s observes this,it resumes the second phase by resubmitting its secondary acknowledgment and the second phase ultimately completes. Thus, the algorithm effectively nests acknowledgments.

Batched acknowledgments

From the description above, it is clear that when multiple acknowledgments are submitted at the same time, only the one with the smallest timestamp completes. This serialization of acknowledgments is alleviated by the second enhancement in which a batch of messages is acknowledged using a single physical acknowledgment. This is done by adding a third component to each of ,, and . These T -values now consist of a message timestamp, a message identi?er and a size ?eld. An acknowledging LP searches its unacknowledged message ρr ρi ∞Φ0,,{}ρi ρr ρ′ρr ρr τs ρ′ρi τi ∞Φ0,,{}τi τ′τs ρr τs ρr ρ′τs ∞Φ0,,{}ρr ρ′ρ′ρ′ρr τs ∞Φ0,,{}ρ′ρr ρi τi ρ′τ′

and receiving messages, saving state, etc.) while the AP’s perform all of the synchronization tasks (i.e., they are responsible for submitting state vectors to the PRN, reading the output of the PRN and executing synchronization algorithms). When an HP causes its state to change (by executing an event, receiving a message, etc.) it communicates this change to its AP. The AP incorporates such changes into state vectors which it submits to the PRN for reduction. The AP also reads the globally reduced output state vector produced by the PRN and makes selected subsets of this available to the HP.

The interface between the H P and the AP consists of two unidirectional channels (implemented as buffers). For information ?owing from the HP to the AP, it is required that none of it be lost and that the AP process the information in the order sent out by the HP. For these two reasons, the IN Buffer is a FIFO. In the other direction, the only requirement is that the most recent version of the AP’s output be available to the HP. Accordingly, the OUT Buffer consists of a single cell buffer, which is repeatedly overwritten by the AP each time it presents new data to the HP. We note here that this con?guration is one of many possible implementations of the global computation model of Figure 1. In particular, the choice of having dedicated processors for simulation and synchronization tasks and their asynchronous operation with respect to each other introduces latencies in the data path which are absent in the model of Figure 1. This requires minor adjustment of synchronization algorithms which are based on the model of Figure 1, as will be seen in Section 5.

As mentioned earlier, the PRN and the AP’s operate asynchronously. Occasionally, the AP’s present state vectors (not necessarily all at the same time) to the PRN for reduction. The PRN operates in reduction cycles, reducing a set of state vectors, one from each processor, to a single output state vector in each cycle. Subsequent reduction cycles re-use state vectors for those AP’s which do not present new state vectors. To satisfy these requirements, the design includes a custom interface between the AP’s and the PRN. The AP’s write new state vectors to the IN registers and read globally reduced state vectors from the OUT registers. Likewise, the PRN reads state vectors from the IN registers, reduces them and writes the global output to the OUT registers. The IN and OUT registers are comprised of three sets of registers each which provide the isolation between the AP and the PRN. The detailed operation of these register interfaces is described in [25].

The PRN is a binary tree of height log2n, where n is the number of processors. Figure 4 shows a PRN for eight processors. Each node of the tree contains a general purpose ALU which performs reduction operations on its two operands. The reductions are performed in parallel across all the ALU’s. The PRN interfaces with each processor through the IN and OUT registers. At each LP, the IN registers present a state vector (which may be the same as the one used in the previous cycle) to the PRN. The PRN starts the reduction by reading the ?rst elements of each state vector and reducing them pair-wise at the top row of ALU’s. These ALU’s then pass the reduced values down to the second row of ALU’s. While the second row of ALU’s reduces these values, the top row reduces the second elements of each state vector. Thus, the PRN operates in a pipelined fashion. The time required for a set of values to pass through a single PRN stage is called the minor cycle time, while the time required for the top row ALU’s to read all the elements of the state vectors is called the input cycle time. Note that an input cycle consists of m minor cycles, where m is

antimessages (when they propagate the results of aggressive processing to other LP’s). In this section, we present our algorithm which is a modi?ed version of the algorithm of Figure 2,modi?ed to incorporate the considerations for the two attributes just mentioned. For exposition,our choice of aggressive protocol is Time Warp [10].

The most important synchronization value required by the LP’s in Time Warp is GVT.GVT is required by the LP’s during a simulation for several reasons: fossil collection in systems with limited state saving space, performing I/O, collecting statistics and detecting termination conditions. To date, GVT computation has been an expensive (time consuming) operation. This is an area in which the framework can signi?cantly bene?t aggressive protocols, by rapidly computing and providing the value of GVT. In order to correctly compute GVT using the framework hardware, we have modi?ed the synchronization algorithm of Figure 2 to correctly maintain two local T -values at each LP , the global reduction of which provides GVT.

3.1Aggressive Framework Algorithm

The Aggressive Framework Algorithm (AFA) listed in Figure 5 is designed to asynchronously compute GVT in a Time Warp system. It assumes the computation model of Figure 1. As mentioned in Section 2.2, the framework hardware differs slightly from this model.We will present AFA adapted for the framework hardware in Section 5 and prove its correctness in Section 6.

We assume familiarity with the Time Warp protocol and associated terminology. By de?nition, GVT is the minimum of two T -values: the smallest of the logical clocks of all the LP’s and the smallest unreceived message time among all the LP’s. The original framework algorithm maintains the minimum unreceived message time,, but does not maintain the smallest logical clock value. Instead, it maintains the smallest next event time,. In a non-aggressive system, we can distinguish between two local T -values:, which is the timestamp of the event being executed (or just executed) by LP i (i.e., the logical clock value of LP i ) and , which is the timestamp of the next event that LP i intends to execute. In such a system, LP i will not execute its next event unless it can ascertain its safety. Each LP i submits its for reduction rather than its , so that the LP’s can identify the next safe event(s). In an aggressive system, since LP’s do not wait to ascertain the safety of an event before executing it, we consider only , the logical clock value of LP i . Thus, the algorithm of Figure 5 maintains the following two T -values:

: the minimum of all the logical clocks : the minimum unreceived message time and GVT is de?ned as .

AFA has two concurrent procedures,PROCESS and DO_ACK . The PROCESS procedure invokes three subroutines,SEND_MSG ,RCV_MSG and ROLLBACK as required. One change in AFA from the framework algorithm of Figure 2 is that RCV_MSG is now called as a subroutine by PROCESS instead of being a concurrent procedure invoked when a message is received. This is υ′η′σi ηi ηi σi σi σ′MIN LP i

σi ()=υ′MIN LP i

υi ()=GVT MIN σ′υ′,()=

messages have arrived in the LP’s logical past (i.e. they have to be inserted in the events list before the event just executed - such messages are called stragglers ), then PROCESS calls ROLLBACK , to perform the rollback. On a rollback, the logical clock of the LP is rolled back to the timestamp of the straggler. In the ROLLBACK subroutine, we have chosen to employ aggressive cancellation [26].We show later that lazy cancellation is easily accommodated in this algorithm. At the end of a rollback, the PROCESS procedure checks to see if any new stragglers have arrived during the rollback process which will cause the LP to roll back further. If so,ROLLBACK is called again. Thus,ROLLBACK is invoked repeatedly until all of the newly received messages are in the LP’s logical future (i.e., the LP has rolled back far enough). These messages are now incorporated into the events list of the LP in the RCV_MSG subroutine. The DO_ACK procedure is the same as the CHKACK procedure in Figure 2; it performs acknowledgments of messages using the PRN.

In order to maintain correctly, the timestamps of antimessages must also be part of the GVT computation. To see why, consider a system of two LP’s with LP 0 having at 1000, LP 1having at 1500 and no messages in transit. Therefore, GVT has a value of 1000. Now, LP 0 sends LP 1 an antimessage with timestamp 1000 and then proceeds to execute its next event with timestamp 2000. While the antimessage is in transit, if antimessages are not acknowledged, we have at 2000, at 1500 and both and at , giving a GVT of 1500. When the antimessage is ?nally received by LP 1, falls to 1000, bringing GVT down to 1000. This is an error since GVT must be strictly non-decreasing. The error occurs because the smallest timestamp in the system, which is the timestamp of the antimessage, is momentarily absent from the GVT computation algorithm. The problem is solved by requiring antimessages also to be acknowledged. To do this, the ROLLBACK procedure calls SEND_MSG to send the antimessages.This guarantees that the timestamps of antimessages are accounted for in the value of . In addition, we see that the received message is marked for acknowledgment independent of its sign in RCV_MSG (i.e. antimessages are also acknowledged).

3.2Event preemption and lazy cancellation

We indicate how two optional features of Time Warp (event preemption and lazy cancellation) can be incorporated into AFA. In AFA as presented in Figure 5, the event processing procedure (PROCESS ) checks to see if new messages have arrived after the execution of each event. When one or more of the messages received during the execution of an event have a smaller timestamp than that of the event, an ef?cient implementation will preempt the execution of the event (since it will be rolled back anyway) rather than wait for the execution to complete. To incorporate this into AFA, the RCV_MSG subroutine is activated as a concurrent procedure with PROCESS and DO_ACK whenever the LP receives a message. In addition, at the start of the RCV_MSG procedure, a check is made to see if the received message is in the LP’s logical past and if so, the ROLLBACK procedure is called. In effect, each iteration of the WHILE and FOR loops in PROCESS is transformed into a concurrent invocation of RCV_MSG .

With lazy cancellation, upon rolling back, an LP sends out antimessages only as required.In Figure 5, the ROLLBACK subroutine implements aggressive cancellation, wherein antimessages are sent out during a rollback regardless of whether they are needed or not. To implement lazy cancellation, the FOR loop is removed from the ROLLBACK subroutine. Instead, the PROCESS υi σ0σ1σ0σ1υ0υ1∞σ1υi

procedure must now perform a check after every event execution, to see if any antimessages need to be sent out. If so, the SEND_MSG subroutine is called to actually send the antimessage.

4Correctness

In this section, we establish the correctness of AFA shown in Figure 5. Throughout this paper we are concerned only with temporal correctness. We assume that the simulators under discussion are functionally accurate. By temporal correctness we mean that none of the committed events violate causality constraints. The following criteria de?ne (temporal) correctness of a parallel simulation:

i)each LP must ultimately execute events in strictly non-decreasing timestamp

order [9]

ii)the simulator must make progress if the application being simulated makes progress

The second criterion incorporates the concepts of freedom from deadlock as well as termination.

4.1Correctness of a Time Warp system

For a Time Warp system, it can be proven that the value of GVT never decreases, by considering all of the possible ways in which GVT may change [10]. H ence, GVT de?nes a commitment horizon, i.e. events with timestamps lower than GVT can be committed. Other uses of GVT are fossil collection, gathering statistical data and detecting the occurrence of termination conditions. It is therefore reasonable to assume that every Time Warp LP needs to know the value of GVT periodically. In a typical Time Warp implementation, this is done by actually computing it during the simulation.

An ideal time warp system is one in which the LP’s obtain all the information they need instantaneously. For instance, in an ideal system, when an LP receives a message sent to it by another, the sending LP becomes aware of the fact that the message has been received at the same instant the receiving LP receives the message. Obviously, no implementation can be ideal but every implementation has a corresponding ideal system. Timestamps often remain in an implementation for some time after they have disappeared from the ideal system. Returning to our previous example, typically, the sending LP is noti?ed of the receipt of the message by some form of acknowledgment. Since this acknowledgment takes time to accomplish, the message remains outstanding for a longer period of time in the implementation than in the ideal system. This lag is especially true of implementations in which the GVT computation proceeds asynchronously with the simulation, i.e., the simulation is not suspended when GVT is computed. As a result, the GVT computed by an implementation is usually an approximation of the value of GVT in the corresponding ideal system. At any instant of real time t, we distinguish between the actual value of GVT, GVT a(t), which is the value of GVT in the ideal system at time t (i.e. the smallest timestamp in the ideal system at time t), and the computed value of GVT, GVT c(t), which is the value of GVT made available by the Time Warp implementation at time t.

4.2.2Properties of AP software

The software executing on the AP’s should have the following properties:

P5periodic read

Each AP (and therefore each LP) reads the output of the PRN periodically ,

i.e. no LP will ignore the output of the PRN for an arbitrary amount of

time.

P6?nite delay

No communication in a FIFO remains unprocessed by the AP for an

arbitrary amount of time. This assumes that the communications are

enqueued at an average rate which is lower than the average rate at

which the communications are processed.

4.2.3Properties of AFA

P7set clock on event

An LP sets its to the timestamp of an event before it executes that

event.

P8set unreceived message time

When an LP sends a message, if its is greater than the timestamp of

the message, its will be set to the timestamp of the message;

otherwise, there is no change in its .

P9update unreceived message time

When LP i receives the primary acknowledgment for a message which has

timestamp , that message is removed from its outstanding message list

and its is set to the smallest among the timestamps of the remaining

messages in its outstanding message list.

P10unreceived is minimum

At any LP i , is equal to the smallest of the timestamps of the messages

in its outstanding message list. This follows directly from P8 (set

unreceived message time) and P9 (update unreceived message time).

P11maintain unreceived message time

When an LP sends a message M , its is less than or equal to the

timestamp of M (from P8 (set unreceived message time)). In addition, it

will remain so until the LP observes the primary acknowledgment for M .

P12set clock on message

When an LP receives a message, its is made less than or equal to the

timestamp of the message before the acknowledgment of the message is

initiated.

σi υi υi υi υi υi υi υi σi

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绚烂的近义词和反义词 导读:星星使天空绚烂夺目;知识使人增长才干。造烛求明,读书求理。如下是小编给大家整理的绚烂的近义词和反义词,希望对大家有所作用。 【近义词】:奇丽、富丽、灿烂、烂漫、瑰丽、璀璨、绚丽、绮丽、艳丽、辉煌、鲜丽、鲜艳、 【反义词】:暗淡光色昏暗灯光暗淡色调暗淡不光明暗淡的前景前途暗淡 绚烂的造句: 1、枫叶是秋天这本大书中最绚烂的一页。 2、春天是初生的旭日,生机勃勃;春天是美丽的姑娘,温柔妩媚;春天是重生的希望,绚烂无比。 3、落叶,是你点缀了秋天的丛林,那丛林就像一幅山水画卷,使秋天更加绚烂,是你像舞蹈家一样快乐的舞蹈,像蝴蝶一样飘飞,才给我们带来了美的感受。 4、摘一根美丽绚烂的花,戴在发间,折一根清翠的柳枝系在腰间。我们常常穿梭于密草林间,寻觅所谓的幸福踪影,其实,幸福时刻萦绕在我们身边。 5、让我们用艺术的水彩渲染浮云的斑澜。让我们用文化的魅力点缀阳光的绚烂。 6、今天是你的生日,让荡漾的春风把烦恼统统送走,让浪漫的春雨把梦想慢慢滋润,让绚烂的春花把生活好好装扮,让温暖的春天

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白领阶层,在遮遮掩掩进进退退中做着迷茫的爱情游戏。 他说,他们不应低估自身能力,也不要自怨自艾,哀叹命运。 碧丝自怨自艾说自己没有孩子,竟然不懂得庆幸没有孩子才是她前世修来的,日子可以过得比较好。 对自己的恶习他常自怨自艾。 人们想让鲍什知道,他的讲话让他们不再自怨自艾,帮助他们走出离婚的阴影,或更加重视家庭。 精彩的年华已经过去,剩下的只是自怨自艾以及年老珠黄时对生命不甘心而又不得不忍受的痛。 10.愚痴邪见即是自造恶业;懒惰懈怠就是自毁前程;自怨自艾终究于事无补;怨天尤人只会更加坏事。 1怜悯中长大的孩子,将来容易自怨自艾。 1我懂了,当你自怨自艾时,幸福就会停靠在别的港湾。

1忠诚就体现在这里,沈从文没有因委屈而自怨自艾,他认从了这不公平的待遇,很平静地转向了文物研究。 1现在的我们不再自怨自艾,不会再陷在改变不了的悲剧里不能自拔。 1但倘有不知道自怨自艾的人,想将这位先生"送进疯人院"去,我可要拚命反对,尽力呼冤的。 1他和我的日程安排者兰迪。怀特都提醒我,在我自怨自艾的同时,我还应该对我的职员以及他们未来的安康表现出更多的关心。 1当我们有朋友陷入情绪低落、自怨自艾和挫折失败的时候,这无疑是一个极好的范例,让我们可学以致用,运用当中的原则来扶立他。 1如果一个人过度放纵自己,就会陷入自怨自艾之中。 1这个问题的关键不是因为你不能事事精通就自怨自艾,对自己苛刻,而是找出你的强项和对你真正重要的东西。 20.花时间整理评估下你目前的状况,但是不要自怨自艾。 2第一个生日,我在忽而愤怒,忽而无限地自怨自艾中度过。

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