The purpose of the exercise is to indicate the general performance characteristics of the banking information processing system.

Queueing Theory: Bomber landing analysis

The following formulas come from the study: Hillier F.S., Lieberman G.J. Instruction of Operation Research. New York: McGrew Hill 1990 oraz Z. Jędrzejczyk, K.Kukuła, J.Skrzypek, A. Walkosz Badania operacyjne w przykładach i zadaniach, Wydawnictwo Naukowe PWN

Arrival rate: λ

These are transactions arriving within a minute. It is a stream of transaction orders sent from bank customers sent online.

import numpy as np


np.random.seed(148)  
Λ1 = (np.random.random_sample(2354)*4690)
Λ2 = (np.random.random_sample(4899)*2319)
Λ3 = (np.random.random_sample(3511)*5491)

a1= sum(Λ1)
p1= len(Λ1)
λ1 = 1/(p1/a1)

a2= sum(Λ2)
p2= len(Λ2)
λ2 = 1/(p2/a2)

a3= sum(Λ3)
p3= len(Λ3)
λ3 = 1/(p3/a3)


print('Number of transactions:     ', p1)
print('Sume of transactions time: 
print('Arrival_λ1: 
print('-----------------------------------------------')
print('Number of transactions:     ', p2)
print('Sume of transactions time: 
print('Arrival_λ2: 
print('-----------------------------------------------')
print('Number of transactions:     ', p3)
print('Sume of transactions time: 
print('Arrival_λ3: 
print('-----------------------------------------------')

Service rate: μ

This parameter represents the execution time of individual banking transactions by the banking transaction system.Each of the three data centers has a number of servers. A single server is treated as an individual information flow channel.

r1 = 5
r2 = 6
r3 = 8

np.random.seed(148)  
M1 = (np.random.random_sample(2354)*959)
M2 = (np.random.random_sample(4899)*790)
M3 = (np.random.random_sample(3511)*903)

a1= sum(M1)
p1= len(M1)
μ1 = 1/(p1/a1)

a2= sum(M2)
p2= len(M2)
μ2 = 1/(p2/a2)

a3= sum(M3)
p3= len(M3)
μ3 = 1/(p3/a3)


print('Number of transactions:     ', p1)
print('Sume of service time: 
print('Service_μ: 
print('-----------------------------------------------')
print('Number of transactions:     ', p2)
print('Sume of service time: 
print('Service_μ: 
print('-----------------------------------------------')
print('Number of transactions:     ', p3)
print('Sume of service time: 
print('Service_μ: 
print('-----------------------------------------------')

Traffic intensity: ϱ

Traffic intensity is the ratio of arrival rate to service rate.

$$\text{Traffic intensity equation:} \\
\bbox[#f3f3f3,12px,border:1px solid black] {
{ϱ}=\frac{λ}{{μ}*r}
} \qquad (1)
$$

ϱ1 = λ1/(μ1*r1)

if ϱ1 >= 1:
        print("The system is unstable, we do not carry out further calculations!")
        print('Arrival rate λ1 > service rate μ1, ϱ1= 
        
else:     
        print("The system is stable, you can calculate the probabilities!")
        print('Arrival rate λ1 > service rate μ1')
        print('Arrival rate λ1:      
        print('Service rate μ1:      
        print('Traffic intensity ϱ1: 

ϱ2 = λ2/(μ2*r2)


if ϱ2 >= 1:
        print("The system is unstable, we do not carry out further calculations!")
        print('Arrival rate λ2 > service rate μ2 ,ϱ2= 
        
else:     
        print("The system is stable, you can calculate the probabilities!")
        print('Arrival rate λ2 > service rate μ2')
        print('Arrival rate λ2:      
        print('Service rate μ2:      
        print('Traffic intensity ϱ2: 

ϱ3 = λ3/(μ3*r3)


if ϱ3 >= 1:
        print("The system is unstable, we do not carry out further calculations!")
        print('Arrival rate λ3 > service rate μ3, ϱ3= 
        
else:     
        print("The system is stable, you can calculate the probabilities!")
        print('Arrival rate λ3 > service rate μ3')
        print('Arrival rate λ3:      
        print('Service rate μ3:      
        print('Traffic intensity ϱ3: 

Multiple service channel

$\begin{align}
λ < rμ &&\text{where r is the number of service channels} \tag 5\\
\end{align}$

$$\text{The probability that there are no queues at r service stations:} \\
\bbox[#f3f3f3,12px,border:1px solid black] {
p(n=0) = \displaystyle\frac{1}{sum_{n=1}^\infty}
} \qquad (6)
$$

The probability that there are no queues at r service stations.

$$\text{The probability that there is no queue n = 0 in the system:} \\
\bbox[#f3f3f3,12px,border:1px solid black] {
p(n=0) =\frac{1}{\displaystyle\sum_{i=0}^{r-1} \frac{?^i}{i!}+\frac{?^r}{{\left(r-{?}\right)}{\left(r-1\right)!}}}
} \qquad (7)
$$

$$\text{Average number of items waiting in queue:} \\
\bbox[#f3f3f3,12px,border:1px solid black] {
ψ =\frac{{?^{r+1}{p(n=0)}}}{{(r-?)}^2{(r-1)!}}
} \qquad (8)
$$

$$\text{The probability that there is no queue n = 0 in the system:} \\
\bbox[#f3f3f3,12px,border:1px solid black] {p(n) =\begin{cases}
\frac{{ψ^n}{p(n=0)}}{n!}, & \text{if $n\le r$}
\frac{{r^{r-n}}{ψ^n}{p(n=0)}}{r!}, & \text{if $n\gt r$}
\end{cases}
} \qquad (9)
$$

The probability that the system has more than zero elements and elements is more than r channels.

$$\text{The probability that there is no queue n = 0 in the system:} \\
\bbox[#f3f3f3,12px,border:1px solid black] {
p(n>n_0) =\frac{{r^{r-n_{0}}}{?^{n_{0}+1}}{p(n=0)}}{(r-{?})r!}
} \qquad (10)
$$

$$\text{The probability that the waiting time of an item in the queue is longer n that t.:} \\
\bbox[#f3f3f3,12px,border:1px solid black] {
p(t>t_0) ={{p{(n>r-1)}}{e^{μt(r-ψ)}}}
} \qquad (11)
$$

λ=λ1
μ=μ1
r=r1

We check for inequality:

λ<rμ

print('Arrival rate λ:      
print('Service rate μ:      
ϱ = λ/(r*μ)
print('Traffic intensity ϱ: 

if ϱ > 1:
        print("The system is unstable, we do not carry out further calculations!")
        print('Arrival rate λ > split service rate μ')
else:
        print("The system is stable, you can calculate the probabilities!")
        print('Arrival rate λ > split service rate μ')

def factorial(n): 
    return 1 if (n==1 or n==0) else n * factorial(n - 1);  

Since there are two airports, the time to accept the plane is 2*μ

Probability of no queues, pattern no. 7

$${
p(n=0) =\frac{1}{\displaystyle\sum_{i=0}^{r-1} \frac{?^i}{i!}+\frac{?^r}{{\left(r-{?}\right)}{\left(r-1\right)!}}}
}\qquad (7)$$

p7=1/((1+ϱ)+(((ϱ**r)/((r-ϱ)*factorial(r-1)))))
print('Probability of no queues ?(?=0)≈ 

Probability that you will have to wait in queue, template no. 10

$${
p(n>n_0) =\frac{{r^{r-n_{0}}}{?^{n_{0}+1}}{p(n=0)}}{(r-{?})r!}
} \qquad (10)$$

if ϱ > 1:
        print("The system is unstable, we do not carry out further calculations!")
        print('Arrival rate λ > split service rate μ')
else:
        n0 = 1
        p10 = ((r**(r-n0))*(?**(n0+1))*p7)/((r-?)*factorial(r))
        print('Probability to wait in queue ?(?>?0)≈ 

Probability of two items in the queue: $p(n>2)$ where $n_0 =2$

n0 = 2

if ϱ > 1:
        print("The system is unstable, we do not carry out further calculations!")
        print('Arrival rate λ > split service rate μ')
else:
        n0 = 2
        p11 = ((r**(r-n0))*(?**(n0+1))*p7)/((r-?)*factorial(r))
        print('Probability of two items in the queue: ?(?>2)≈ 

The probability that the item will have to queue over k minutes: $p(t>?_0)$

$${
p(n>n_0) =frac{{r^{r-n_{0}}}{?^{n_{0}+1}}{p(n=0)}}{(r-{?})r!}
} qquad (10)$$$${
p(t>t_0) ={{p{(n>r-1)}}{e^{μt(r-ψ)}}}
} qquad (11)
$$

## time in minutes
t0=0.05
t0=t0/60

print('n - unspecified number of units in the queue')
print('number of channels r =', r)

$ p(n>r-1) = p(n>1)$
equation ⟹ 10 for $p(n>1)$

n0=1
p12 = ((r**(r-n0))*(?**(n0+1))*p7)/((r-?)*factorial(r))
p12   

0.6233845932783932

# wzór 11
if ϱ > 1:
        print("The system is unstable, we do not carry out further calculations!")
        print('Arrival rate λ > split service rate μ')
else:
        p13 = p12*(np.e**(-μ*t0*(r-?)))
        print('The probability that the item will have to queue over n minutes: ?(?>?0)≈ 

What is the average number of transactions  waiting?!

${
ψ =\frac{{?^{r+1}{p(n=0)}}}{{(r-?)}^2{(r-1)!}}
} \qquad (8)$

if ϱ > 1:
        print("The system is unstable, we do not carry out further calculations!")
        print('Arrival rate λ > split service rate μ')
else:
        p14 = ((?**(r+1))*p7)/(((r-?)**2)*factorial(r-1))
        print('The average number of items waiting ψ ≈ 

def queueing_test(λ,μ,r):
    
    def factorial(n): 
        return 1 if (n==1 or n==0) else n * factorial(n - 1);  
    
    
    print("-----------------------------------------------------------------------")
    print('Arrival rate λ:      
    print('Service rate μ:      
    ϱ = λ/(μ*r)
    print('Traffic intensity ϱ: 
    print("-----------------------------------------------------------------------")
    if ϱ > 1:
        print("The system is unstable, we do not carry out further calculations!")
        print('Arrival rate λ > split service rate μ')
    else:
        print("The system is stable, you can calculate the probabilities!")
        print('Arrival rate λ > split service rate μ')
    print("-----------------------------------------------------------------------")    
    p7=1/((1+ϱ)+(((ϱ**r)/((r-ϱ)*factorial(r-1)))))
    print('P7 Probability of no queues ?(?=0)≈ 
    print("-----------------------------------------------------------------------")
    n0 = 2
    p11 = ((r**(r-n0))*(?**(n0+1))*p7)/((r-?)*factorial(r))
    print('P11 Probability of two items in the queue: ?(?>2)≈ 
    print("-----------------------------------------------------------------------")
    n0=1
    p12 = ((r**(r-n0))*(?**(n0+1))*p7)/((r-?)*factorial(r))
    

    
    ## time in minutes
    t1=0.0165  ## 1 SEKUNDA PRZESTÓJ
    t1=t1/60
  
    p13A = p12*(np.e**(-μ*t1*(r-?)))
    print('P13 The probability that the item will have to queue over 1.0 SEC: ?(?>?0)≈ 
    
    ## time in minutes
    t0=0.05  ## 3 SEKUNDY PRZESTÓJ
    t0=t0/60
  
    p13B = p12*(np.e**(-μ*t0*(r-?)))
    print('P13 The probability that the item will have to queue over 3.0 SEC: ?(?>?0)≈ 
    
    
    
    ## time in minutes
    t2=0.0834  ## 5 SEKUNDA PRZESTÓJ
    t2=t2/60
  
    p13C = p12*(np.e**(-μ*t2*(r-?)))
    print('P13 The probability that the item will have to queue over 5.0 SEC: ?(?>?0)≈ 
      
      
    print("-----------------------------------------------------------------------")
    p14 = ((?**(r+1))*p7)/(((r-?)**2)*factorial(r-1))
    print('P14 The average number of items waiting ψ ≈ 
    print("-----------------------------------------------------------------------")   
    

First server room

queueing_test(λ1,μ1,r1)

Second server room

queueing_test(λ2,μ2,r2)

Third server room

queueing_test(λ3,μ3,r3)

As you can see above, we tested the performance of three bank information processing centers. The probability of no queue in the first center is p7 = 50