I am Hardworking individual with high interest in applied statistics research and data science. I am always ready to face new challenges head on as far as statistical science is concerned.
International Journal of Applied Mathematics and Statistics, 2020
Regular and correct hand hygiene is one of the most important measures to prevent infection with ... more Regular and correct hand hygiene is one of the most important measures to prevent infection with the COVID-19 virus. WASH practitioners should work to enable more frequent and regular hand hygiene by improving access to hand hygiene facilities to support good hand hygiene behavior. Performing hand hygiene at the right time, using the right technique with either an alcohol-based hand rub and soap and water is critical. It makes water to be an essential resource in the fight against the pandemic. This article ventured into the analysis of water demand by Kisii County householders. This article employed a triple exponential smoothing method. The exponential smoothing methods usually applied in the analysis of univariate time series data. This study employed the Cox-Stuart method to determine the trend of the data. Since p-value = 0.00001141 < the significance level (α) = 0.05, this study concluded that the water data has a trend. The parameters of the triple exponential smoothing were identified to be α=0.2358, β=0.0028 and γ = 0.0976. They were determined in such a way that the mean squared error (MSE) of the error is minimized. In-sample forecasting was employed. No significant difference was noted. The exponential smoothing model was employed in out of sample forecasting, and it was realized that the water demand was expected to decrease. This study recommends the use of other statistical models to establish if the same results could be realized.
international journal of statistics and applied mathematics, 2020
Markovian models basically refers to models which, largely, relies on the present state to predic... more Markovian models basically refers to models which, largely, relies on the present state to predict the future. A birth process refers to the arrival of a customer to a system and a death process explains the departure of a customer from a system. This study modelled queues at the revenue collection points using the Markovian birth-death analytical models. The traffic intensity and waiting times were estimated from the secondary data collected from the revenue collection point. The focus was on Markovian queuing systems with infinite capacity that is M/M/m models, where m ≥ 1. The queuing model employed at Bus Park Revenue collection point was identified to be M/M/2. The traffic intensity of the service station was estimated and it was discovered that the Birth rate > death rate, that is ρ 2 = 1.029412, indicating that the system was unstable. This indicated that the queue could grow indefinitely. As a result it was difficult to obtain other performance parameters. As a remedy, this study resorted to addition of a server with same rate of service, 70 customers/hr, to the system. This study assumed that the service distribution was the same for all the servers. On addition of a server the estimated number of customers in the system reduced significantly. After conducting queue analysis, this study concluded that the system at Bus Park Revenue collection point was not stable. This study recommended the addition of a server to the revenue collection point. Additionally, future researchers should conduct a non Markovian analysis of a birth-death process to determine if the same results could be realized. Keywords: Markovian, birth, death, M/M/m, traffic intensity 1. Introduction A Markovian process is a stochastic process with the property that the current/present state is not influenced by the previous/past states. In other words, the chance of any future behavior of processes, when its current/present state is known, is not altered by additional knowledge concerning its past behavior [1]. A birth-death process is a CTMC where the state transitions are only of two types, that is Births (inputs) and deaths (outputs). When births occur in a system with n objects, it is presumed that the state transitions from + 1 that is the state increase by one. When a death occurs, it is also presumed that the state transitions from − 1 that is the state decrease by one [2]. Birth-Death processes have many applications in real life situation. The Birth-death process can be applied in epidemiological studies, demography, queuing theory, and in engineering sector. In a non-random environment, the birth-death process in queuing models tend to be long term averages, so the average rate of arrival is given as B and the average service rate given as D [3]. In M/M/s queuing systems the arrivals into a service facility are considered to be inputs (Births) and the numbers served from a service facility are considered to be outputs (Deaths) [4]. A birth-death process is a special kind of CTMC that has applications in queuing systems. If the arrivals to a queuing system are according to a Poisson process and the service times are exponentially distributed then the resulting queuing system is a birth-death process [2, 3, 5]. Most of the queuing models fit the birth-death process. A birth refers to a customer arrival-this leads to an increase in the number of customers in the system from n to n+1 and a death occurs when a customer leaves the system after being attended to leading to a decrease in the number of customers in the system from n to n-1. A queuing system based on the birth-death process is in state E n at time t if the number of customers is then n, that is N(t)=n [3, 6] .
Fitting a Markovian queuing model to bus park revenue collection point in Kisii town,Kenya, 2019
In queuing theory one deals with the mathematical analysis of the performance of queuing systems.... more In queuing theory one deals with the mathematical analysis of the performance of queuing systems. In our daily lives customers encounter queues while seeking services in institutions. The increase in the number of customers has resulted to congestion at revenue collection points in Kenyan towns. There is therefore need to study the queuing systems to identify possible remedies. This study sought to fit a queuing model to bus park revenue collection point as a preliminary action in studying the congestion problem in Kisii town, Kenya. The study considered and collected data on the inter-arrival times, service times and the number of servers at Kisii Bus Park Revenue Collection Point. The inter-arrival and service times were plotted and compared to a plot of a theoretical exponential distribution. The inter-arrival times resembled a theoretical exponential distribution with a parameter 1.022 and the service times resembled a theoretical exponential distribution with a parameter 1.209. Further, Kolmogorov Smirnov and Anderson Darling goodness of fit tests were conducted to determine if the inter-arrival and service times were exponentially distributed. In both cases, the test statistics were less than the critical value. The study therefore established that the inter-arrival and the service times could be modeled as exponential hence Markovian. The revenue collection point used two servers. This study assumed that the servers followed the same service distribution. This study concluded that the inter-arrival and the service times had an exponential distribution and the queuing model used was M/M/2. 1. Introduction Queues are experienced in our daily lives in business situations where customers have to wait for services to be delivered to them for example: in telephone exchange, in banks, in public transportation, in a supermarket, at the county revenue collection points, at a petrol station, waiting to use an automated teller machine (ATM) and people queuing to wait for their turn to vote [1, 2]. When using phones to conduct daily businesses, sometimes one may be put on hold and wait for their turn to receive the services. In modern life, queues are not experienced by humans only. Modern communication systems transmit messages, like emails, from one device to another by queuing them up inside the network. Modern communication systems maintain queues called inventories of raw materials, partly finished goods, and finished goods throughout the manufacturing process of a business institution. The supply chain management in businesses is nothing but the management of queues [3]. Queuing models help in the design process by predicting system performance in organizations. For example queuing models might be used to evaluate the costs and benefits of adding a server (s) to an existing system of an institution. The models enable institutions/organizations to compute the system performance measures in terms of more basic quantities like the waiting times, traffic intensity and the queue lengths in the systems. Some crucial measures of system (s) performance are: the mean queue length, probability of a customer waiting for service in the system, the probability of finding the system being idle, the probability distribution of the number of customers in the system, the utilization of the server (s) and customer waiting time [2]. Queuing systems have a wide range of applications in modern times [3]. The first queuing theory problem was found in the telephone exchange congestion studied by A.K. Erlang (1878-1929).
In queuing theory one deals with the mathematical analysis of the performance of queuing systems.... more In queuing theory one deals with the mathematical analysis of the performance of queuing systems. In our daily lives customers encounter queues while seeking services in institutions. The increase in the number of customers has resulted to congestion at revenue collection points in Kenyan towns. There is therefore need to study the queuing systems to identify possible remedies. This study sought to fit a queuing model to bus park revenue collection point as a preliminary action in studying the congestion problem in Kisii town, Kenya. The study considered and collected data on the inter-arrival times, service times and the number of servers at Kisii Bus Park Revenue Collection Point. The inter-arrival and service times were plotted and compared to a plot of a theoretical exponential distribution. The inter-arrival times resembled a theoretical exponential distribution with a parameter 1.022 and the service times resembled a theoretical exponential distribution with a parameter 1.209....
Regular and correct hand hygiene is one of the most important measures to prevent infection with ... more Regular and correct hand hygiene is one of the most important measures to prevent infection with the COVID-19 virus. WASH practitioners should work to enable more frequent and regular hand hygiene by improving access to hand hygiene facilities to support good hand hygiene behavior. Performing hand hygiene at the right time, using the right technique with either an alcohol-based hand rub and soap and water is critical. It makes water to be an essential resource in the fight against the pandemic. This article ventured into the analysis of water demand by Kisii County householders. This article employed a triple exponential smoothing method. The exponential smoothing methods usually applied in the analysis of univariate time series data. This study employed the Cox-Stuart method to determine the trend of the data. Since p-value = 0.00001141 < the significance level (α) = 0.05, this study concluded that the water data has a trend. The parameters of the triple exponential smoothing we...
Markovian models basically refers to models which, largely, relies on the present state to predic... more Markovian models basically refers to models which, largely, relies on the present state to predict the future. A birth process refers to the arrival of a customer to a system and a death process explains the departure of a customer from a system. This study modelled queues at the revenue collection points using the Markovian birth-death analytical models. The traffic intensity and waiting times were estimated from the secondary data collected from the revenue collection point. The focus was on Markovian queuing systems with infinite capacity that is M/M/m models, where m≥1. The queuing model employed at Bus Park Revenue collection point was identified to be M/M/2. The traffic intensity of the service station was estimated and it was discovered that the Birth rate > death rate, that is ρ2 =1.029412, indicating that the system was unstable. This indicated that the queue could grow indefinitely. As a result it was difficult to obtain other performance parameters. As a remedy, this s...
International Journal of Applied Mathematics and Statistics, 2020
Regular and correct hand hygiene is one of the most important measures to prevent infection with ... more Regular and correct hand hygiene is one of the most important measures to prevent infection with the COVID-19 virus. WASH practitioners should work to enable more frequent and regular hand hygiene by improving access to hand hygiene facilities to support good hand hygiene behavior. Performing hand hygiene at the right time, using the right technique with either an alcohol-based hand rub and soap and water is critical. It makes water to be an essential resource in the fight against the pandemic. This article ventured into the analysis of water demand by Kisii County householders. This article employed a triple exponential smoothing method. The exponential smoothing methods usually applied in the analysis of univariate time series data. This study employed the Cox-Stuart method to determine the trend of the data. Since p-value = 0.00001141 < the significance level (α) = 0.05, this study concluded that the water data has a trend. The parameters of the triple exponential smoothing were identified to be α=0.2358, β=0.0028 and γ = 0.0976. They were determined in such a way that the mean squared error (MSE) of the error is minimized. In-sample forecasting was employed. No significant difference was noted. The exponential smoothing model was employed in out of sample forecasting, and it was realized that the water demand was expected to decrease. This study recommends the use of other statistical models to establish if the same results could be realized.
International journal of Statistics and applied Mathematics, 2020
Markovian models basically refers to models which, largely, relies on the present state to predic... more Markovian models basically refers to models which, largely, relies on the present state to predict the future. A birth process refers to the arrival of a customer to a system and a death process explains the departure of a customer from a system. This study modelled queues at the revenue collection points using the Markovian birth-death analytical models. The traffic intensity and waiting times were estimated from the secondary data collected from the revenue collection point. The focus was on Markovian queuing systems with infinite capacity that is M/M/m models, where m ≥ 1. The queuing model employed at Bus Park Revenue collection point was identified to be M/M/2. The traffic intensity of the service station was estimated and it was discovered that the Birth rate > death rate, that is ρ 2 = 1.029412, indicating that the system was unstable. This indicated that the queue could grow indefinitely. As a result it was difficult to obtain other performance parameters. As a remedy, this study resorted to addition of a server with same rate of service, 70 customers/hr, to the system. This study assumed that the service distribution was the same for all the servers. On addition of a server the estimated number of customers in the system reduced significantly. After conducting queue analysis, this study concluded that the system at Bus Park Revenue collection point was not stable. This study recommended the addition of a server to the revenue collection point. Additionally, future researchers should conduct a non Markovian analysis of a birth-death process to determine if the same results could be realized. Keywords: Markovian, birth, death, M/M/m, traffic intensity 1. Introduction A Markovian process is a stochastic process with the property that the current/present state is not influenced by the previous/past states. In other words, the chance of any future behavior of processes, when its current/present state is known, is not altered by additional knowledge concerning its past behavior [1]. A birth-death process is a CTMC where the state transitions are only of two types, that is Births (inputs) and deaths (outputs). When births occur in a system with n objects, it is presumed that the state transitions from + 1 that is the state increase by one. When a death occurs, it is also presumed that the state transitions from − 1 that is the state decrease by one [2]. Birth-Death processes have many applications in real life situation. The Birth-death process can be applied in epidemiological studies, demography, queuing theory, and in engineering sector. In a non-random environment, the birth-death process in queuing models tend to be long term averages, so the average rate of arrival is given as B and the average service rate given as D [3]. In M/M/s queuing systems the arrivals into a service facility are considered to be inputs (Births) and the numbers served from a service facility are considered to be outputs (Deaths) [4]. A birth-death process is a special kind of CTMC that has applications in queuing systems. If the arrivals to a queuing system are according to a Poisson process and the service times are exponentially distributed then the resulting queuing system is a birth-death process [2, 3, 5]. Most of the queuing models fit the birth-death process. A birth refers to a customer arrival-this leads to an increase in the number of customers in the system from n to n+1 and a death occurs when a customer leaves the system after being attended to leading to a decrease in the number of customers in the system from n to n-1. A queuing system based on the birth-death process is in state E n at time t if the number of customers is then n, that is N(t)=n [3, 6] .
International Journal of Applied Mathematics and Statistics, 2020
Regular and correct hand hygiene is one of the most important measures to prevent infection with ... more Regular and correct hand hygiene is one of the most important measures to prevent infection with the COVID-19 virus. WASH practitioners should work to enable more frequent and regular hand hygiene by improving access to hand hygiene facilities to support good hand hygiene behavior. Performing hand hygiene at the right time, using the right technique with either an alcohol-based hand rub and soap and water is critical. It makes water to be an essential resource in the fight against the pandemic. This article ventured into the analysis of water demand by Kisii County householders. This article employed a triple exponential smoothing method. The exponential smoothing methods usually applied in the analysis of univariate time series data. This study employed the Cox-Stuart method to determine the trend of the data. Since p-value = 0.00001141 < the significance level (α) = 0.05, this study concluded that the water data has a trend. The parameters of the triple exponential smoothing were identified to be α=0.2358, β=0.0028 and γ = 0.0976. They were determined in such a way that the mean squared error (MSE) of the error is minimized. In-sample forecasting was employed. No significant difference was noted. The exponential smoothing model was employed in out of sample forecasting, and it was realized that the water demand was expected to decrease. This study recommends the use of other statistical models to establish if the same results could be realized.
international journal of statistics and applied mathematics, 2020
Markovian models basically refers to models which, largely, relies on the present state to predic... more Markovian models basically refers to models which, largely, relies on the present state to predict the future. A birth process refers to the arrival of a customer to a system and a death process explains the departure of a customer from a system. This study modelled queues at the revenue collection points using the Markovian birth-death analytical models. The traffic intensity and waiting times were estimated from the secondary data collected from the revenue collection point. The focus was on Markovian queuing systems with infinite capacity that is M/M/m models, where m ≥ 1. The queuing model employed at Bus Park Revenue collection point was identified to be M/M/2. The traffic intensity of the service station was estimated and it was discovered that the Birth rate > death rate, that is ρ 2 = 1.029412, indicating that the system was unstable. This indicated that the queue could grow indefinitely. As a result it was difficult to obtain other performance parameters. As a remedy, this study resorted to addition of a server with same rate of service, 70 customers/hr, to the system. This study assumed that the service distribution was the same for all the servers. On addition of a server the estimated number of customers in the system reduced significantly. After conducting queue analysis, this study concluded that the system at Bus Park Revenue collection point was not stable. This study recommended the addition of a server to the revenue collection point. Additionally, future researchers should conduct a non Markovian analysis of a birth-death process to determine if the same results could be realized. Keywords: Markovian, birth, death, M/M/m, traffic intensity 1. Introduction A Markovian process is a stochastic process with the property that the current/present state is not influenced by the previous/past states. In other words, the chance of any future behavior of processes, when its current/present state is known, is not altered by additional knowledge concerning its past behavior [1]. A birth-death process is a CTMC where the state transitions are only of two types, that is Births (inputs) and deaths (outputs). When births occur in a system with n objects, it is presumed that the state transitions from + 1 that is the state increase by one. When a death occurs, it is also presumed that the state transitions from − 1 that is the state decrease by one [2]. Birth-Death processes have many applications in real life situation. The Birth-death process can be applied in epidemiological studies, demography, queuing theory, and in engineering sector. In a non-random environment, the birth-death process in queuing models tend to be long term averages, so the average rate of arrival is given as B and the average service rate given as D [3]. In M/M/s queuing systems the arrivals into a service facility are considered to be inputs (Births) and the numbers served from a service facility are considered to be outputs (Deaths) [4]. A birth-death process is a special kind of CTMC that has applications in queuing systems. If the arrivals to a queuing system are according to a Poisson process and the service times are exponentially distributed then the resulting queuing system is a birth-death process [2, 3, 5]. Most of the queuing models fit the birth-death process. A birth refers to a customer arrival-this leads to an increase in the number of customers in the system from n to n+1 and a death occurs when a customer leaves the system after being attended to leading to a decrease in the number of customers in the system from n to n-1. A queuing system based on the birth-death process is in state E n at time t if the number of customers is then n, that is N(t)=n [3, 6] .
Fitting a Markovian queuing model to bus park revenue collection point in Kisii town,Kenya, 2019
In queuing theory one deals with the mathematical analysis of the performance of queuing systems.... more In queuing theory one deals with the mathematical analysis of the performance of queuing systems. In our daily lives customers encounter queues while seeking services in institutions. The increase in the number of customers has resulted to congestion at revenue collection points in Kenyan towns. There is therefore need to study the queuing systems to identify possible remedies. This study sought to fit a queuing model to bus park revenue collection point as a preliminary action in studying the congestion problem in Kisii town, Kenya. The study considered and collected data on the inter-arrival times, service times and the number of servers at Kisii Bus Park Revenue Collection Point. The inter-arrival and service times were plotted and compared to a plot of a theoretical exponential distribution. The inter-arrival times resembled a theoretical exponential distribution with a parameter 1.022 and the service times resembled a theoretical exponential distribution with a parameter 1.209. Further, Kolmogorov Smirnov and Anderson Darling goodness of fit tests were conducted to determine if the inter-arrival and service times were exponentially distributed. In both cases, the test statistics were less than the critical value. The study therefore established that the inter-arrival and the service times could be modeled as exponential hence Markovian. The revenue collection point used two servers. This study assumed that the servers followed the same service distribution. This study concluded that the inter-arrival and the service times had an exponential distribution and the queuing model used was M/M/2. 1. Introduction Queues are experienced in our daily lives in business situations where customers have to wait for services to be delivered to them for example: in telephone exchange, in banks, in public transportation, in a supermarket, at the county revenue collection points, at a petrol station, waiting to use an automated teller machine (ATM) and people queuing to wait for their turn to vote [1, 2]. When using phones to conduct daily businesses, sometimes one may be put on hold and wait for their turn to receive the services. In modern life, queues are not experienced by humans only. Modern communication systems transmit messages, like emails, from one device to another by queuing them up inside the network. Modern communication systems maintain queues called inventories of raw materials, partly finished goods, and finished goods throughout the manufacturing process of a business institution. The supply chain management in businesses is nothing but the management of queues [3]. Queuing models help in the design process by predicting system performance in organizations. For example queuing models might be used to evaluate the costs and benefits of adding a server (s) to an existing system of an institution. The models enable institutions/organizations to compute the system performance measures in terms of more basic quantities like the waiting times, traffic intensity and the queue lengths in the systems. Some crucial measures of system (s) performance are: the mean queue length, probability of a customer waiting for service in the system, the probability of finding the system being idle, the probability distribution of the number of customers in the system, the utilization of the server (s) and customer waiting time [2]. Queuing systems have a wide range of applications in modern times [3]. The first queuing theory problem was found in the telephone exchange congestion studied by A.K. Erlang (1878-1929).
In queuing theory one deals with the mathematical analysis of the performance of queuing systems.... more In queuing theory one deals with the mathematical analysis of the performance of queuing systems. In our daily lives customers encounter queues while seeking services in institutions. The increase in the number of customers has resulted to congestion at revenue collection points in Kenyan towns. There is therefore need to study the queuing systems to identify possible remedies. This study sought to fit a queuing model to bus park revenue collection point as a preliminary action in studying the congestion problem in Kisii town, Kenya. The study considered and collected data on the inter-arrival times, service times and the number of servers at Kisii Bus Park Revenue Collection Point. The inter-arrival and service times were plotted and compared to a plot of a theoretical exponential distribution. The inter-arrival times resembled a theoretical exponential distribution with a parameter 1.022 and the service times resembled a theoretical exponential distribution with a parameter 1.209....
Regular and correct hand hygiene is one of the most important measures to prevent infection with ... more Regular and correct hand hygiene is one of the most important measures to prevent infection with the COVID-19 virus. WASH practitioners should work to enable more frequent and regular hand hygiene by improving access to hand hygiene facilities to support good hand hygiene behavior. Performing hand hygiene at the right time, using the right technique with either an alcohol-based hand rub and soap and water is critical. It makes water to be an essential resource in the fight against the pandemic. This article ventured into the analysis of water demand by Kisii County householders. This article employed a triple exponential smoothing method. The exponential smoothing methods usually applied in the analysis of univariate time series data. This study employed the Cox-Stuart method to determine the trend of the data. Since p-value = 0.00001141 < the significance level (α) = 0.05, this study concluded that the water data has a trend. The parameters of the triple exponential smoothing we...
Markovian models basically refers to models which, largely, relies on the present state to predic... more Markovian models basically refers to models which, largely, relies on the present state to predict the future. A birth process refers to the arrival of a customer to a system and a death process explains the departure of a customer from a system. This study modelled queues at the revenue collection points using the Markovian birth-death analytical models. The traffic intensity and waiting times were estimated from the secondary data collected from the revenue collection point. The focus was on Markovian queuing systems with infinite capacity that is M/M/m models, where m≥1. The queuing model employed at Bus Park Revenue collection point was identified to be M/M/2. The traffic intensity of the service station was estimated and it was discovered that the Birth rate > death rate, that is ρ2 =1.029412, indicating that the system was unstable. This indicated that the queue could grow indefinitely. As a result it was difficult to obtain other performance parameters. As a remedy, this s...
International Journal of Applied Mathematics and Statistics, 2020
Regular and correct hand hygiene is one of the most important measures to prevent infection with ... more Regular and correct hand hygiene is one of the most important measures to prevent infection with the COVID-19 virus. WASH practitioners should work to enable more frequent and regular hand hygiene by improving access to hand hygiene facilities to support good hand hygiene behavior. Performing hand hygiene at the right time, using the right technique with either an alcohol-based hand rub and soap and water is critical. It makes water to be an essential resource in the fight against the pandemic. This article ventured into the analysis of water demand by Kisii County householders. This article employed a triple exponential smoothing method. The exponential smoothing methods usually applied in the analysis of univariate time series data. This study employed the Cox-Stuart method to determine the trend of the data. Since p-value = 0.00001141 < the significance level (α) = 0.05, this study concluded that the water data has a trend. The parameters of the triple exponential smoothing were identified to be α=0.2358, β=0.0028 and γ = 0.0976. They were determined in such a way that the mean squared error (MSE) of the error is minimized. In-sample forecasting was employed. No significant difference was noted. The exponential smoothing model was employed in out of sample forecasting, and it was realized that the water demand was expected to decrease. This study recommends the use of other statistical models to establish if the same results could be realized.
International journal of Statistics and applied Mathematics, 2020
Markovian models basically refers to models which, largely, relies on the present state to predic... more Markovian models basically refers to models which, largely, relies on the present state to predict the future. A birth process refers to the arrival of a customer to a system and a death process explains the departure of a customer from a system. This study modelled queues at the revenue collection points using the Markovian birth-death analytical models. The traffic intensity and waiting times were estimated from the secondary data collected from the revenue collection point. The focus was on Markovian queuing systems with infinite capacity that is M/M/m models, where m ≥ 1. The queuing model employed at Bus Park Revenue collection point was identified to be M/M/2. The traffic intensity of the service station was estimated and it was discovered that the Birth rate > death rate, that is ρ 2 = 1.029412, indicating that the system was unstable. This indicated that the queue could grow indefinitely. As a result it was difficult to obtain other performance parameters. As a remedy, this study resorted to addition of a server with same rate of service, 70 customers/hr, to the system. This study assumed that the service distribution was the same for all the servers. On addition of a server the estimated number of customers in the system reduced significantly. After conducting queue analysis, this study concluded that the system at Bus Park Revenue collection point was not stable. This study recommended the addition of a server to the revenue collection point. Additionally, future researchers should conduct a non Markovian analysis of a birth-death process to determine if the same results could be realized. Keywords: Markovian, birth, death, M/M/m, traffic intensity 1. Introduction A Markovian process is a stochastic process with the property that the current/present state is not influenced by the previous/past states. In other words, the chance of any future behavior of processes, when its current/present state is known, is not altered by additional knowledge concerning its past behavior [1]. A birth-death process is a CTMC where the state transitions are only of two types, that is Births (inputs) and deaths (outputs). When births occur in a system with n objects, it is presumed that the state transitions from + 1 that is the state increase by one. When a death occurs, it is also presumed that the state transitions from − 1 that is the state decrease by one [2]. Birth-Death processes have many applications in real life situation. The Birth-death process can be applied in epidemiological studies, demography, queuing theory, and in engineering sector. In a non-random environment, the birth-death process in queuing models tend to be long term averages, so the average rate of arrival is given as B and the average service rate given as D [3]. In M/M/s queuing systems the arrivals into a service facility are considered to be inputs (Births) and the numbers served from a service facility are considered to be outputs (Deaths) [4]. A birth-death process is a special kind of CTMC that has applications in queuing systems. If the arrivals to a queuing system are according to a Poisson process and the service times are exponentially distributed then the resulting queuing system is a birth-death process [2, 3, 5]. Most of the queuing models fit the birth-death process. A birth refers to a customer arrival-this leads to an increase in the number of customers in the system from n to n+1 and a death occurs when a customer leaves the system after being attended to leading to a decrease in the number of customers in the system from n to n-1. A queuing system based on the birth-death process is in state E n at time t if the number of customers is then n, that is N(t)=n [3, 6] .
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