Modeling of sufficiency of the mobile emergency units in case of situations of high complexity

Abstract

The system of response on the emergency situations is represented here. This article also represents complicated cases when involvement of several types of mobile units is required to react on different emergency situations. Firemen, rescuers, ambulance or police can be involved as mobile units. Mathematical models which describe the work of this system with involvement of two or three units of different types are developed here. These models allow to define necessary amount of mobile units to liquidate dangerous situations with guaranteed probability and these models also allow to define the estimate of probability of the system conditions “dangerous situations - emergency services units”. This object is solved by using the mathematical apparatus of exploring the operations and theory of mass service. Solving of this object intends to arrange the list of conditions { S } of the explored system. The time average { t } of transition from one condition to another is also defined from the analysis of the system. If we set that the quantities { t } are accidental and they are subordinated by the exponential laws of arrangements, we can get vectors of intensity of transitions {l} and {m}, which are the characteristics of the exponential law of arrangements. Vector {l} determines the frequency of occurrence of dangerous situations. It consists of: l₁ - frequency of occurrence of situations which demand involving one mobile unit; l₂ - two mobile units; l₃ - three mobile units etc. Vector {m} reflects speeds of liquidation of dangerous situations and it contains quantities: m₁ - speed of liquidation of situations which demands involving one mobile unit; m₂ - speed of liquidation of situations which demands involving two mobile units; m₃ - three mobile units etc. Diagrams of transition for two or three different types of involving units are shown in this article. The system of equations in the presumption about stationarity of processes is shown in this article. They are corresponded to the diagrams of transitions and they contain probabilities { p } of conditions { S }. Critical probabilities of refusals are also shown here: p_refusal - probability of condition S_refusal , when all the units are occupied and no mobile units can be sent to emersed dangerous situations. The possibility of solving the objects of analysis and synthesis of the given system is also shown in this article.