A shared cluster contains multiple autonomous schedulers, which compete for shared but limited resources and have the incentive to request more resource containers to increase its QoS as we discussed in Section 5.1.2. As such, the scheduling scenario in a shared cluster is best modelled as an non-cooperative
game among rational and strategic players (schedulers). The players are rational because they want to maximize their own gain. They are strategic because they can choose their strategies(i.e., scheduling decisions) that influence other players.
In the game theory, the payoff of a player depends not only on his own strategy, but also on another player’s strategy. A popular way of characterizing this dynamics is through Nash Equilibrium (N.E). The payoff of one player is
5. Job Scheduling in the Cloud
dependent on the choice of strategies. A player might decide to unilaterally switch his strategy to improve his payoff. This switch in the strategy will affect other players by changing their payoff. Therefore, other players might decide to shift their strategies as well, which in turn affects the player who originates the change of strategy. The collection of players is regarded as being at the N.E if no player can improve his payoff by unilaterally switching his strategy.
Pure & Mixed Strategy
Although each player has a set of available strategies to choose, sometimes the player will only choose one of the strategies as it has the maximum payoff, which is called the pure strategy in game theory. However, the agreement on pure strategy among players is not always guaranteed. In this situation, the players can select a strategy by randomizing over the set of pure strategies based on a certain probability distribution. This is called the mixed strategy, in which the N.E is guaranteed. We define the set of mixed strategies for playeri to be
Si =Q(Ai). Then, the set of mixed strategy profile is simply the Cartesian
product of the individual mixed strategy set,{S1× · · · ×Sn}. si(ai) denotes the
probability that a pure strategy will be selected under the mixed strategy si.
The set of the pure strategies that are assigned positive probability forms the mixed strategy si, which is called the support of si. Namely, the support of a
mixed strategysi is the set of pure strategies{ai|si(ai)≥0 andPsi(ai) = 1}.
Expected Utility Payoff
Due to the randomness of mixed strategy, we use the idea of expected utility from decision theory to represent the payoff of a mixed strategy. For a given game,G(k, ~A, ~u), wherekis the number of players in this game,A~ is the vector of each player’s strategies set, and~uis the vector of expected utility cost for each player with mixed strategy profile{si= (s1,· · · , sn)}. Therefore, the expected
5. Job Scheduling in the Cloud ui(si) = X ai∈Ai ui(ai) n Y j=1 sj(aj) (5.1)
Note that a pure strategy is a special case of a mixed strategy, because it assigned one specific strategy with probability one and others with probability zero. The expected utility payoff of pure strategy can be treated in the same way. Therefore, we focus on getting the mixed strategy as our strategy profile in this work.
Strategy Profile with Nash Equilibrium
Now we will look at the game from the perspective of an individual player instead of the outside supervisor. The purpose of each player is to maximize his expected payoff. This expected payoff not only depends on the strategy chosen by himself, but also on the strategy chosen by his competitors. Thus, this player would know how to choose his best response if he knows the strategies that his competitors are going to play. Specifically, player i’s best response tos−i, his
competitors’ strategy profile, is a strategy profiles∗
i ∈si, and the expected off ui(s∗i, s−i)≥ui(si, s−i).
Because none of the players can know what strategies his competitors would adopt, it is not practical to deliver the best response for the players. However, we can leverage the idea of best response to define the most important concept in the non-cooperative game,N.E, which is a strategy profile{s= (s∗1,· · ·, s∗k)}
if and only if all player play his best response to others. For example, a given two-player(p1andp2) game, each player has two strategies with his own pay-off matrixAor B, respectively. The pair of mixed strategies (p, q), one for player
p1 and one for player p2, respectively, is a N.E if all other mixed strategiesp0
for playerp1 will bep0·A·q≤p·A·qand for all other mixed strategiesq0 for
player p2 will bep·B·q0 ≤p·B·q. Two equations indicate those two players
cannot improve its payoffs by switching their mixed strategy fromp, q to any other mixed strategy p0 or q0.
5. Job Scheduling in the Cloud job decision scheduled job ini.alisa.on job
execu.on comple.onjob
conflicted/rejected
running runningupdate Scheduling cost Servicing cost
.me line
Figure 5.5: The life cycle of submitted job in the shared cluster
Therefore, the N.E is a stable strategy profile we want to achieve in this strategical scenario as no player would want to alter his strategy and all of the players play the best response to against others’ strategy.