Queuing theory is a classical mathematical method to analyze the congestions and delays of waiting in line. It deals with number of customers waiting in line and many other aspects of queues. Our objective of the study has to develop a new fuzzy queuing model. The study intends to examine the long-run behavior and determination of the steadiness of queuing system. For this purpose the Buckley method has been applied. In this study Buckley method has extended using the trapezoidal shaped fuzzy numbers which are introduced to find the explicit formula for membership function of the fuzzy estimation to develop a new fuzzy queuing model. Fuzzy approach has been applied for measuring different linguistic factors in queuing systems. The results concluded that fuzzy queueing model is stable and bounds obtained by Buckley estimates are at least good as compare to bounds obtained by membership function.