A retrial queueing model in which the inter arrival times follow Markovian Arrival Process(MAP) and the service times follow phase type distribution is studied. At the end of receiving the service, the customer has two options namely, either he may go to orbit with probability ?? 1 to get the service again, if he is not satisfied or with probability ?? 1, he may depart the system. Similarly, at the end of providing service, the server can either opt to take vacation with probability ?? 2 or be idle with probability ?? 2. During the busy period, the server may experience breakdown. Both the breakdown times and repair times of the server follow exponential distribution with parameter ?? and ?? respectively. The resulting QBD process is analysed in the steady state by employing matrix analytic method. The busy period analysis of our model has also been done. Finally, the numerical and graphical illustration of our model has been given.

Анотація наукової статті за медичними технологіями, автор наукової роботи - G. Ayyappan, R. Gowthami


Область наук:

  • Медичні технології

  • Рік видавництва: 2019


    Журнал: Reliability: Theory & Applications


    Наукова стаття на тему 'Analysis of MAP / PH / 1 Retrial Queue With Constant Retrial Rate, Bernoulli Schedule Vacation, Bernoulli Feedback, Breakdown and Repair'

    Текст наукової роботи на тему «Analysis of MAP / PH / 1 Retrial Queue With Constant Retrial Rate, Bernoulli Schedule Vacation, Bernoulli Feedback, Breakdown and Repair»

    ?Analysis of MAP / PH / 1 Retrial Queue With Constant Retrial Rate, Bernoulli Schedule Vacation, Bernoulli Feedback,

    Breakdown and Repair

    G. Ayyappan, R. Gowthami

    Department of Mathematics, Pondicherry Engineering College Ця електронна адреса захищена від спам-ботів. Вам потрібно увімкнути JavaScript, щоб побачити її., Ця електронна адреса захищена від спам-ботів. Вам потрібно увімкнути JavaScript, щоб побачити її.

    Abstract

    A retrial queueing model in which the inter arrival times follow Markovian Arrival Process (MAP) and the service times follow phase type distribution is studied. At the end of receiving the service, the customer has two options namely, either he may go to orbit with probability q1 to get the service again, if he is not satisfied or with probability p1, he may depart the system. Similarly, at the end of providing service, the server can either opt to take vacation with probability p2 or be idle with probability q2. During the busy period, the server may experience breakdown. Both the breakdown times and repair times of the server follow exponential distribution with parameter a and 5 respectively. The resulting QBD process is analysed in the steady state by employing matrix analytic method. The busy period analysis of our model has also been done. Finally, the numerical and graphical illustration of our model has been given.

    Keywords: Markovian arrival process, Phase type distribution, Retrial queues, Bernoulli vacation, Bernoulli feedback, Breakdown and Repair.

    I Introduction

    On contrary to the normal queueing model, customers arriving to the retrial queueing system join the orbit, if all the servers are busy. They stay in the orbit for some amount of time which is usually exponentially distributed and then they try to know whether there is a possibility to receive service. If the server is free at that moment, they start to receive the service; if not, they come back to the orbit and repeat the process again. In this situation, if the server is available, then there is also a possibility for the new arrival to receive service directly without joining the orbit. Due to this complexity, it is usually tedious to derive analytic results for the retrial models. However, a vast amount of numerical and approximation methods are available to study the retrial queueing systems.

    One of the most versatile modelling tools in the theory of point processes is the Markovian Arrival Process (MAP). Neuts (1979) introduced a new concept namely Versatile Markovian PointProcesses (VMPP) to model the arrival processes which are not essentially renewal processes. In order to understand VMPP in a clear and simpler way, Lucantoni et al (1990) coined two new terms, namely MAP and Batch MAP. Chakravarthy (2010) have greatly discussed about MAP in the Encyclopaedia of Operations Research and Management Science. A MAP is usually characterized by the parameter matrices (D0, D1), each of which of dimension n in which D0 governs the change over related to no arrivals whereas D1 governs the change over related to

    arrivals. The generator matrix of the resulting continuous time Markov chain is given by Q = D0 + The point process characterized by the MAP is a peculiar class of semi-Markov process whose transition probability matrix is ​​given by

    ? e ° oXD1dx = [I - eD ° t] (- D0) -1D1. Let the transition probability vector of the generator matrix D = D0 + D1 be denoted by n such that nQ = 0 and ne = 1. Then, the average arrival count per unit time in the stable version of the MAP, also termed as the fundamental rate is defined as A = nD1em. Latouche et al (1999) have deeply discussed about PH-distributions and QBD.

    If the duration between consecutive retrials are exponentially distributed with rate na, where n is the size of the orbit, then such retrial queues are said to follow classical retrial policy and most of the retrial queues follow this policy. But, present situation in communication protocols and local area networks indicates that there are a vast number of queueing situations in which the retrial rate does not depend on the orbit size. This is known as constant retrial policy and it was first studied by Fayolle (1986) while examining the telephone exchange model as a classical single server Markovian retrial queue. He has modelled his system in such a way that the orbital customers make a queue and the request for service can be made by the foremost customer in the waiting line and the retrial times follow exponential distribution with rate a. Artalejo et al (2000) studied the Markovian retrial queueing system with multiserver and constant retrial rate.

    If a server gets breakdown, then the customer who is currently receiving service has the choice of being in the system till the repair process gets over or permanently move out of the system or go back to the orbit to resume the service. Such situations are mostly seen in computer and communication networks, at airport with stacked aircraft and in retail shops. Aissani (1998), Kulkarni and Choi (1990) have studied about the effect of unreliable server and the repair process on retrial queues. Reliability of the non-Markovian queueing system when the server is prone to breakdown was analysed by Aissani and Artalejo (1998).

    Wang et al (2009) have greatly analyzed the discrete time retrial queue in which server is subject to breakdown and repair and obtained generating function for both system size and orbit size. Retrial queues with batch Markovian Arrival Process, breakdown and repair has been investigated by Li et al (2006) They studied the system by combining matrix-analytic method and the censoring technique with the supplementary variable method. By using Generalized Stochastic Petri Nets (GSPN), Gharbi et al (2011) have proposed an approach for analyzing the finite source retrial systems in which server is prone to breakdown and repair. Efrosin et al (2011) have examined the Markovian retrial queueing model with constant retrial rate and an unreliable server. Dimitriou et al (2010) have investigated the repairable retrial model and derived the stability condition for it.

    The concept of Bernoulli schedule vacation is that after providing service to the customer, the server may opt to take vacation with probability p or starts the service to next customer with probability 1 - p. Keilson et al (1986) have analyzed the non-Markovian vacation queueing model with Bernoulli schedules. The average waiting time for the non-Markovian cyclic service queues with Bernoulli schedules has been derived by Servi (1986).

    The concept of Bernoulli feedback is that after receiving the service, the customer can go to orbit with probability p to get the service again or departs the system permanently with probability q (p + q = 1). While investigating the single server non-Markovian model, Tak'acs (1963) has introduced the concept of feedback mechanism. Krishnakumar et al (2010) have studied the single server Markovian retrial queueing system with feedback and derived the joint distribution of the server state and the orbit size by using generating function technique. Choudhury et al (2005) have derived the distribution for queue size and busy period for the non-Markovian queueing model with two stages of non-homogeneous services and Bernoulli feedback mechanism. Chen et al (2015) have greatly analysed the concept of Bernoulli vacation policy and Bernoulli feedback for the retrial model in discrete time. Badamchi Zadeh et al (2008) have

    discussed about the non-Markovian queueing system with Bernoulli feedback and Bernoulli vacation. Retrial queues in which the server is prone to experience breakdown and repair have been studied by Kulkarni and Choi (1990). The steady state probability vectors of our model are computed by employing matrix geometric method which was introduced by M. F. Neuts (1981). The rate matrix is ​​obtained by means of the logarithmic reduction algorithm specified by Latouche et al (1999).

    Hunter (1983) has shown that the queue length process examined at every distinct embedding is a Markov renewal process. He has given more emphasis on finding discrete time outputs for the queue length processes. Finally, he has examined the Markovian models with finite / infinite waiting space and the instantaneous Bernoulli feedback. A non-Markovian queueing model with Bernoulli feedback mechanism has been studied by Disney et al. (1980). They have shown that the output of their model is also Markov-renewal. Foley et al. (1983) have studied the non-Markovian queueing system with two servers and delayed feedback mechanism. They have given a choice for the customers who have received service from the lower server, either to depart the system or to go to the upper server to get the re-service, They have shown that their system is uniquely weakly lumpable to a Poisson process.

    A non-Markovian batch arrival queueing model with Bernoulli vacation under multiple vacation policy has been investigated by Choudhury et al. (2018). The busy period distribution and the waiting time distribution have been derived for their model under steady state. Chakravarthy (2008) has studied the multi server queueing model in which arrival follows MAP and the vacation times follow phase type distribution. He has obtained the waiting time distribution and has presented various numerical examples. Chang et al. (2018) have studied the multi server Markovian retrial queueing model with feedback customers and unreliable servers. They have calculated the stationary distribution by developing a new recursive algorithm and have derived the cost function. Finally, they have made a comparison to validate the exactness of the approximate optimal solution.

    The main motivating factor for our model is from online shopping. Nowadays, most of the people give more preferences to online shopping. Every company is selling different varieties of goods like dresses, electronic gadgets, house hold articles, etc ,. These items may be considered as different phases. A customer may get service (booking a product) from any of these phases. If a customer is arriving during the busy period, he / she has to wait for some time in the invisible queue (orbit). After providing service to the customer, the server may either go for other jobs like updating their websites, launching new products, etc., (which may be considered as vacation) or the server may remain idle. During busy period, the server may get breakdown (the server problem). The customer who is receiving service at that moment has to join with the orbital customers and retry to get the service again. Moreover, after booking a particular item, if the customer is not satisfied with it (that is in the sense of price or quality of the item), the customer may again join the orbit and retry to get the service. We have framed our model in such a way that it will match with this situation.

    On analysing the literature of queueing theory, we could find many articles that have discussed about the Bernoulli schedule vacation, Bernoulli feedback, breakdown and repair individually and also in different combinations for various arrival and service patterns. In our work, we have modelled our system in such a way that with all these attributes, we have considered MAP for arrival and phase type distribution for service times.

    The rest of our work is structured in this manner: Section 2 briefly discusses about the model under study. The infinitesimal generator matrix is ​​obtained in the Section 3. The analysis of this paper has been done in the steady state in the Section 4. The busy period analysis of our model has been done in the Section 5. Some of the performance measures are evaluated in Section 6. Finally, under the Section 7, the behavioural aspects of our queueing model has been analysed with the support of numerical values ​​and graphical representations.

    II Model Description

    We consider a retrial queueing model with single server in which the customers come to the system as indicated by the Markovian Arrival Process with D0 and D1 as its parameter matrices of dimension n. The service times are supposed to follow phase type distribution with representation (a, T) of order m such that T0 + Te = 0. On an arrival of a customer, if the server is available, then he provides service instantly. If not, the customer has to join the orbit of infinite capacity. Irrespective of the orbit size, each unit makes retrial at a constant rate from the orbit. The inter retrial times follow exponential distribution with parameter / j .. At the end of providing service to the customer, the server may either go for vacation with probability p2 or remains idle with probability q2 where p2 + q2 = 1. Similarly, after receiving the service, if the customer is satisfied, then he leaves the system with probability p1. Otherwise, if the customer is not satisfied, then he joins the orbit with probability ^ to get the service, where p1 + q1 = 1. But, he has no priority over the orbital customers and he has to compete with them to get the service . During the busy period, the server may get breakdown. As a result, the customer who is receiving service at that time has to join the orbit of infinite capacity. After the completion of repair process, the server becomes idle. The breakdown times, the repair times and the vacation times are all supposed to follow exponential distribution with parameters a, 5 and q respectively.

    III The generator matrix

    The formulation of the generator matrix for our queueing model has been done in this section. We will begin our study by describing the following notations which are needed.

    Notations:

    • N (t): Number of customers in the orbit

    • In: An n-dimensional identity matrix

    • e: A column vector (of needed dimension) with each of its entries as 1

    • Kronecker multiplication of two matrices

    • ©: Kronecker addition of two matrices

    • Y (t) - Nature of the server at time t, where

    Y (t) = {0, the server is in vacation.

    1, the server is idle.

    2, the server is offering service.

    3, the server is in breakdown.

    • S (t): Phase of the service process at time t

    • M (t): Phase of the Markovian Arrival Process at time t

    • A: Rate of arrival and is defined as A = nD1e where n is the invariant probability vector

    of

    the generator matrix D = D0 + D1

    • y: Rate of service, where y = [a (-T) -1e] -1

    It is obvious and can proved that {(N (t), Y (t), S (t), M (t)): t 0} is a continuous time Markov chain (CTMC) whose state space is given below:

    Y = l * J l (i)

    where

    l (i) = {(i, 0,1): i>1,1<l<n} J {(i, 1,1): i>1,1<l<n} J {(?, 2, k, l): i > 1,1 < k < m, 1 <

    I < n}

    J {(i, 3, l): i >1,1<l<n}

    The generator matrix of the Markov chain is as follows:

    rB00 B01 0 0 0 0 0 B10 A1 A0

    0 0 0 0 0 A2 A1 A0 0 0

    0 0 0 A2 A1 A0 0 0 ......

    Q =

    where,

    Boo = [^ o- ^ In ^ n 00 Do a®fl1p1p2 ^ 0® I "P1 ^ 0®In r0 (Do-ffI")] Bo1 =

    1

    [D1 0 0 00 0 0 0 ^ 7 ° ®I "q ^ 0® ^ Im ® A em ® ctI"]

    Do - ^ In 0

    B10 = [0 0 00 0 ^ a ® I "0 0 00 0 0] A1 = 00 a®?>1 0p1p2 ^ ° ® I "® In r0 (D ° -CTIn) 00

    5L

    0

    A) -

    A =

    0 0 0 00 0 ^ a ® I "00 0 0 00 000

    A =

    |D1 0 0 00 0 0 0 ^ 7 ° ®In q ^ 0® ^ Im ® A em ® ctI "0 0 0

    IV Analysis of the System

    The analysis of the our model has been discussed in this section under steady state.

    4.1 Stability Condition

    Let us define A = A ° + A1 + A2. Then, clearly A is an infinitesimal generator matrix and as a result, we can find an invariant probability vector V of A which obeys

    VA = 0; Ve = 1 where the vector V is given by V = ^ 3).

    The vector V, partitioned as V = is computed by solving the following

    equations:

    + ^ [A) - MU + ^ r0 ® I "] + ^ 3 [« In] = 0 ^ 1 [a ® ^ + ^ a ® I "] + © (D - CTIJ] = 0

    subject to

    ^ 0 + + ^ 2 + ^ 3 = 1.

    The necessary and sufficient condition required by the system to attain stability is < V42e i.e.,

    ^ 0 [? * 1en] + ^ fa ^ 0 ® e "+ em ® D ^ n + em ® aen] + ^ D ^ n <

    4.2 The Transition Probability Vector

    Let the transition probability vector of the infinitesimal generator Q be indicated as x. This probability vector can be partitioned as: x = (x0, x1, x2, ...), where x0 is of dimension 2n + mn and Xj is of dimension 3n + mn, for i > 1.

    Since x is a transition probability vector of Q, the following two conditions will be satisfied

    by it:

    xQ = 0andxe = 1

    Once the condition for the system to be stable is achieved, the invariant probability vector x can be computed using

    xi + 1 = x1R ', i > 0

    and the remaining vectors namely, x0andx1 can be evaluated by solving the equations given below:

    X0B00 + X1B10 = 0

    X0B01 + XJA1 + RA2] = 0

    Step0: H ^ (-A1) -1A0, L ^ (-A1) -1A2, G = LandT = H.

    based on the normalizing condition

    x0e2n + mn + X1 [I - R] 1e3n + mn = 1

    The rate matrix R can be evaluated by making use of Logarithmic Reduction Algorithm proposed by Latouche et al. (1999).

    Logarithmic Reduction Algorithm:

    1) A2,

    Stepl: U = HL + LH M = H2 H ^ (I - U) -1M

    M ^ L2 L ^ (I - U) -1M G ^ G + TL T ^ TH

    continueStepluntiI || e - Ge ||?< e. (A1 + a0G

    Step2: R = -A0 (A1 + A0G) -1

    V Analysis of the busy period

    Under this section, we perform the analysis of busy period of our system.

    The duration of time between the customer's arrival to the void system and the instant at which the system size reaches zero for the first time is termed as the busy period. Hence, the first passage time from level 1 to level 0 is the analogue of the busy period.

    The first return time to level zero with the condition that there should be atleast one visit to a state in any other levels is termed as the busy cycle. Let us first bring in the idea of ​​fundamental period for the purpose of analysing the busy period. As far as the QBD process is concerned, it is nothing but the first passage time from the level i to the level i - 1, where i > 2. The discussion has to be done separately for the boundary states (i.e.) for the cases = 0,1. It can be easily seen that there are 3n + mn states for each level i where i > 1. Therefore, while arranging the states in the lexicographic order, jth state of level i may be indicated as (i, j).

    NOTATIONS:

    • Gjj, (k, x) - The conditional probability that the QBD process enters the level i - 1 by making precisely k transitions to the left and also by entering the state (i, j ') given that it started in the state ( ,) at time = 0.

    • Gjp (z, s) = e-sxdGjp (k, x): Izl < 1, Re (s) > 0

    • G (z, s) - Thematrix (Gjj, (z, s))

    • G = (Gjj,) = G (1,0) - The matrix which concerns about the first passage times without including the boundary states.

    • G (p0 (k, x) - The conditional probability that the QBD process enters the level 0 by making precisely k transitions to the left given that it started in the level 1 at time = 0.

    • G (p0 (k, x) - The first return time to the level 0.

    • E ^ - The expected first passage time from the level i to the level i - 1, given that at time = 0, the process is in the state (,).

    • E1 - The column vector with E ^ as its entries.

    • E2j - The expected number of customers who received service during the first passage

    time from the level to the level - 1, given that the first passage time begins in the state (,).

    • IE2 - The column vector with E2j- as its entries.

    • IE (1'0) - The vector which gives the expected first passage times from the level 1 to the

    level 0.

    • E (1'0:> - The vector which gives the expected number of service completions in the first passage time from the level 1 to the level 0.

    • E (0'0:> - The expected first return time to the level 0.

    • E (0'0:> - The expected number of service completions in the course of first return time to

    the level 0.

    It can be easily seen that the matrix G (z, s) satisfies the following equation: G (z, s) = z [sI - A1] -1A2 + [sI-A1] -1A0G2 (z, s) Once the rate matrix R is evaluated, we can easily find the matrix G by making use of the result

    G = - [A1 + RA2] -1A2 The matrix G may also be evaluated by employing logarithmic reduction algorithm.

    As far as the boundary states are concerned, namely 0 and 1, we have the following equations which are satisfied by G (1,0) (z, s) and G (0,0) (z, s) respectively.

    G (1'0) (z, s) = z [sI - A1] -1B10 + [sI - A1] -1A0G (z, s) G (1'0) (z, s)

    G (0,0) (z, s) = [sI-B00] -1B01G (1,0) (z, s). Since, the three matrices namely, G, G (1,0) (1,0) and G (0,0) (1,0) are stochastic, we may easily evaluate the following moments:

    d

    E1 = --G (z, s) | s = 0.z = 1 = - [A1 + A0 (G + I)] - 1e

    d

    E2 = ^ G (z, s) | s = 0; z = 1, = - [A1 + A0 (G + I)] - 1A2e

    (1,0) d

    E1 = -? DG (1'0) (z, s) ls = 0; z = 1 = - [A1 + A0G] -1 [A0IE1 + e]

    I2 (1,0) = ± G (1,0) (z, s) | s = 0; z = 1 = - [A1 + A0G] -1 [B10e + A0E2]

    I1 (0'0) = -? DG (0,0) (z, s) | s = 0; z = 1 = -B-01 ^ + B01E1 (1'0)] IE2 (0'0) = ^ G (0,0) (z, s) | s = 0; z = 1 = -B00-1B01E2 (1'0)

    VI Performance Measures

    In order to examine the behaviour of our model in the steady state, a few performance measures for our model are enumerated in this section. • Probability of orbit being empty:

    ^ Empty = ^ 1 = 0 ?? = 1 +? Fc = 1 ?? = 1 X02M

    Probability of server to be idle:

    ) Yn = 0? I = 1

    ^? E = ?? = 0? "= 1 xi1i

    Probability of server to be in vacation: ^ Vacation ?? = 0? I = 1 * i0i

    Probability of server to be busy:

    n _ yra ™ vm

    rfcusy =? i = 0? fc = 1? i = 1 Xj

    Probability of server to be in breakdown:

    ) yn

    = 1 Li = 1

    P = Ln v

    r breakdown? -? i = 1? -? 1 = 1 A-i3l

    • Probability of a new arrival getting into service directly:

    Ps = 1sr = 0 XnD1e}

    • Probability of a new arrival getting to receive service directly with a minimum of one customer waiting in the orbit:

    Psw = 1 {Li = 1 xi1 ^ 1e}

    The total retrial rate at which the orbital customers appeal for service:

    .,* - niY ^ Yn f -i- Ym Yn f -i- Ym Ym Yn v -i- Ym Yn H- = ?? {Li = 1 Li = 1 xi0i + Li = 1 Li = 1 xni + Li = 1 Lk = 1 Li = 1 xi2ki + Li = 1 Li = 1

    The effective retrial rate:

    TOT yn

    n = 1 L 1 = 1

    Ps = P {Li = 1 L = 1 xi1l}

    Expected orbit size

    Eorbit Li = 1 ixi&3n + mn

    Average system size:

    Esystem Eorbit + Pbusy

    Probability of a successful retrial:

    v + x 1

    Psr =,. . J {Li = 1 L = 1 xi1l}

    Mean number of successful retrial: Esrt = 1 ^ x {Li = 1 L = 1 i-xi1l}

    VII Numerical Results

    In this section, we will analyse the behaviour of our model numerically as well as graphically. The following five different MAP representations, all of which have the same mean, say 1, are taken into consideration for the arrival process. Erlang of order 2:

    D0 = [-2 20 -2]; D1 = [0 02 0]

    Exponential:

    D0 = [-1]; D1 = [1]

    Hyperexponential:

    D0 = [-1.90 00 -0.19]; D1 = [1.710 0.1900.171 0.019] Since, all these three arrival process are renewal, their correlation is zero.

    Consider the following three phase type distribution for the service times. Erlang of order 2:

    a = (1,0); T = [-12 120 -12]

    Exponential:

    a = (1); T = [-6]

    Hyperexponential:

    a = (0.8,0.2); T = [-16.8 00 -1.68]

    Illustrative Example 1:

    In the following tables, we examine the impact of the retrial rate ^ against the expected orbit size. Fix X = 1; y = 6; p1 = 0.6; ^ = 2; p2 = 0.4; a = 1; q1 = 0.4; 5 = 2; q2 = 0.6.

    Table 1: Expected orbit size - Exponential Service

    ? ARRIVAL

    EXPONENTIAL ERLANG HYPEREXPONENTIAL

    11 10.0011675 7.48381067 28.7824173

    13 7.70793519 5.79663583 21.8380220

    15 6.53032933 4.92315841 18.2956968

    17 5.81352572 4.38926486 16.1464306

    19 5.33131245 4.02923081 14.7031629

    21 4.98472155 3.77006250 13.6669608

    Table 2: Expected orbit size - Erlang Service

    ARRIVAL

    EXPONENTIAL ERLANG HYPEREXPONENTIAL

    11 12.7430548 9.45396003 37.5253785

    13 9.29498782 6.94117250 26.9239288

    15 7.66359572 5.73985615 21.9503036

    17 6.71251431 5.03600320 19.0617824

    19 6.08961722 4.57372333 17.1738503

    21 5.65003242 4.24691754 15.8431278

    Table 3: Expected orbit size - Hyperexponential Service

    ? ARRIVAL

    EXPONENTIAL ERLANG HYPEREXPONENTIAL

    11 4.86515668 3.72736456 12.8474069

    13 4.18208704 3.21276505 10.8735917

    15 3.76653401 2.89837776 9.67760920

    17 3.48707744 2.68644585 8.87528324

    19 3.18123908 2.53393179 8.29971350

    21 3.13499689 2.41893523 7.86667776

    From the Table 1-3, we have the following observations.

    • While maximizing the retrial rate, the expected orbit size minimizes for different arrangements of service and arrival times.

    • While comparing to Erlang and Exponential arrival times, the expected orbit size decreases more rapidly in the case of hyperexponential arrival time. Similarly, the expected orbit size decreases slowly in the case of Erlang arrival time.

    Illustrative Example 2:

    In the following tables, we examine the impact of the service rate y on the expected orbit size. Fix X = 1; n = 8; p1 = 0.60; 77 = 2; p2 = 0.40; a = 1; q1 = 0.40; 5 = 2; q2 = 0.60.

    Table 4: Expected orbit size - Exponential Service

    Y ARRIVAL

    EXPONENTIAL ERLANG HYPEREXPONENTIAL

    7 8.62406523 6.38645821 24.9106128

    8 5.34880881 3.98252157 14.9745343

    9 3.99796043 2.98468674 10.9073389

    10 3.26376783 2.44125073 8.70686572

    11 2.80380376 2.10058231 7.33376198

    12 2.48925463 1.86760598 6.39840730

    13 2.26091389 1.69851986 5.72205463

    Table 5: Expected orbit size - Erlang Service

    Y ARRIVAL

    EXPONENTIAL ERLANG HYPEREXPONENTIAL

    7 10.0886405 7.41091127 29.7073972

    8 5.78766750 4.28690000 16.4457714

    9 4.20009203 3.12293216 11.6004235

    10 3.37742675 2.51789282 9.10389213

    11 2.87562366 2.14837061 7.58839020

    12 2.53827076 1.89982294 6.57424954

    13 2.29625267 1.72148762 5.85002618

    Table 6: Expected orbit size - Hyperexponential Service

    Y ARRIVAL

    EXPONENTIAL ERLANG HYPEREXPONENTIAL

    7 4.97313477 3.77351943 13.3210642

    8 3.82708589 2.90365725 10.0520703

    9 3.17297465 2.40685867 8.19104494

    10 2.75103836 2.08653058 6.99299693

    11 2.45682845 1.86336640 6.15918869

    12 2.24027528 1.69928391 5.54661758

    13 2.07439882 1.57374500 5.07829007

    From the Table 4-6, we have the following observations.

    • While maximizing the service rate, the expected orbit size minimizes for various possible arrangements of arrival and service times.

    • While comparing to Erlang and Exponential arrival times, expected orbit size decreases more rapidly forhyperexponential arrival time. Similarly, the expected orbit size reduces gradually for Erlang arrival time.

    Illustrative Example 3:

    In the following tables, we examine the impact of the vacation rate q on expected orbit size. Fix A = 1; y = 6; p1 = 0.60; n = 8; p2 = 0.40; a = 1; q1 = 0.40; 5 = 2; q2 = 0.60.

    Table 7: Expected orbit size - Exponential Service

    n ARRIVAL

    EXPONENTIAL ERLANG HYPEREXPONENTIAL

    4 3.37760655 2.48794146 9.31560352

    6 2.38366783 1.76179679 6.27202950

    8 2.04477644 1.51618610 5.22962692

    10 1.87519953 1.39392122 4.70752570

    12 1.77370727 1.32100067 4.39510928

    14 1.70624181 1.27264837 4.18755323

    16 1.65818186 1.23826766 4.03979561

    Table 8: Expected orbit size - Erlang Service

    n ARRIVAL

    EXPONENTIAL ERLANG HYPEREXPONENTIAL

    4 3.76004704 2.74496640 10.6614094

    6 2.59592138 1.90098311 7.03743508

    8 2.20977550 1.62305018 5.82862515

    10 2.01854731 1.48608798 5.22879429

    12 1.90472438 1.40484025 4.87154080

    14 1.82932015 1.35114669 4.63485557

    16 1.77572971 1.31305560 4.46666909

    Table 9: Expected orbit size - Hyperexponential Service

    n ARRIVAL

    EXPONENTIAL ERLANG HYPEREXPONENTIAL

    4 2.20929153 1.67412072 5.46231400

    6 1.65814092 1.26336768 3.87581037

    8 1.45463836 1.11330793 3.29231358

    10 1.34958578 1.03632469 2.99270300

    12 1.28565431 0.989662976 2.81122828

    14 1.24271322 0.958408469 2.68981153

    16 1.21190604 0.936030889 2.60298246

    From the Table 7-9, we have the following observations.

    • While maximizing the vacation rate, the expected orbit size minimizes for various possible arrangements of arrival and service times.

    • While comparing to Erlang and Exponential arrival times, the expected orbit size decreases more rapidly for hyperexponential arrival time. Similarly, the expected orbit size reduces gradually for Erlang arrival time.

    Illustrative Example 4:

    In the following tables, we examine the impact of the repair rate 5 against the expected orbit size. Fix X = 1; y = 6; p1 = 0.60; ^ = 2; p2 = 0.40; a = 1; q1 = 0.40; ^ = 8; q2 = 0.60.

    Table 10: Expected orbit size - Exponential Service

    S ARRIVAL

    EXPONENTIAL ERLANG HYPEREXPONENTIAL

    4 7.59759409 5.51219114 22.7455157

    6 5.87750594 4.25241592 17.5011470

    8 5.24414894 3.78930406 15.5590737

    10 4.91583865 3.54965983 14.5484770

    12 4.71517696 3.40339073 13.9291648

    14 4.57989976 3.30488730 13.5108519

    16 4.48255049 3.23406112 13.2093878

    Table 11: Expected orbit size - Erlang Service

    S ARRIVAL

    EXPONENTIAL ERLANG HYPEREXPONENTIAL

    4 9.01354432 6.47340005 27.6296447

    6 6.68529225 4.79331402 20.3620356

    8 5.87341675 4.20725586 17.8186074

    10 5.46167788 3.91034115 16.5247530

    12 5.21302166 3.73120654 15.7416246

    14 5.04665662 3.61145470 15.2167986

    16 4.92756297 3.52578803 14.8406217

    Table 12: Expected orbit size - Hyperexponential Service

    S ARRIVAL

    EXPONENTIAL ERLANG HYPEREXPONENTIAL

    4 4.31052599 3.21966516 11.8178685

    6 3.70054138 2.75276167 10.1062609

    8 3.44417977 2.55760891 9.37940438

    10 3.30337635 2.45078331 8.97785988

    12 3.21444535 2.38346547 8.72330076

    14 3.15320451 2.33718412 8.54755003

    16 3.10847255 2.30342084 8.41893285

    From the Table 10-12, we have the following observations.

    • While maximizing the repair rate, the expected orbit size minimizes for various possible arrangements of arrival and service times.

    • While comparing to Erlang and Exponential arrival times, the expected orbit size decreases more rapidly in the case hyperexponential arrival time. Similarly, the expected orbit size decreases gradually for Erlang arrival time.

    Illustrative Example 5:

    In the following tables, we examine the impact of the breakdown rate a against expected orbit size. Fix A = 1; y = 6; p1 = 0.60; r \ = 2; p2 = 0.40; 5 = 2; q1 = 0.40; n = Q; q2 = 0.60.

    Table 13: Expected orbit size - Exponential Service

    a ARRIVAL

    EXPONENTIAL ERLANG HYPEREXPONENTIAL

    0.2 4.08040357 2.96155493 11.8191902

    0.3 4.73131254 3.44689911 13.7479083

    0.4 5.54494873 4.05191775 16.1683587

    0.5 6.59105242 4.82715640 19.2937651

    0.6 7.98585710 5.85628764 23.4817132

    0.7 9.93858363 7.28856770 29.3811856

    0.8 12.8676733 9.41875439 38.3052798

    Table 14: Expected orbit size - Erlang Service

    a ARRIVAL

    EXPONENTIAL ERLANG HYPEREXPONENTIAL

    0.2 4.08450150 2.94457362 11.9867048

    0.3 4.80393168 3.47566946 14.1600116

    0.4 5.73618355 4.16175105 16.9879714

    0.5 6.99170138 5.08201017 20.8145996

    0.6 8.77333791 6.38066758 26.2765620

    0.7 11.4985662 8.35096411 34.6983724

    0.8 16.1853911 11.6946701 49.3649830

    Table 15: Expected orbit size - Hyperexponential Service

    a ARRIVAL

    EXPONENTIAL ERLANG HYPEREXPONENTIAL

    0.2 4.02198460 3.02084111 10.8591758

    0.3 4.32444739 3.25474173 11.6877493

    0.4 4.65258139 3.50785258 12.5914617

    0.5 5.01076886 3.78348382 13.5827069

    0.6 5.40425830 4.08558108 14.6764439

    0.7 5.83942470 4.41891647 15.8909556

    0.8 6.32412046 4.78934454 17.2488808

    From the Table 13-15, we have the following observations.

    • While maximizing the breakdown rate, the mean orbit size also maximizes for all possible arrangements of arrival and service times.

    • While comparing to exponential and hyperexponential service times, the mean orbit size increases more rapidly for Erlang service time. Similarly, the mean orbit size increases slowly for hyperexponential service time.

    Illustrative Example 6:

    In the following figures, we observe the impact of both the retrial rate ^ and vacation rate ^ against expected system size. Fix A = 1; p1 = 0.60; y = 6; p2 = 0.40; a = 1; q1 = 0.40; 5 = 2; q2 = 0.60.

    Figure 1: Exponential arrival

    Figure 2: Erlang arrival

    Figure 3: Hyperexponential arrival

    A quick observation of Figure 1-3 discloses the fact that expected system size decreases while maximizing both the retrial rate and vacation rate for all possible arrangements of arrival and service times.

    Illustrative Example 7:

    In the following figures, we analyse the influence of both the repair rate 5 and the service rate y against the probability of server being idle.

    We fix X = 1; p1 = 0.60; ^ = 2; p2 = 0.40; n = 8; q1 = 0.40; a = 1q2 = 0.60.

    "« I «• | •» M&.-I Om>-

    Figure 4: Exponential arrival

    CtiWI Gtwh CfcTWl Сгщ * \

    Figure 5: Erlang arrival

    • "mm)

    Figure 6: Hyperexponential arrival

    A quick view of Figure 4-6 reveals the fact that the probability of server being idle increases while increasing both the repair rate and the service rate for all possible arrangements of arrival and service times.

    VIII Conclusion

    In our work, we have discussed about the retrial queuing system in which arrival follows MAP and service time follows PH-distribution together with Bernoulli schedule vacation, Bernoulli feedback, breakdown and repair. The effect of retrial rate, repair rate, vacation rate, breakdown rate and service rate on the average orbit size has been analysed through numerical values. Also, the influence of both the repair rate and the service rate against the probability of server being idle and the effect of retrial rate and the vacation rate on the average system size have been clearly visualized with the support of graphical representations. We have also analysed the busy period for our model. Our work can be extended to queueing models in which arrival follows BMAP.

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    Ключові слова: Markovian arrival process /Phase type distribution /Retrial queues /Bernoulli vacation /Bernoulli feedback /Breakdown and Repair.

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