Neutrosophic Event-Based Queueing Model

In this paper we have defined the concept of neutrosophic queueing systems and defined its neutrosophic performance measures. An important application of neutrosophic logic in queueing systems we face in real life were discussed, that is the neutrosophic events accuring times, because of its wide applications in networking and simulating communication systems specialy when probability distribution is not known, and because it’s more realistic to consider and to not ignore the imprecise events times. Event-based table of a neutrosophic queueing system was presented and its neutrosophic performance measures were derived, i.e. neutrosophic mean waiting time in queue, neutrosophic mean waiting time in system, neutrosophic expected number of customers in queue and neutrosophic expected number of customers in system. Neutrosophic Little’s Formulas (NLF) were also defined which is a main tool in queueing systems problems to make it easier finding performance measures from each other.

based table and NLF was introduced, then a solved example was presented to show the power of neutrosophic crisp sets [3,5,18] in solving problems better and more accurate than in classical crisp logic.

Definitions and notions
Suppose that the queueing system consists of one waiting line. Arriving customer takes his service from only one server. If the server is busy then arriving customer waits in the queue until the server is empty. The server can serve one customer at time according to first comes first served policy. The customer departs after he takes his service.
Let by definition:  ( ) denotes the neutrosophic time of customer (n) arrival. denotes the neutrosophic expected number of customers in system. NLF will be then (based on classical little's formula in [14,15]): = * = * Where is the arrival rate, that is total number of arrivals divided by the time between last departure and starting time.
Among this paper, neutrosophic statistical numbers presented in [19] will be used, which is an effective method to describe indeterminacy in data specially data that it is not useful to be presented in the form (T, I, F) like parameters of distributions, statistical data, statistical data tables, frequency tables, time series, etc [4, 5, 10 ,19, 20].

Example
Suppose that we have the following event-based table of a queueing system representing arriving customers to an ATM (all times are in minutes):

Neutrosophic Solution:
For customer 1: we start when the system is empty so ( ) = 0 that is the arrival time of ther first customer.
Since the system is empty then the arrival customer will be served without any waiting so ( ) = 0. Interarrival time between first and next customer is For customer 2: ( ) = [3,4] that is the arrival time of the second customer.
The service starting time of this customer (customer 2) will be its arriving time if the server is empty or it will be immediately after the departure time of the previous customer if the server is busy, so it's Interarrival time between third and forth customer is And so on.. We can form the following table:  Which means that customer will stay in the system between 2.17 and 2.833 that is between 2 mins and 10 secs and 2 mins and 50 secs from the time he arrives until he departs from the system.
We have also = = 0.375 that is the arrival rate because we totally have 6 customers arrived during 16 minutes. That means expected number of customers in the system will be between 0.81375 and 1.062375 customers. That means expected number of custoemrs in queue will range between 0.12375 and 0.25125 customers.

Crisp Solution:
To solve this kind of problems using crisp queues we may take midpoints of intervals, so:  Table 3: Input data converted into crisp data For customer 1: Since the system is empty then the arrival customer will be served without any waiting so ( ) = 0.
The service time for this customer is approximatley ( ) = 1.5 so the departure time will be approximately ( ) = ( ) + ( ) = 0 + 1.5 = 1.5 that is the apprpximated time of start serving the customer plus the approximated serving time.
The waiting time in queue is the difference between starting serving the customer ( ) and arriving time of this customer ( ) so : Interarrival time between first and next customer is approximately ( ) = ( ) − ( ) = 3.5 − 0 = 3.5 For customer 2: ( ) = 3.5 that is the approximated arrival time of the second customer.
The service starting time of this customer will be its arriving time if the server is empty or it will be immediately after the departure time of the previous customer if the server is busy, so it's ( ) = max ( ) , ( ) = max(1.5,3.5) = 3.5 The service time for this customer is ( ) = 4 so the departure time will be ( ) = ( ) + ( ) = 3.5 + 4 = 7.5 .
The waiting time in queue is the difference between starting serving the customer ( ) and arriving time of this customer ( ) so :  We notice that neutrosophic solutions are more accurate and realistic than crisp solutions.

Conclusions
In this article, we discussed basics of neutrosophic queueing theory and have shown the power of neutrosophic logic in modelling queues with imprecise and incomplete inputs in even-based tables of queues. We solved an example which contains indetermined times of arrivals, departures, services and services starting times and calculated the neutrosophic mean waiting time in queue, neutrosophic mean waiting time in system, neutrosophic expected number of customers in queue and neutrosophic expected number of customers in system then compared it to crisp solutions.
We found that neutrosophic solutions can be considered as an extension to crisp solutions, also neutrosophic solutions are more accurate than crisp solutions.
In future work author looking forward to study more complex applications of neutrosophic logic to more genereal queueing problems like bulk queues, balk queues, networks of queues and impatient customers behaviour in queues like jockeying and reneging.