In this article an M/G/1 queueing model with single server, Poisson input, k-phases of heterogeneous services and Bernoulli feedback design has been considered. For this model, we derive the steady-state probability generating function (PGF) of queue size at the random epoch and at the service completion epoch. Then, we derive the Laplace-Stieltjes Transform (LST) of the distribution of response time, the means of response time, number of customers in the system and busy period.
M/G/1 Queue; Bernoulli Feedback; Queue Size; Waiting Time and Busy Period1. Introduction
The M/G/1 queueing model is one of the famous and applied models in which the distribution of service times is unknown. For this reason, many of real models could be considered by this model. Multi-phase service is important, because some of systems have more than one phase service, for example manufacture production lines. Feedback is also important, because in some queueing models, some customers, after completion the service, may need to go back to the end of queue to take the service again. It means that the service customer is not acceptable and must go to the end of queue. According to these conditions, we have considered an M/G/1 queueing model with k-Phase Optional Services and Bernoulli feedback.
For this model, first we find the steady-state probability generating function (PGF) of queue size at the random epoch and at the service completion epoch. Then, we derive the Laplace-Stieltjes Transform (LST) of the distribution of response time. The means of response time, number of customers in the system and busy period will be derived by using the PGF and LST.
In relation of this model, [1-9,11] have derived some results. The model that they have considered is two phases. But, in this article, we will consider a k-phase queue with optional service and Bernoulli feedback in all phases. Of course, [10] studied an M/G/1 queue with k-phase services and vacation, but without feedback that is different from this paper.
Following, in section 2 we describe the model and give some definitions. In section 3, the PGF of the system size will be derived. In section 4, we will find some measures of effectiveness. At the final section, we provide a conclusion.
2. The Mathematical Model and Definitions
In this model, the server provides first phase of regular service to all the customers. As soon as the i-th (for i = 1, ∙∙∙, k − 1) phase of service of a customer is completed , it may leave the system or immediately go for (i + 1)-th phase of optional service. However, after receiving each phase of unsuccessful service by a unit, then it may immediately join to the end of tail of the original queue as feedback customer to take service again. Thus, the assumptions of the model are:
1) Customers arrive at the system to a Poisson process with rate.
2) The service discipline of the system is FCFS1.
3) The server provides k-phases of heterogeneous service for any customer. The service times for k-phases are independent random variable that denoted by with distribution functions and LST of these distributions are. These variables have finite moments, that is for.
4) As soon as the i-th phase of service of a customer is completed, the customer may go to the (i + 1)-th phase of service with probability.
5) After completion of the i-th phase, if the customer is dissatisfied with its service for certain reason or it received unsuccessful service, in this case the customer may immediately joins the end of the original queue as a feedback customer for receiving the service again with probability, for, otherwise the customer may depart the system with probability.
Definition 2.1. The modified service time or the time required by a customer to complete the service cycle is given by
(2.1)
where and. Then the LST of B is given by
and
As we know, in the queueing systems the utilization factor is and denoted by. This measure says if the system is in steady state or not. In this article, we study the model in equiblirium. It happens when.
Definition 2.2. The elapsed of i-th phases service at time “t” is denoted by for. We introduce the random variable as follow
Thus, we have a bivariate Markov process , where if and if, for. Now, we define probabilities as
for and
We know that and, for . Also is continuous at. Then we have the hazard rate functions of as
where, and is the conditional probability of completion of i-th phase of service during the time interval, given that the elapsed service time is.
By assuming that the system is in the steady state, we let
for.
In the next section, we find the PGF of these probabilities.
3. The PGF of the System Size
Now, for, the PGF of the probabilities that explained by (2-5), are defined as
For finding the steady-state PGF from Kolmogorov forward equations, for, we can write the steady-state equations as
Also
It is clear that, for. Now, at, the boundary conditions are
Note that, the normalizing condition is
which we use this condition to find the relation between and.
In the next Lemma, we derive the relation between and.
Lemma 3.1. From relation (3-3), we have
Proof. See [4]. □
Proposition 3.2. By the z-transform of that is
we have
(3.10)
Proof. By multiplying the relation (3.7) in and summation from to, and using (3.6), the proof is completed.
Proposition 3.3. For, we have
Proof. By multiplying (3.8) in and summation on to , we can obtain (3.11).
Corollary 3.4. By proposition 3.3, we have
where
Now, by (3.10) and (3.13), we obtain
(3.14)
Corollary 3.5. If, for , we have
(3.15)
and for
(3.16)
Now, if, and is the PGF of queue size distribution at a random epoch, then from (3.15) and (3.16), we have
(3.17)
and the PGF of the queue size at the departure epoch is
(3.18)
In the next section, we find some measures of effectiveness as the means system size, response time and busy period.
4. The Measures of Effectiveness
This section includes three sub-sections. In the sub-section 4.1, we find the mean system size. In the sub-section 4.2, the mean response time is obtained and in the last sub-section we calculate the mean busy period.
4.1. The Mean System Size
If be the mean number of customers in the queue, then we have
For calculating we use the following lemma.
Lemma 4.1.1. By (3.13) and for, we have 1)(4.1.1)
2) (4.1.2)
Now, we write in form ofwhere
and
Since, then by using the L’Hopital rule , we have
where
hence, in which
and
4.2. The Mean Response Time
Here, we show the response time variable with. For finding the mean of, first we need to obtain the LST of the distribution of waiting time in the queue, then by using this, we find the LST of the response time distribution. Then we can find the mean response time.
From classical formula in the queueing theory, we know that, where
Now, if we put and using the (3.18), we have
On the other hand, the response time is defined by, where is queueing time and is the service time. Then, by using the convolution property, the LST of is
in which by some simple computation we find that
where
and
The mean response time, for an arbitrary customer, is
But, then, by using the L’Hopital rule, we have
On the other hand, and, then
with replacing in (4.2.6), it results that
where
and
4.3. The Mean Busy Period
In this section, according to the definition of the busy period, we obtain the mean busy period of the model that we have studied here. Suppose that, and are the busy and idle periods, respectively. Now, according to the renewal theory, these variables are renewal processes and we have
From this, it results
or
or
thus
Now, suppose that is the probability that the server is servicing in i-th phase that is P [the server is busy with (ps)i], for
we have
Also, we know, then
4.4. Special Cases
In this article we have obtained some results for an M/G/1 queue with k-phases of heterogeneous services and Bernoulli feedback design. Now, we consider some special cases of this model that are agreement with the models which have been studied by [8,10]. These special cases are followed.
Special case 4.4.1. If, and the last phase be the vacation for server and without feedback, then, and are the same as those have been obtained by [10].
Special case 4.4.2. If and, then, , and are equal to the [4].
5. Conclusion
In this paper, we considered an M/G/1 queueing model with single server, Poisson input, k-phases of heterogeneous services and Bernoulli feedback design. In this model, as soon as the i-th (for i = 1, ∙∙∙, k − 1) phase of service of a customer is completed , it may leave the system or immediately go for (i + 1)-th phase of optional service. However, after receiving each phase of unsuccessful service by a unit, then it may immediately join to the end of tail of the original queue as feedback customer to take service again. We analyzed this mode via obtaining the steady-state probability generating function (PGF) of queue size at the random epoch and at the service completion epoch. Then, we derived the Laplace-Stieltjes Transform (LST) of the distribution of response time, the means of response time, number of customers in the system and busy period-model, the server provides first phase of regular service to all the customers.
REFERENCESNOTESReferencesD. Bertsimas and X. Papaconstantinou, “On the Steady State Solution of the M/C2(a;b)/S Queueing System,” Transportation Sciences, Vol. 22, No. 2, 1988, pp. 125138. doi:10.1287/trsc.22.2.125O. J. Boxma and U. Yechiali, “An M/G/1 Queue with Multiple Type of Feedback and Gated Vacations,” Journal of Applied Probability, Vol. 34, No. 3, 1997, pp. 773784. doi:10.2307/3215102G. Choudhury, “A Batch Arrival Queueing System with an Additional Service Channel,” International Journal of Information and Management Sciences, Vol. 14, No. 2, 2003, pp. 17-30.G. Choudhury and P. Madhuchanda, “A Two Phase Queueing System with Bernolli Feedback,” Information and Management Sciences, Vol. 16, No. 1, 2005, pp. 35-52. B. D. Choi, B. Kim and S. H. Choi, “An M/G/1 Queue with Multiple Types of Feedback, Gated Vacations and FCFS Policy,” Computers and Operations Research, Vol. 30, No. 9, 2003, pp. 1289-1309.
doi:10.1016/S0305-0548(02)00071-0R. L. Disney, “A Note on sojourn Times in M/G/1 Queue with Instantaneous Bernoulli Feed-Back,” Naval Research Logistics Quarterly, Vol. 28, No. 4, 1981, pp. 679-684.
doi:10.1002/nav.3800280415B. Krishna Kumar, A. Vijaykumar and D. Arivudainambi, “An M/G/1 Retrial Queueing System with two Phase Service and Preemptive Resume,” Annals of Operations Research, Vol. 113, No. 1-4, 2002, pp. 61-79.
doi:10.1023/A:1020901710087K. C. Madan, “An M/G/1 Queue with Second Optional Service,” Queueing Systems, Vol. 34, No. 1-4, 2000, pp. 37-46. doi:10.1023/A:1019144716929J. Medhi, “A Single Server Poisson Input Queue with a Second Optional Channel,” Queueing Systems, Vol. 42, No. 3, 2002, pp. 239-242. doi:10.1023/A:1020519830116G. H. Shahkar and A. Badamchizadeh, “A Single Server Queue with k-Phase of Heterogeneous Service under Bernoulli Schedule and a General Vacation Time,” Journal of Theoretical Statistic, Vol. 20, No. 2, 2006, pp. 151-162.H. Takagi, “A Note on the Response Time in M/G/1 Queue with Service in Random Order and Bernoulli Feedback,” Journal of Operational Research Society of Japan, Vol. 39, No. 4, 1996, pp. 486-500.
.
with distribution functions 
and LST of these distributions are
. These variables have finite moments, that is 
for
.
.
, for
, otherwise the customer may depart the system with probability
.
(2.1)
and
. Then the LST of B is given by
and denoted by
. This measure says if the system is in steady state or not. In this article, we study the model in equiblirium. It happens when
.
at time “t” is denoted by 
for
. We introduce the random variable 
as follow
, where 
if 
and 
if
, for
. Now, we define probabilities as
and
and
, for 
. Also 
is continuous at
. Then we have the hazard rate functions of 
as
, and 
is the conditional probability of completion of i-th phase of service during the time interval
, given that the elapsed service time is
.
.
, the PGF of the probabilities that explained by (2-5), are defined as
, we can write the steady-state equations as
, for
. Now, at
, the boundary conditions are
and
.
and
.
that is
(3.10)
and summation from 
to
, and using (3.6), the proof is completed.
, we have
and summation on 
to 
, we can obtain (3.11).
(3.14)
, for 
, we have
(3.15)
(3.16)
, 
and 
is the PGF of queue size distribution at a random epoch, then from (3.15) and (3.16), we have
(3.17)
(3.18)
be the mean number of customers in the queue, then we have
we use the following lemma.
, we have 1)
(4.1.1)
(4.1.2)
in form of
where
, then by using the L’Hopital rule , we have
, in which
. For finding the mean of
, first we need to obtain the LST of the distribution of waiting time in the queue, then by using this, we find the LST of the response time distribution. Then we can find the mean response time.
, where
and using the (3.18), we have
, where 
is queueing time and 
is the service time. Then, by using the convolution property, the LST of 
is
, then, by using the L’Hopital rule, we have
and
, then
and 
are the busy and idle periods, respectively. Now, according to the renewal theory, these variables are renewal processes and we have 
is the probability that the server is servicing in i-th phase that is P [the server is busy with (ps)i]
, for 
, then
, 
and the last phase be the vacation for server and without feedback, then
, 
and 
are the same as those have been obtained by [
and
, then
, 
, 
and 
are equal to the [