The Gate Distribution

Gates that transfer objects within their capacity, without formation of queues, are considered in a combinatorial framework. The number of different ways, I m,n, ( ) , n objects can pass randomly an array of m gates, each with constrained capacity , is used to characterize its performance. Two gate distributions were formulated and their properties studied. The first, n dependent distribution was found to be symmetric, whereas the second, dependent distribution is skew. These properties persist also in the case of multivariate gate distributions. In the latter case, the expectation and variance turn to have additive properties, respectively. INTRODUCTION AND STATEMENT OF PROBLEM Many service systems provide access or exist in the form of gates. This is typical of arrays of cashiers in large retail and discount stores, such as supermarkets and warehouses, cashiers and gates in stadiums and amusement parks, toll plazas, tellers in banks, security checks in airports, and services provided by local and national authorities to the public in municipality and government agencies. Gates are commonplace in computational hardware. Systems that transfer information and facilitate communication, e.g. telephone exchange centers, can be also viewed as transfer gates for incoming calls. Devices for purification of materials often rely on differential capacity for mass transfer, whereby the slower material is separated across a barrier from the faster one. Gate systems can be characterized by their number, capacity per gate and variability, e.g. if there are different gates in the array. We define an ideal gate system as one that operates within its capacity range so that no queues, delays or blockage of passage can occur. This is an ideal situation that guarantees customer or user satisfaction, and hence must be the target of any advanced design. If the approach to the array of gates is random, then its performance is enhanced with an increase in the number of ways it can be passed within the limits of its capacity. Thus, we are interested in the conditions that provide the maximum number of ways in which an assembly of identical objects can pass the gate array randomly, without exceeding its capacity and avoiding the formation of queues. We explore the sensitivity of the array to a unit change in the object transfer capacity per gate. We also consider heterogeneous arrays comprising sections of gates with different capacities. The studied gate systems is viewed in the context of two new distributions, which are functions of the number of gates, their specific capacity, and the number of passing objects or entities. In the first gate distribution, the number of passing objects serves as the random variable, while the number of *Address correspondence to this author at the Department of Civil and Environmental Engineering, Technion – Israel Institute of Technology, Haifa 32000, Israel; E-mail: [email protected] gates and the capacity per gate are set as system parameters. In the second gate distribution, the capacity per gate serves as the random variable, while the number of gates and passing objects turn system parameters. The first gate distribution is shown to be symmetric for uniform as well as heterogeneous gate arrays. In contrast, the second gate distribution is skew, irrespective of the gate array being uniform or not. Furthermore, it is also self shifting with respect to the range wherein it is defined. The symmetry of the first gate distribution facilitates a good fit, in a wide range of the number of gates, capacity limit and passing objects, with the normal distribution having the same expectation and variance. Finally, these gate distribution facilitate analysis and optimization of performance regarding passage of objects in a mode of queueless operations. OBJECTIVE OF WORK 1. In how many ways, I(m,n, ), can n objects pass m gates per unit time subject to the constraint that the maximum allowable capacity of each gate is objects per unit time. Note that the variable time can be replaced by any other measuring unit such as length and area, depending on the nature of the problem. 2. Identify and formulate the relevant distributions associated with I(m,n, ), as a function of m,n, , with respect to their significance and properties. In item 1, it is implied that as long as the capacity of a gate is not exceeded, there is no queuing. A queue is formed once the capacity is exceeded, but according to the definition of I(m,n, ), this is not allowed. THEORY The expression of I(m, n , ), I m,n, ( ) = 1 ( ) i i m ( ) m 1 m+n i 1 , = +1 , n m i=0 n