BARRIER FUNCTIONS AND THEIR MODIFICATIONS
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BACKWARD CHAINING See Expert systems.
BACKWARD KOLMOGOROV EQUATIONS In a continuous-time Markov chain with state X(t) at time t, define p;j(t) as the probability that X(t + s) = j, given thatX(s) = i, s, t;:::: 0, and r;i as the transition rate out of state i to state j. Then Kolmogorov's backward equations say that, for all states i, j and times t ;:::: 0, the derivatives dp;j(t)ldt = Lk.,irik Pkj(t) - V;p;j(t), where V; is the transition rate out of state i, V; = Ljrij· See Markov chains; Markov processes.
BACKWARD CHAINING An approach to reasoning in which an inference engine endeavors to find a value for an overall goal by recursively finding values for subgoals. At any point in the recursion, the effort of finding a value for the immediate goal involves examining rule conclusions to identify those rules that could possibly establish a value for that goal. An unknown variable in the premise of one of these candidate rules becomes a new subgoal for recursion purposes.
BACKWARD-RECURRENCE TIME Suppose events occur at times Tb T2 , • • • such that the interevent times Tk - Tk-l are mutually independent, positive random variables with a common cumulative distribution function. Choose an arbitrary
time t. The backward recurrence time at t is the elapsed time since the most recent occurrence of an event prior to t.
BALANCE EQUATIONS (1) In probability modeling, steady-state systems of equations for the state probabilities of a stochastic process found by equating transition rates. For Markov chains, such equations can be derived from the Kolmogorov differential equations or from the fact that the flow rate into a system state or level must equal the rate out of that state or level for steady state to be achieved. (2) In linear programming (usually referring to a production process model), constraints that express the equality of inflows and outflows of material. See Markov chains.
BALKING When customers arriving at a queueing system decide not to join the line and instead go away because they anticipate too long a wait.
BANKING Stavros A. Zenios University of Cyprus, Nicosia & University of Pennsylvania, Philadelphia
Operations research and management science techniques find applications in numerous and diverse areas of operation in a banking institution. Applications include, for example, the use of data-driven models to measure the operating efficiency of bank branches through data envelopment analysis, the use of image recognition techniques for check processing, the use of artificial neural networks for evaluating loan applications, and the use of facility location theory for opening new branches and placing automatic teller machines (e.g., Harker and Zenios, 1999). A primary area of application, however, is that of financial risk control in developing broad asset/ liability management strategies. Papers that summarize these areas are Zenios (1993), Jarrow et al. (1994), and Ziemba and Mulvey (1998). This work can be classified into three categories: (1) pricing
S. I. Gass et.al., (eds.), Encyclopedia of
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