Steadystate characteristics of threechannel queueing systems with Erlangian service times
Bohdan Kopytko
,Kostyantyn Zhernovyi
Journal of Applied Mathematics and Computational Mechanics 
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STEADYSTATE CHARACTERISTICS OF THREECHANNEL QUEUEING SYSTEMS WITH ERLANGIAN SERVICE TIMES
Bohdan Kopytko^{ 1}, Kostyantyn Zhernovyi^{ 2}
^{1} Institute of Mathematics,
Czestochowa University of Technology
Częstochowa, Poland
^{2} Educational and Scientific
Institute, Banking University
Lviv, Ukraine
bohdan.kopytko@im.pcz.pl, k.zhernovyi@yahoo.com
Abstract. We propose a method of study the M/E_{2}/3/∞ queueing systems: standard system and systems with the threshold and hysteretic strategies of the random dropping of customers in order to control the input flow. Recurrence relations to compute the stationary distribution of the number of customers and the steadystate characteristics are obtained. The developed algorithms are tested on examples using simulation models constructed with the assistance of the GPSS World tools.
Keywords: threechannel queueing system, Poisson input, Erlangian service times, random dropping of customers, fictitious phase method, recurrence relations
1. Introduction
There are currently no created analytical methods of study of the M/G/n/m and M/G/n/∞ queueing systems with the number of channels The M/D/n/∞ system is one of the few exceptions. It is possible to obtain some of the characteristics associated with the number of customers in the system [1].
To investigate the singlechannel systems with Erlangian service times, in particular the M/E_{s}/1/∞ system [1], the method of fictitious phases, developed by Erlang [2], was applied. The Erlangian service times of the order means that each customer runs sequentially service phases, the duration of which is distributed exponentially with parameters_{ }_{ }respectively.
The objective of this work is the construction with the help of the method of fictitious phase recurrence algorithms for computing the stationary distribution of the number of customers in the threechannel queueing system M/E_{2}/3/∞, as well as in the systems of the same type with threshold and hysteretic strategies of the random dropping of customers. The random dropping of arrivals is a powerful tool for parameter control of a queueing system. Each arriving customer can be accepted for service with a probability depending on the queue length at the time of arrival of the customer, even if the buffer is not completely full [36].
2. The M/E_{2}/3/∞ system
We consider the M/E_{2}/3/∞ system. Let be a parameter of the exponential distribution of the time intervals between moments of arrival of customers. Suppose that the service time of each customer is distributed under the generalized Erlang law of the second order, that is, the service time is the sum of two independent random variables exponentially distributed with parameters_{ }_{ }and_{ }_{ }respectively.
Let denote the number of customers in the system and let be the number of busy channels. In accordance with the method of phases, let us enumerate the system’s states as follows: corresponds to the empty system; is the state, when and the service occurs in the first phase; is the state, when and the service occurs in the second phase; is the state, when and the services occur in the first phase; is the state, when and the services occur in the first and second phase respectively; is the state, when and the services occur in the second phase; is the state, when and the services occur in the first phase; is the state, when and the services in two channels occur in the first phase and in one channel the service occurs in the second phase; is the state, when and the service in two channels occur in the second phase and in one channel the service occurs in the first phase; is the state, when and the service occurs in the second phase. We denote by and respectively, steadystate probabilities that the system is in the each of these states. Assuming that to calculate the steadystate probabilities, we obtain the system of equations:
(1) 
(2) 
(3) 
The steadystate distribution of the number of customers in the M/E_{2}/3/∞ system exists under the condition that where is the average service time per customer.
Introducing the notation
using equations (1) we find: 
(4) 
where
Recurrence relations (4) contain the parameter_{ } to be determined.
From equations (2) we obtain the recurrence relations:
(5) 
where
The system (1)(3) consists of an infinite number of equations. Let be a sufficiently large natural number. Writing the normalization condition (3) in the form
(6) 
we determine the approximate values of the stationary probabilities using equations
(7) 
Here is the steadystate probability that is the approximate value of the probability obtained after replacing condition (3) by equality (6).
Recurrence relations (4) and (5) allow us to calculate and consistently obtain the expression for and as functions of the unknown parameter To determine we use the condition that in the steady state, the average number of busy channels is equal to the system load factor writing it in the form
Using the formulas
we find approximate values of the average queue length and the average waiting time in the steady state. The number is chosen so large that the condition
(8) 
holds, where is a positive number specifying the required accuracy of calculations.
3. The M/E_{2}/3/∞ system with the threshold strategy of the random dropping of customers
We consider for the M/E_{2}/3/∞ system a strategy of random dropping of customers performed according to the rule: if at the time of the arrival of a customer (not taking into account the arrived customer), then the customer is accepted for service with probability and leaves the system (is discarded) with probability Confining ourselves to a simplified version of the strategy, let’s fix a threshold value and suppose that for and for The intensity of the simplest flow of customers received for service as a result of random dropping, is equal to
The system of equations for determining the steadystate probabilities contains the equation (1), (3) and the following equations:
(9) 
(10) 
(11) 
Using equations (1), we have the equalities (4) and from the equations (9) we obtain the recurrence relations:
(12) 
Using equation (10), we find:
(13) 
where From equations (11) we obtain:
(14) 
where
Recurrence relations (4), (12)(14) allow us to calculate and consistently obtain the expression for and as functions of the unknown parameter_{ }_{ }To determine_{ }_{ }we use the condition that in the steady state the average number of busy channels is equal to the system load factor The steadystate probabilities exist under the condition that We find the approximate values of steadystate probabilities by the formulas (7) where the number_{ }_{ }is chosen so large that the condition (8) holds.
The approximate values of the average queue length average waiting time and probability of service in the steady state
Using the formulas
(15) 
we find the approximate values of the average queue length average waiting time and probability of service in the steady state. The formula for is obtained as the ratio of the average number of serviced customers per unit of time to the average arrival rate of customers.
4. The M/E_{2}/3/∞ system with the hysteretic strategy of the random dropping of customers
We consider for the M/E_{2}/3/∞ system a hysteretic strategy of a random dropping of customers with two thresholds_{ }_{ }and_{ }_{ }() and with two operation modes: basic mode and dropping mode. Assume that for the basic mode, and for the dropping mode. Here is the number of customers in the system at the time of the arrival of a customer (not taking into account the arrived customer). If at the time of the arrival of a customer, condition_{ }_{ }is satisfied, then the mode is not changed. The dropping mode operates from the time when the number of customers in the system (on service and in the queue) reaches the value of to the time when the number of customers decreases to
We introduce the following notation for the states of the system in the basic mode: the states correspond to the notation introduced above, is the state, when and the services occur in the first phase; is the state, when and the services in two channels occur in the first phase and in one channel the service occurs in the second phase; is the state, when and the service in two channels occur in the second phase and in one channel the service occurs in the first phase;_{ } is the state, when and the service occurs in the second phase. We denote by and respectively, the steadystate probabilities that the system is in each of mentioned states. We denote by and analogous states of the system in the dropping mode and let and denote the steadystate probabilities that the system is in each of these states.
For determining the steadystate probabilities, we obtain the system containing the equation (1) and the following equations:
(16) 
(17) 
(18) 
(19) 
(20) 
(21) 
(22) 
(23) 
(24) 
Introduce the notation
Using equations (1), we have the equalities (4) and from the equations (16) we obtain the recurrence relations:
(25) 
Using equations (17), (21), (22) and (18), we find:
(26) 
To determine and as functions of the parameter we use the equations (19) and (20).
The equations (23) allow us to find
(27) 
To determine we use the condition that in the steady state the average number of busy channels is equal to the system load factor
The steadystate probabilities exist under the condition that Using the recurrence relations (25)(27) and the normalization condition (24), we obtain the approximate values of the steadystate probabilities from the equalities:
Using the formulas (15) we find the approximate values of the stationary characteristics and
5. Examples for the calculating of stationary characteristics
Introduce the notation for the studied queueing systems. Let the M/E_{2}/3/∞ system be the System 1, the M/E_{2}/3/∞ system with the threshold strategy of the random dropping of customers be the System 2 and the M/E_{2}/3/∞ system with the hysteretic strategy of the random dropping of customers be the System 3.
For all the systems we put: Let for the System 1, for the Systems 2 and 3, for the System 2, for the System 3.
The values of the steadystate probabilities and stationary characteristics of the systems 13, found using the recurrence relations obtained in this paper, are presented in Tables 1 and 2. In order to verify the obtained values, the tables contain the computing results evaluated by the GPSS World simulation system [7] for the time value
In calculating the approximate values of the steadystate probabilities the number_{ }_{ }is chosen so large that the condition
(28) 
holds. The obtained minimum values of for which the condition (28) is satisfied, are equal to 156, 164 and 162 for the Systems 1, 2 and 3, respectively.
Table 1
Stationary distribution of the number of customers in the system (1  analytical method, 2  GPSS World)
k 
Values of the steadystate probabilities p_{k} 

System 1 
System 2 
System 3 

1, N = 156 
2 
1, N = 164 
2 
1, N = 162 
2 

0 
0.0241532 
0.0241463 
0.0019693 
0.0019621 
0.0028508 
0.0028780 
1 
0.0666245 
0.0664944 
0.0068771 
0.0068758 
0.0099553 
0.0100608 
2 
0.0942915 
0.0942690 
0.0124350 
0.0124441 
0.0180009 
0.0180814 
3 
0.0954951 
0.0954166 
0.0163353 
0.0163913 
0.0236470 
0.0237228 
4 
0.0881153 
0.0881351 
0.0197881 
0.0198060 
0.0286452 
0.0287503 
5 
0.0786458 
0.0785520 
0.0233579 
0.0233281 
0.0338128 
0.0338232 
6 
0.0693057 
0.0694374 
0.0273314 
0.0272895 
0.0395640 
0.0395121 
7 
0.0607719 
0.0609322 
0.0318850 
0.0317958 
0.0461397 
0.0460776 
8 
0.0531851 
0.0530799 
0.0371591 
0.0368896 
0.0536338 
0.0536840 
9 
0.0465099 
0.0465122 
0.0432901 
0.0432911 
0.0618105 
0.0619795 
10 
0.0406604 
0.0405606 
0.0504217 
0.0503929 
0.0667520 
0.0670598 
20 
0.0105844 
0.0105622 
0.0296765 
0.0297640 
0.0236203 
0.0235319 
30 
0.0027549 
0.0027422 
0.0077243 
0.0078382 
0.0061479 
0.0060459 
40 
0.0007170 
0.0007173 
0.0020105 
0.0019613 
0.0016002 
0.0016040 
50 
0.0001866 
0.0002080 
0.0005233 
0.0005068 
0.0004165 
0.0004464 
60 
0.0000486 
0.0000583 
0.0001362 
0.0001378 
0.0001084 
0.0001119 
70 
0.0000126 
0.0000162 
0.0000355 
0.0000264 
0.0000282 
0.0000277 
80 
0.0000033 
0.0000062 
0.0000092 
0.0000095 
0.0000073 
0.0000074 
90 
8.566∙10^{–7} 
0.0000008 
0.0000024 
0.0000014 
0.0000019 
0.0000011 
100 
2.229∙10^{–7} 
0.0000008 
6.251∙10^{–7} 
0.0000013 
4.975∙10^{–7} 
0.0000002 
110 
5.803∙10^{–8} 
0.0000000 
1.626∙10^{–7} 
0.0000009 
1.295∙10^{–7} 
0.0000000 
130 
3.931∙10^{–9} 

1.102∙10^{–8} 
0.0000000 
8.773∙10^{–9} 

150 
2.663∙10^{–10} 

7.467∙10^{–10} 

5.943∙10^{–10} 

Table 2
Stationary characteristics of the system
The system number 
Method 
E(Q) 
E(W) 
P_{sv} 
1 
analytical 
5.7429114 
3.1905063 
1 
1 
GPSS World 
5.745 
3.193 
1 
2 
analytical 
12.3769008 
6.2553772 
0.8793786 
2 
GPSS World 
12.378 
6.256 
0.879 
3 
analytical 
10.7951977 
5.4825119 
0.8751218 
3 
GPSS World 
10.790 
5.478 
0.875 
6. Conclusions
The numerical algorithm for solving a system of linear algebraic equations for the steadystate probabilities, proposed in this paper, is constructed taking into account the structural features of the system, in particular the presence of three or four unknown in most of its equations. The obtained recurrence relations are used for the direct calculation of the solutions of the system, that allow us to reduce the number of calculations in comparison with application of one of the classical methods (direct or iterative). Direct methods require the implementation of preliminary transformations of the matrix of the system, and only in the second stage of computing the solutions is consistently defined. When using iterative methods, the calculations are performed repeatedly until a solution is found with a given accuracy. The use of iterative methods is possible only under the condition of diagonal dominance of the matrix of the system.
References
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