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| Graph of Optimization of cost |
Characteristics of Queuing Model
- System : The place where the customer arrives in order to get service is called system.
- Arrival Rate or Arrival Pattern: The average number of customers arriving per unit time within the system to get service is called arrival rate. Arrival is random and if we assume that the arrival is governed by Poisson's distribution then the time between arrivals is exponentially distributed.
For a given arrival rate (λ) ,
\[ P(x) = \frac{e^{-λ} λ^x}{x!}\]
For x = 0,1,2,3,......
P(x) : Probability of x arrival
x: Number of arrival per unit time
The corresponding exponential distribution for inter arrival time is given by
\[P(t) = λ e^{-λt}\]
- Service Rate or Service Pattern : The number of customers serviced per unit time is called service rate. There is no standard probability distribution for service pattern but it is assumed to follow exponential distribution for simplicity.
- Service Rule or Service Discipline: It is about how customers are picked from queue for service. Examples: First Come First Serve (FCFS), Last in first Out(LIFO), Service in Random Order (SIRO) etc.
- Calling Population: The entire sample of customers from which a few visit the service facility is known as calling population. It is assumed finite if arrival rate depends on the number of customers being served or waiting and assumed if arrival rate is independent of number of customers being served or waiting.
Customers Behavior
- Jockey: When customer switches the queue in hope to get service faster.
- Reneging: Customer leaves the system without getting service.
- Balking: If queue is very long, customer decides not to join the queue.
- Cheater: Customer takes illegal means like bribing in hope to to get service faster.
Representation of Queuing Model
Queuing models are represented by Kendall Lee Notaion:
(a/b/c) : (d/e/f)
where, a= probability distribution for arrival pattern
b= probability distribution for service pattern
c = number of servers
d= service rule or service discipline
e= size of system
f= size calling population
Single Server Queuing Model (M/M/1) : (FIFO/∞/∞)
λ = Arrival rate (Poisson Distribution)
\( \frac{1}{λ}\) = Inter arrival rate (Exponential) or time gap between two consecutive arrivals
μ = Service rate (exponential)
If λ > μ then queue length will keep on increasing and at last system will fail.
If λ < μ then System works:
Traffic Intensity(ρ)
The ratio of arrival rate to service rate is called traffic intensity. It is also known as utilization factor or channel efficiency and clearing ratio.
\[ ρ = \frac{λ}{μ}\]
It tells the percentage time server is busy and probability that a customer has to wait.
Some Important formulas
1. Probability that the service facility is idle or probability of zero customers in the system
\[ P_o = 1 - ρ\]
2. Probability of 'n' customers in the system
\[ P_n = ρ^n . P_o\]
\[P_o + P_1 + P_2 + P_3 + .......... = 1\]
3. Average number of customers in the system: It includes both the customers waiting in the queue and customers which are provided service.
\[ L_s = \sum_{n=0}^{\infty} n. P_n\]
\[ L_s = \frac{ρ}{1 - ρ}\]
4. Average number of customers in the queue: It doesn't include the customers which are provided service.
\[ L_q = \frac{ρ^2}{1 - ρ}\]
5. Average length of non-empty queue or at least one customer in the queue
\[ L_n = \frac{1}{1 - ρ}\]
Little's Law
It states that average number of customers in the queue or system is equal to average customers arrival rate multiplied by average waiting time in the queue or system respectively.
\[L_q = λ.W_q , L_s = λ.W_s\]
6. Waiting time of the customer in system
\[ W_s = \frac{L_s}{λ} = \frac{1}{μ - λ}\]
7. Waiting time time of the customer in queue
\[ W_q = \frac{L_q}{λ} = W_s - \frac{1}{μ}\]
7. Probability of 'n' arrivals in time t''
\[ P(n, t) = \frac{e^{-λt}{λt}^n}{n!}\]
8. Probability that the waiting time in the queue is greater than or equal to 't'
\[ P(W_q \geq t) = \frac{λ}{μ}e^{-(μ - λ)t}\]
9. Probability that the waiting time in the system is greater than or equal to 't'
\[ P(W_s \geq t) = e^{-(μ - λ)t}\]
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