Application-based Statistical Approach To Identifying Queuing Model
NAIMEH HIRBAWI BADIE SARTAWI [email protected]@alquds.edu
Al-Quds University, Computer Science Department, Jerusalem, Palestine July 2014
ABSTRACT
Queuing theory is used to model many systems in different fields in our life; it is used in simple systems and complex ones. The main idea in the application of a mathematical model is to measure the performance measures of the application in order to improve the performance of the system. This paper will present an analytical method and simulation modelin estimation the appropriate model for applications and support to identify the variable parameters that affect the performance measures of the application. The analytical method depends on the mathematical parameters of the real application and according to theirvalues we define the characteristics of the queuing model.Simulation is designed to support and help the decision-makers whether in computer science or mathematics to define the appropriate queuing model and then it will calculate the performance measures for the model.
Keywords: Queuing theory, queuing model, performance measures, mean, variance, Coefficient of Variance.
1.
INTRODUCTION
Queuing theory is used to deal with systems that include queues (waitinglines).It enables mathematical analysis of the behavior of systems in order to evaluate the performance measures of the systems including waiting time in queue, numbers of servers, and utilization of server and so on. Queuing theory is used in many fields to predict how long customers will wait in line and is useful in determining the optimal number of servers. It is also important to computer networking because it can predict the length of time to wait for the data it requests. To solve practical problems the first step is to identify the appropriate queuing model. In queuing theory they always start with a simple model and then if the results do not fit to the problem they continue with a
more complicated one [1].In simple models we use simple formulas that can easily expect the effect of a given parameter on the performance measure but if we use other distributions the mathematical models will be much more complicated.
There are many queuing models in the world but the most common are the stochastic models, we use queuing models to approximate a real application, so we can analyze mathematically the queuing behavior. In queuing models we can define numbers of important steady stateperformance measures such as the average number in the queue or the system, the average time spent in the queue or the system, the distribution of those numbers or times, the probability of the queue either it is full or empty and the probability of the system in a particular
state[1][2][3]. These performance measures are important to improve the system and predict the effect of suggested changes the system. The aim of this research is how to identify the appropriate queuing model for application and then calculates the performance measures. It will help to develop more efficient queuing systems that reduce customer wait times and increase the number of customers that can be served; also it will help analysts not to move from one model to another in order to improve the performance for their systems and by using our simulation model no need for mathematical background in probabilities to define the distribution for the characteristics of queuing process.
This research is based on historical data (Previous papers) which analyzes the behavior of the application/system and define the approximate appropriate queuing model by using mathematical methodand simulation model. We estimate the arrival rate distribution and service ratedistributionfor the queuing model using mathematical parameter. The simulation presentsthe appropriate queuing model for the application with an easier use interface also doing the calculations directly for the performance measures of the model. We use mathematical and simulation tools to capture and analyze data. As a result, this mathematical model and simulation model can provide a good approximation of the queuing model in the real application.
2. BACKGROUND
Queuing theory was invented in the 1909s by A.K. Erlang to improve the development of telephony applications. In the 1950s Leonard Kleinrock applied queuing theory to computer networking, James Jackson studied queuing theory in network of
multiple nodes. In 1953, G. Kendall introduced the notation A/B/X/Y/Z type of queuing theory.
In 1957 Jackson presented open queuing networks (external arrivals and departures) with exponential servers which are a class of queuing networks where the steady-state distribution is simple to compute as the network has a product-form solution. In 1961 J. little proved a formula with dependency of mean number of jobs in queue (and systems) from mean response time (waiting time). In 1963 Jackson introduced queuing networks that rely on the state of the system and closed queuing networks (no external arrivals or departures) with exponential servers[4]. In 70’s queuing theory was used to evaluate the computer performance. In 1973 J.P. Buzen proposed the convolution algorithm for computation of the normalization constant .In 2000 William Stallings provide a practical guide of queuing analysis [2], Manish K. Govil and Michael present paper about the contributions and applications of queuing theory in the field of discrete part manufacturing. RahavDor, Joseph M. Lancaster, Mark A. Franklin, Jeremy Buhler, and Roger D. Chamberlain use queuing theory to model streaming applications[5]. In recent years the progress of queuing theory usesfor software modeling and now they interesting in modeling the insurance systems and retrial queues.
M.Reiser and H.Kobayashi considered variance of service time distribution to develop analytical models for computing systems but in their study they cover the models with exponential arrival distribution and single server (for network systems).
In Literature there are a few references about the behavior of mean and variance because they depend on graph analysis to determine the distribution.
2.1The main characteristics of queuing
process
Arrival Pattern of customers: It is important to estimate the probability distribution of the arrival times between successive customer arrivals (inter-arrival times).
Service time Pattern: The probability distribution of service times is also important to estimate. It depends on many factors such as number of servers and the number of customers waiting in queue for service.
Number of Servers: This refers to the number of servers we need for the system.
System Capacity: The queuing system can be finite or infinite. The capacity refers to the physical limitation of the system such as awaiting room. When the waiting room is full the next customer must leave since there is not enough room to wait.
Queue Discipline: The discipline of the queue explains the way in which the customers are served in the queue. The most popular disciplines are FCFS – First Come First Served, LCFS – Last Come First served and RSS – Random Selection Service.
2.2 Model Notation
Queuing models can be described by Kendall’s notation:
A/B/X/Y/Z
A –The interarrival time distribution
B –The service time distribution. X – The number of servers. Y – The system capacity. Z – The queue discipline.
Common Distributions
M: Exponential
Ek: Erlang with parameter k
Hk: Hyper exponential with parameter k (mixture of k exponentials)
D: Deterministic (constant) G: General
2.3Queuing Models
In this section we will discuss the types of queuing models and some of the most common models for each type and their application.
2.3.1 Markovian models (systems)
In these models the arrival process is Poisson and exponential service times [1][2]. The mean arrival rate (per unit of time) is denoted by (λ). The service time rate is denoted by (μ). From the equilibrium probabilities we can derive the performance measures for the system such as the mean number of customers in the queue (and system) and the mean time spent in the system. Here are some of the queuing models that are common in queuing theory.The M/M/1 or M/M/1/∞is the most common one in systems in which customers arrive according to a Poisson process and the service times of the customers are independent and identically exponentially distributed.M/M/1 is a good approximation for a large number of queuing systems such as customer services environment, banks, and phonequeuing systems and so on.The M/M/c or M/M/c/∞arrivals in this model are a Poisson process and Service time is exponentially distributed with (c) multi servers.This type of models arises in many systems including lines at a bank, phone queuing systems, the application of computer resources, etc.
Other types are the M/M/c/c loss system and M/M/∞, in the first one arriving customer is served if at least one server is available. If all servers are busy then the newly arriving customer is lost. In the second one there is no queue because it has infinite servers so
each arriving customer receives service and a customer never has to wait for service.
2.3.2 Non-Markovian models (systems)
In non-Markovian models either the service time or the interarrival time has to be nonexponentially distributed .In these models the analysis and the computation for the performance measures are more complexes .The most common models in non- markovian models are G/M/1 and M/G/1[6] [10] which is classical modeland widely used in networking applications and a large number of real-life computer. Other non-markovian models are M/D/1 and D/M/1.
2.3.3 Network models (systems)
The queuing networks have become important tools in the design and analysis of computer systems because for many applications network models fulfill a convenient balance between accuracy and efficiency. We have two kind of queuing network, open and closed network [7] [8] [9]. In an open queuing network, customers enter the network from outside, receive service at systems and then depart the network. In a closed network the number of customers is constant. A new customer can enter the network exactly at the same time when one customer leaves, in this section we will mention about some of the famous queuing network.A Jackson network is the simplest model of queuing networks .It has only one customer class and infinite number of jobs [7]. This model supposes that the external arrival is identified by a Poisson arrival process. All systems have one or more servers with exponential service times. The service rates can depend on the number of customers at the system. In all systems customers are served in order of arrival (FIFO). Another model Gordon-Newell networks, also called closed Jackson’s networks, it achieves the same assumption
of Jackson’s networks except one that customer can neither enter nor depart the network. The number of jobs of this type of queuing networks is always fixe
3.
METHODOLOGY
To improve the performance of any system, firstwe have to identify the appropriate queuing model according to the behavior of the system and then calculate the performance measures of the system. In this paper we present mathematical approachesto identify the approximate queuing model for the application.
3.1 Assumptions of this study:
1- Equilibrium analysis is used not transient analysis because we can get exact analytical results of the mean performance parameters under equilibrium conditions also in some special cases, we can also get results of higher moments such as variance or probability distributions in spite Transient analysis is not generally practical, only for some very simple cases, so simulation methods are preferred. 2- Departure and arrival rate are state are in
steady-state which mean arrival rate is less than service rate.
3- Assume the applications have one server especially when we calculate the service rate–so if the application has more than one server we have to divide the service rate on number of servers.
4- Assume that the buffer size is infinite.
3.2 Parameters of queuing models
N: Number of total (customers/requests) in the system (in queue plus in service)
λ: Arrival rate (1 / (average number of customers arriving in each queue in a system in period of time( hour ,minute,…etc.))
µ: Serving rate or service rate (1 / (average number of customers being served at a server per period of time (hour, minute, etc.))
cµ: serving rate when c> 1 in a system (c: number of servers)
ρ: Traffic intensity or load, utilization factor (λ/(cµ)) (the expected factor of time the server is busy that is, service capability being utilized on the average arriving customers)
We must take into consideration the three main key characteristics of queuing process to model the real system. The level of modeling heavily depends on the assumptions and the nature of application and the analysis of it; most of the models assumed the involved random variables are exponentially distributed and independent of each other for service time. Even that this assumption is artificial since in practice the exponential distribution is seldom but the main reason for this assumption is the loss of empirical evidence to theopposite, which leads one to favor convenience .Indeed the memoryless (Markov) property of the exponential distribution makes the mathematical analysis simple and only has one parameter, especially when combined with the assumption that arrival processes is Poisson. Although it provides a good fit for interarrival times if the service provided is random than if it involves a fixed set of tasks. It is common in client/server systems. Memoryless property refers to the state of the system at future time which is decided by the system state at the present time and does not depend on the state at earlier time instants .So the most popular queuing models are the M/M/C/K/FCFS
which M refers to a Markovian process that assumes the arrival or service rate follows a Poisson distribution and the time between arrivals or service time follows an exponential distribution. The C refers to the number of servers which in the simplest case is one. For this system the system capacity is infinite and queue discipline is FCFS (First Come First Served). So if we want to check which model is most appropriate to the real system, we have to estimate each of the three main characteristics of the real system in accordance with the characteristics of queuing process arrival distribution,service time distribution and number of servers.
As the most popular and widely used queuing model is M/M/C, So we first check whether the arrival distribution is Poisson distribution and the service time is exponential according to their properties, if not we have to use the mathematical method that determines the arrival distribution and the service time distribution.
The properties (conditions) for arrival rates are: the arrivals must be independent of other arrivals and processed in a sequence, not concurrent, we cannot predict the number of arrivals if we know the time of the arrivals and arrivals can have known peaks. If these conditions are achieved then the distribution of the arrival distribution is Poisson.
The properties (conditions) for service time are: the service time bases on the contents of the arrivals and most arrivals have different service times, and then the service time distribution is Exponential.
Approaches of the study:
In our study we used two approaches to define the queuing model for probability distribution of arrival rate and probability distribution of service rate:
1- Mathematical method which depends on comparing mean and variance for arrival rate, and the ratio of co-efficient of variance for service rate.
2- Simulation model which defines the queuing model and calculates the performance measures for the model.
3.3Mathematical Method
We used mathematical method to define the most popular models that are used in queuing theory. In this stage we collected the historical data (previous researches in queuing theory), and used mathematical tools to analyze again the behavior of the application, and then we adopted our methodology (the mathematical formulas) to estimate the approximate probability distribution of arrival rate and service rate. Then we compared our results with their results.
Evaluating Arrival distribution:
The estimation of approximate arrival rate isdepending on the values of the mean and the variance.We calculate the mean and variance for different applications from previous papers using the following formulas :
Mean (M) =∑
Variance (V) = (∑ ( **2*f) - (N*mean**2))/ (N-1)
Where:
N is the total observations.
is the arrival rate.
f is the frequency observed for an arrival rate over the longer time period.
The table below contains the results for arrival distribution, the predicting distribution of our study and the predicting distribution ofprevious studies.
Figure 2.This table contains values of the mean and variance for applications from previous researches:
Application Name
λ µ Mean variance Distribution
predicted in paper Distribution predicted by our study conclusion
seaport 5.67 0.18 5.68 5.57 Poisson Poisson Mean=variance
Health care 28 14 6.2772 7.071719 Poisson Poisson Mean=variance
Electronic data
system
According to our analysis we found these results which can determine the approximate arrival rate whether Poisson, constant or general:
If the mean of arrival rate (λ)and the variance of it are approximately equal, a Poisson distribution can be used.
If the arrival rates (λ) of the applicationdo not change, the distribution will approximate deterministic distribution.
If there is no style of arrival rate (λ), then the arrival rate distribution will be a General distribution.
Evaluating Service Time distribution
We determined the approximate service time distribution by calculating standard deviation and coefficient of the variance of the service times, using the following formulas:
Average Service Time (mean) = ∑
Standard Deviation (σ) = SQRT ( ∑
Variance=(∑ Coefficient of Variance (CV) = σ / mean.
Where:
n is the number of observations for that service time.
Si is the service time for observation i.
For more accuracy results we used the confidence intervals for service time.
95% Confidence Intervals for Service Time:
Mean (service time) - 1.96 (SE (service time)) Mean (service time) + 1.96 (SE (service time)) SE = σ/SQRT (n)
According to the values of ourcalculationsfor Coefficient of Variance we found:
If the Coefficient of Variance is close to zero, the service distribution is
Deterministic.
If the Coefficient of Variance is close to one, the service distribution is Poisson. Close to one can be formally tested using a Chi-squared test or approximated using a rule of thumb. The rule of thumb says Coefficient of Variance is close to one if it falls in the range 0.7 to 1.3.
If the Coefficient of Variance is between zero and one, but not closes enough to either zero or one for either of the first two choices, then the distribution is Erlang distribution.
If the Coefficient of Variance is greater than one, then the service distribution is Hyper-exponential.
If Coefficient of Variance is none of these then the general model can be used which is the complex one.
Figure 2. Table of results for Service Time distribution of different applications, the predicting distribution by our study and the predicting distribution by previous studies :
Application name Mean Standard derivation Coefficient µ Coefficient of Confidence Of mean 95% predicted Distributio n by our study predicted Distributio n by previous paper Bank(free 120 32.86335 0.273861 [ Determinist Determinist
days) 0.008333 0.234121 - 0.32985 ] ic ic Bank(busy days) 95 75 0.789474 0.010526 [0.741121- 0.844576] Exponentia l Exponentia l Bank(free days) 108 33.40659 0.30932 0.009259 [ 0.259558- 0.382689] Determinist ic Determinist ic seaport 5.58 1.43 0.25627240 1
0.18 N/A Erlang Erlang
Electronic data system 8.857142 857 6.12788874 2 0.61185840 6 0.11290322 6 [ 0.5811243-0.633453] Erlang Erlang Supermarket system 0.01818 0.0146 0.80310 55 [0.7884 - 1.1388] Exponentia l Exponentia l Telephone Call Center 201 248 1.233 0.004951 N/A Exponentia l Exponentia l
3.4 Simulation Model
We present a novel approach for the estimation of queuing model and evaluation the performance measurement of queuing model. This technique is very useful in queuing theory and by using it no need for analyzing the probability distribution to determine which distribution to use for arrival rate and service rate .We will simulate the real application via simulation model.
Figure 3.Logical Model for Original Queuing Theory
Simulation model is divided as follows: i) to analyze the behavior of the system and evaluate define approximate arrival rate and service rate distribution; ii) define the queuing model; and iii) evaluate the performance measures of the model.
In simulation model we have to enter the mean(M) and variance (V)for arrival rate and the co- efficient variance (CV) of service rate then the model will make the comparison and estimate the probability distribution of the arrival rate and service rate, also will compute the arrival rate (λ) andthe service rate (µ).After that the model will compute the performance measures; the utilization for the server, waiting time in queue, waiting time in system and queue length.
Figure 5. Simulation model defining the arrival rate and service rate distribution.
3.5 Validation
This paper is depending on pervious papers so we used two approaches of validations:
First the Arrival and service distribution are defined according to the mean and variance in our study and comparing them with arrival and service distribution of the queuing model that defined in previous studies. Second the performance measures for the queuing model are computed and then we did comparison between the actual and prediction performance measures for the different applications.
We did analysis for previous papers and defined the approximate arrival rate and service distribution bymathematical formulas and via simulation model. We found some interesting remarks for previous papers:
1- Most of the previous papers had Markovian models.
2- Most of them when they define the queuing model they follow the previous papers and
choose the same model without trying to analyze the behavior of the system again. 3- Many errors in their analysis the behavior of
the system :
1- Their study based on the assumption of steady state but we found that it did not achieve the condition of steady state 2- We computed the performance
measures for the application according to their proposed model and found differences in comparison.
In our opinions they choose Markovian model because it is the easier one to analysis and compute the performance measures, and Exponential distribution has only one parameter.
CONCLUSION
In this research we defined the appropriate queuing model for the application. We used two methods in our study to define the approximate arrival rate
distribution and service time distribution, the first one was mathematical method which depends on mathematical equations and the second one wassimulation model which is easy to use to define the queuing model. This research will help in predicting the performance measures of the real application and improve the performance of the system.
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