Study of Fuzziefied Single Server Queueing Systems with Vacation

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Single Server with Vacations

A queuing system in which a server may be turned off is said to be a system with vacations.

[pic 3]

Figure 1. Queuing system with vacations

Inter arrival Rate ():- The arrival rate is the number of arrivals per unit of time. The inter arrival time is the time between each arrival into the system and the next.[pic 4]

Service Rate ():- Service rate denotes the rate at which customers are being served in a system. It is the reciprocal of the service time.[pic 5]

Service Cost ():- Cost-of-service pricing is the setting of a price for a service based on the costs incurred in providing it.[pic 6]

Switch-on Cost ():- When we turn on the server after a vacation then this cost occurs.[pic 7]

Holding Cost ():- The cost of carrying or holding inventory is the sum of the following costs: Money tied up in inventory, such as the cost of capital or the opportunity cost of the money. Physical space occupied by the inventory including rent, depreciation, utility costs, insurance, taxes, etc.[pic 8]

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Heyman Queuing Theory

Heyman in 1968 has proved that it is optimal to keep the server always on when:-

[pic 9]

Figure 2. Heyman Queuing Equation

Where, is the traffic intensity. It’s beneficial to keep the server always on when the switch-off cost is sufficiently larger than the service cost rate .[pic 10][pic 11][pic 12]

State Evaluation

At any time , the system may be empty or non-empty and the server may be on (busy or idle) or off. In the latter case, we say that the server is dormant. The server is idle if it is kept on although there are on customers in the system. [pic 13]

is the state of the server at time , taking the value 1 if the server is on or 0 if the server is off.[pic 14][pic 15]

is the number of customers in the system at time .[pic 16][pic 17]

The state variable changes whenever a new customer arrives or the server completes service. The state variable changes whenever we decide to switch the server on or off. [pic 18][pic 19]

The average cost rate of the system over an infinite time horizon can be expressed as:-

[pic 20]

In order to treat switching costs, we introduce one more parameter , which is the accumulated holding cost within the current server state. The cost is given by:-[pic 21][pic 22]

[pic 23]

Where is the total time the server rests in the current state starting from the last time it was switched to that state, is the th consecutive time unit within the current server state.[pic 24][pic 25][pic 26][pic 27]

Derivation of Decision Criteria

Three Distinct Cases:

- If there are no switching costs, it is trivially optimal to turn on the server when a customer arrives. The existence of a switch-on cost may lead to a delay in turning on the server even though there may be customers present in the system. The optimal turn-on time will be determined by the relationships between holding costs and switching costs. When the accumulation of holding cost during a vacation period is high enough to compete with the switching cost, it is optimal to turn on the server.[pic 28]

- The higher the , the easier it is to make the decision to turn the server on. [pic 29]

- If , keeping the server always off achieves the minimum cost , although the number of customers in the system explodes as . Again, this is a trivial solution; hence, we assume that . The higher the , the easier it is to make the decision to turn the server on. [pic 30][pic 31][pic 32][pic 33][pic 34]

It is interesting to note that the rule base is independent of the service cost rate . This is because of the property of the removable server model where the single server under a long-run criterion is busy with probability at any time instant, independent of the service cost rate.[pic 35][pic 36]

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Rule Base

The inputs of rule base are the parameters , and . Each parameter is represented by four linguistic values, ZO, PS, PM, and PB, which stand for zero, positive small, positive medium and positive big. The complete rule base comprises 64 rules. The output of each rule, denoted by , is the decision concerning whether to turn the server on and it is represented by YES and NO.[pic 37][pic 38][pic 39][pic 40]

To complete the rule base, we applied the following heuristic procedure. To each linguistic value we assign an integer weight: ZO →0, PS →1, PM →2, PB →3. For each rule we calculate the sum of weights of its input parameters. We set d = NO, if sum is less than or equal to 2; otherwise we set d = YES.

[pic 41]

Figure 3. Single Server queueing model with vacation rule base

Mamdami’s implication can be used to determine the grade of output parameter along with height method of defuzzification to determine the crisp output data. Here, I’m using Mamdani implication due to it is widely acceptance.

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Fuzzy Inference System

Fuzzy inference is a method that interprets the values in the input vector and, based on

User-defined rules, assigns values to the output vector. Using the graphical user interface (GUI) editors and viewers in the Fuzzy Logic Toolbox, one can build the rules set, define the membership functions, and analyse the behaviour of a fuzzy inference system (FIS). The following editors and viewers are provided:

FIS