BDR Modelling of Passenger Queues at Nnamdi Azikiwe International Airport, Abuja, Nigeria

Veiw figure View Table

It is a type of model that deals with study of a single queue in equilibrium. There is more than one server and each server provides same type of service or provides identical parallel service. Passengers wait in one queue until one of the service channels is ready to take them in for servicing at the rate of a customer at a time per server. Each characteristic is shown in Table 1.

2.2. Methods

The Birth and Death Rate modeling was developed to predict the arrival and departure rates based on the performance of service level, stochastic/probabilistic processes, exponential and Chi-square distributions. As this work is a follow up to the work of Ademoh and Anosike (2014), similar distribution assumptions were used to enable close comparison. According to Mehri et al (2009), study assumptions are that;

(a) Arrivals come from an infinite or a very large population.

(b) Arrivals are Poisson distributed.

(c) Arrivals are treated on a first in first out basis and that passengers do not balk or renege; i.e. that an arriving passenger is patient and waits in the queue until he is served and that he does not switch between lines. People that balk are those who refuse to join queue. People that renege are those who enter the queue but later leave due to impatience. As the situation in queuing processes may be complicated by these and more issues, queuing theory//waiting lines analysis are modified as follows:

(a) Service time follows negative exponential

(b) Distributions are constant

2.2.1. Performance Characteristics of Queuing System

Blumenfeld (2001) adopted Little’s law on performance of queue system as represented in equations 2.1 and 2.2:

(2.1)

(2.2)

(2.3)

(2.4)

(2.5)

(2.6)

(2.7)

Where:- = number of passengers in system

= probability of exactly passenger in queueing system at time

= average queue length (average number of passengers in queue)

= average system length (average number of passengers including those being served)

= average waiting time in queue (average time a passenger spends in a queue)

= average time in system (average time a passenger spends in a queue plus the service time)

= total number of passengers in system at a particular time

= time that a passenger spends in the system

= number of servers

= arrival rate (number of passengers arriving per unit time)

= mean inter-arrival time

= service rate per unit server (number of passengers served per unit time)

= mean service time

= traffic intensity.

2.2.2. Birth-and-death Rate Modeling Development

In accordance with the work of Hillier and Liebermann (2001), birth refers to arrival of a new customer into queuing system. Death refers to departure of served customer. An illustration of queuing system is in Figure 1.

Figure 1. Illustration of passengers entering or leaving NAIA, Abuja

Download as

Veiw figure View Figure

The pattern of system with passenger is given by steady state balancing equation.

Mean entering rate of passengers = mean leaving rate of passengers

(2.8)

(2.9)

(2.10)

From equation (2.3), if , we have:

(2.11)

Let for

(2.12)

Therefore equation (2.11) becomes:

(2.13)

Thus, steady-state probability will be given by (Hillier and Liebermann, 2001):

(2.14)

(2.15a)

(2.15b)

(2.16)

(2.17)

(2.18a)

(2.18b)

Where: =Average arrival rate (Trani, 2011): =Steady service rate

BDR solves queuing problems one at a time. Equations 2.10-2.18 were adopted and used to determine passenger arrival and departure rates and the system’s service level of performance.

2.2.3. Assumptions

These are as follows:

(a) Given:-N(t)=λ_n;i.e. exponential probability distribution with parameter λ_n(n=0,1,2,3…) till the next birth (i.e. the next passenger arrival).

(b) Given:- N(t)=µ_n;i.e. exponential probability distribution with parameter µ_n (n=0,1,2,3…) until next death (i.e. when passenger service is completed and he departs).

(c) The random variable of assumption (i); and assumption (ii) are mutually independent. The next transition in system would be either:

2.3. Statistical Testing of Modeling

Arrival and departure assumes Poisson’s distribution as done by Asmussen (1987):

(2.19)

Where:- = Probability of sample

Mean value; Expected value.

Statistical testing is conducted on the modeling using Chi-square distributional assumption as given in equation (2.20):

(2.20)

i.e reject, otherwise accept an option.

(2.21)

And expected sampling data is given as below:

(2.22)

Where:- = Statistical testing

= number of rows; = Number of columns

= Total number of data

= chi-square distribution

= Observed frequency;

= Expected frequency.

These are inputted into the modeling equations and tested to generate simulated results and then compared with the results obtained with Multi server approach (Ademoh and Anosike, 2014) for solving same NAIA queuing problem to enable careful conclusion on the differences and the better of these two approaches.

3. Results and Discussion

3.1. Presentation of Results

Flight data for year 2013 collected from four airlines companies including the British Airways, Ethiopian Airline, Arik Airline and Aero Contractors that use NAIA adopted for the modeling simulation are as shown in Table 2 -Table 5. Table 2 presents passenger data for British Airways for Abuja-London with the expected number of passengers averaged at 188 per day.

Table 2. British airways passengers in 2013

Download as

Veiw figure View Table

Table 3. Ethiopian Airline Passengers in 2013

Download as

Veiw figure View Table

Table 3 is Ethiopian Airline data for Abuja to Addis Ababa route with 120 passenger average daily expectation. Table 4 is that on passengers for year 2013 for Arik Airline for Abuja-Lagos with daily expected passenger average of 1573.

Table 4. Arik Airline Ltd Passengers in 2013

Download as

Veiw figure View Table

Table 5 shows passenger data collected from Aero Contractor with expected passengers/day at an average of 1034. Table 6 presents average of passengers in system. Flight data were not made available by four airline operators used for simulating modeling for months of August and September. This was observed as general trend for most airlines that operate in Nigeria.

Table 5. Aero Contractor Passengers in 2013

Download as

Veiw figure View Table

Table 6. Average No. of Passenger in System

Download as

Veiw figure View Table

3.1.1. Simulation of Queuing Modeling

Figure 2 shows Matlab graphical user interface program developed for modeling. The software was Installed was done in the following steps:

(a) Installation of MATLAB 2009 version or above into personal computer system.

(b) Loading file called queue fig onto screen

(d) Pressing of calculate button to display the result of the simulation.

(e) Press quit button once satisfied with result.

The GUI’s calculate button in the software will automatically compute and display the result of the program based on the modeling equations 2.1-2.22 once the button is pressed after inputs.

Figure 2. Graphical user interface for modelling the NAIA, Abuja system

Download as

Veiw figure View Figure

3.1.2. Required Basic Parameter

Required basic parameters for the model to work referred to as boundary conditions are given in Table 7 as adopted by Ademoh and Anosike (2014).

Table 7. Boundary Conditions

Download as

Veiw figure View Table

3.1.3. BDR Modeling of the Arrival And Departure Rate with Service Factor 0.5

Arrival rate of passenger into the system was modelled with the BDR method at service factor 0.5 per month. The result is presented in Table 8.

Table 8. Modeling result of arrival/departure rate at 0.5 service factor

Download as

Veiw figure View Table

3.1.4. Statistical Testing of Arrival and Departure Rate at 0.5 Service Factor

The result showed that arrival and departure rates of the modeling was same with Multi server approach (Ademoh and Anosike, 2014) except for server capacity that BDR gave 486 while Multi server gave 243 at all the utilisation factors. In Table 9 analysis of basic requirement for testing BDR method at service factor 0.5 showed it required more servers (aircrafts) at any given period. By it, modelling 160 passengers per server showed a requirement of 3 servers while MS (Ademoh and Anosike, 2014) required 2 aircrafts. BDR modeling gave an improvement of service level as 50% more passengers in MS modeling will be served by BDR at a specified period.

Table 9. Requirement of Service Factor 0.5

Download as

Veiw figure View Table

The expected arrival and departure at a service factor of 0.5 is presented in Table 10. Null hypothesis criterion was used to analyse model result at service factor of 0.5. The result is as in Table 10. Table 11, the null hypothesis, shows that the arrival and departure rate should be within range 0-13 passengers per hour from tested criterion, otherwise it would be rejected.

Table 10. Expected arrival and departure at service factor 0.5

Download as

Veiw figure View Table

Null hypothesis criterion was used to analyse result at service factor of 0.5 and Table 11 was generated as result of passenger arrival at a rate of 31 travelers/hr as follows:

The departure rate is given by;

Table 11. Criterion of Null hypothesis at service factor 0.5

Download as

Veiw figure View Table

The readily acceptable hypothesis showed that passengers arrived at a rate of 1-13 persons/hr; that is passengers arriving at 5% significance satisfied the condition at a service factor of 0.5. The passenger arrival and departure rates were rejected at service factor of 0.9 as it was above 13 persons. The hypothesis showed that arrival and departure rate must fall in a range of 0-13 passengers/hr from criterion, otherwise it will be rejected. To satisfy the condition each of the simulated airlines must fly2 more aircrafts/day.

3.1.5. Statistical Testing of Arrival and Departure Rate at 0.9 Service Factor

The result the BDR modeling simulation at service factor 0.9 is similar to that of MS modeling (Ademoh and Anosike 2014) except for server capacity in which BDR gave 690 but MS modeling gave 345 at all utilisation factors. Result of arrival rate of passengers to NAIA at service factor of 0.9 in a month is presented in Table 12 and Table 13. Table 12 shows expected arriva/departure at service factor 0.9. Table 13 shows requirement of service factor 0.9 which showed modeling 160passengers/server needed 4 planes for BDR while MS modeling required 2 aircrafts.

Table 12. Arrival/departure rate service factor 0.9

Download as

Veiw figure View Table

Table 13. Requirement of service factor 0.9

Download as

Veiw figure View Table

BDR modeling had better of service level than MS modeling as double of the passengers of MS modeling will be served by BDR at any given period. Requirement for arrival/departure Expected arrival and departure at service factor 0.9 is as shown in Table 14.

Table 14. Expected arrival/departure at service of factor 0.9

Download as

Veiw figure View Table

Effect of service factor on arrival/departure rate for modeling was shown in Table 10 and Table 14 for service factors 0.5 and 0.9. Table 10 predicted average maximum arrival/departure rates of 10 passengers/hr and 25 passengers/hr respectively at 0.5 service factor with utilization factor 0.2. Arrival/departure rates were 14 passengers/hr and 36 passengers/hr at service factor 0.9 and same utilization factor 0.2. It showed average of 10 departing passengers/hr at service factor 0.5 and average of 21 departing passengers/hr at service factor 0.9. Utilization factor affected departure rates at both service factors 0.5 and 0.9 as rates decreased with increased utilization factor. Aarrival rate was given by:

And the departure rate was also given by:

The analyses result presented in Table 15 shows that both arrival and departure rate should be in a range of 0-13 passengers/hour even at service factor 0.9 as modeling was tested using the chi-distribution assumption. Considering the result in Table 6 it was only the international airlines that were able to meet the required standard. Arriving passengers of the two local airlines didn’t satisfy the condition because there were more travelers and over-utilisation that servers experienced. For a better service by the system, average arrival rate of 31passengers/hr were rejected as local airlines used only 2 aircrafts.

Table 15. Null hypothesis criterion at factor 0.9

Download as

Veiw figure View Table

3.1.6. Expected Passengers in System

Table 16 shows result of expected passengers into the NAIA system based on arrival/departure rate at service factor 0.5 per month. It was estimated between the initial probability of 0.6667; final probability of 0.3333 and of expected 21863 passengers/month of initial evaluation. Initial probability was possibility of passengers; final probability was probability of passengers that actually entered from the expected passengers. Table 17 is expected passengers into NAIA.

Table 16. Expected passengers at 0.5 service factor

Download as

Veiw figure View Table

Table 17. Expected passengers at 0.9 service factor

Download as

Veiw figure View Table

3.1.7. Performance of Service level of System

Passengers entering and leaving NAIA system was analyzed/tested for performance of service level time. The results are in Table 18 and Table 19.

Table 18. Service level’s performance at 0.5 factor

Download as

Veiw figure View Table

Table 19. Service level’s performance at 0.9 Factor

Download as

Veiw figure View Table

3.1.8. Required Servers Per Day

The raw data (Table 2-Table 5) showed that 2 servers carrying 160 passengers/server were required per day. When this BDR modeling was applied it gave a result of 156passengers/server per day. This exactly agreed with that of MS modeling (Ademoh and Anosike, 2014). Estimation was average arrival and departure rates of passengers in Table 10 and Table 14 for 0.5/0.9 service factors respectively. This server capacity was modeled to predict the required servers per airline; result of which is presented in Table 20 that clearly show that any airline carrying more than 1000 passengers/day required more than 5 aircrafts/daily operations.

Table 20. Average passengers and the daily number of required aircrafts in NAIA system

Download as

Veiw figure View Table

3.2. Discussion

The effect of service factor on arrival/departure rates are shown in Figure 3 and Figure 4 to illustrate interrelationships between the two modeling approaches; BDR approach of this work and MS approach of Ademoh and Anosike (2014). Maximum server capacity was 312travelers per day for BDR modeling. Service factor 0.5 had maximum rate of 240 arriving passengers/day and factor 0.9 had 312 arriving passengers/day.

Figure 3. Effect of service factor on arrival rate

Download as

Veiw figure View Figure

Figure 4. Effect of utilisation factor on departure rate

Download as

Veiw figure View Figure

Figure 5. Expected passengers per server

Download as

Veiw figure View Figure