On a Single Server Vacation Queue with Two Types of Service and Two
Types of Vacation
KAILASH C. MADAN
Department of Mathematical Sciences,
Ahlia University,
P. O. Box 10878, Manama,
BAHRAIN
Abstract: - We study a single server queueing system that receives singly arriving customers according to a
Poisson process. The server offers one of the two types of heterogeneous services. Before the beginning of a
service, , the customer can choose an exponential service with probability or a deterministic service with
probability , where Immediately after a service is completed, the server has a choice of taking a
vacation with probability
δ
, or, with probability
1-δ
, the server may continue staying in the system. We further
assume that if the server opts to take a vacation, then with probability
1
α
, he may take a vacation of an
exponential duration with mean vacation time
1/υ (υ > 0)
or with probability
2
α
he may want to take a
deterministic vacation with constant duration d>0, where
12
α +α = 1
. After a vacation is complete, the server
instantly starts providing service if there is at least one customer in the system or the server remains idle in the
system till a new customer arrives for service. We find a steady state solution in terms of the generating
function of the queue length as well as the steady state probabilities for all different states of the system.
Key-Words: - Single server, Poisson arrivals; exponential service, deterministic service; exponential vacation,
deterministic vacation; generating function; queue length, States of the system, steady state.
1 Introduction
In the majority of queueing systems, the server
provides the same kind of service to customers and
the service time follows the same distribution, [1], [2],
[3]. In addition, as it happens in the majority of the
vacation queueing systems, the server’s vacation
follows the same distribution, [4], [5], [6], [7], [8]. In
the last couple of decades, vacation queues have been
studied extensively. The work done by all these, and
many other authors deals with service interruptions
either due to random system failures or due to
optional server vacations with many different
vacation policies. Queueing systems with
deterministic service or deterministic vacations have
been studied by many authors including, [9], in which
the author deals with a queueing system which
allows the server to opt for either an exponential
vacation or for a deterministic vacation. In the present
paper, we extend the idea in [9] and study a queueing
system in which the customer has a choice of either
taking a service with exponential duration or a
deterministic duration in addition to the server having
the choice of taking a vacation of an exponential
length or a deterministic vacation or no vacation after
each service. Symbolically, we denote our system as
queueing system. We find
steady state generating functions of queue lengths of
all different states of the system and derive results
corresponding to various interesting special cases
including the earlier known results of the systems
M/M/D/1, M/D/M/1, M/M1 and M/D/1.
2 Model Description
Customers arrive at the system one by one
according to a Poisson process with mean arrival
rate
Before his service starts, a customer can opt for
an exponential service with mean service time
with probability or a deterministic
service of constant duration ‘k’ with probability
, where .
As soon as a service of a customer is complete,
the server may decide to go on a vacation with
probability
δ
, or may not take a vacation with
probability .. Next, we assume that if the
server decides to take a vacation, then with
probability
1
α
, he may take a vacation of random
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Kailash C. Madan
E-ISSN: 2224-2678
98
Volume 23, 2024
length which follows an exponential distribution
with mean vacation time ( or with
probability
2,α
he may take a deterministic
vacation with constant duration ‘
d
’. Where
12
α +α = 1
.
As soon as his vacation is over, the server
immediately takes up a customer at the head of
the queue for service, if a customer is waiting in
the queue. However, if on returning the server
finds the queue empty, then he still joins the
system and remains idle until a new customer
arrives in the system.
All stochastic processes involved in the system
are independent of each other.
3 Definitions and Equations
We assume that is the steady state probability
that there are
n( 0)
customers in the queue excluding
one customer in service and the server is providing
exponential service, is the steady state
probability that there are
n( 0)
customers in the
queue and the server is providing a deterministic
service,
1
n
V
is the steady state probability that there
are
n( 0)
customers in the queue and the server is on
exponential vacation, is the steady state
probability that there are
n( 0)
customers in the
queue and the server is on deterministic vacation.
Further, let be
the steady state probability that there are
n( 0)
customers in the queue irrespective of whether
the server is providing any type of service or is on
any type of vacation.
Next, we define
Q
to be the steady-state probability
that there is no customer in the system and the server
is idle.
We further assume that is the probability of
r
arrivals during the period of deterministic service
time k and therefore,
(1)
Next, we assume that is the probability of
r
arrivals during the period of deterministic
vacation d and therefore,
(2)
Now, we define the following Probability Generating
Functions (PGFs):
(3)
(4)
(5)
(6)
=
, (7)
.
(8)
4 Equations Governing the System
We use probability reasoning to obtain the following
steady-state equations:
+
+
(9)
+ ( +
( (10)
+ (
+ (
(11)
+ (
+ (
(12)
Kailash C. Madan
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Volume 23, 2024
+
, (13)
(14)
+
, (15)
Q =
. (16)
5 Steady State Solution
Use of the standard generating function approach,
equations (9), (10) give.
=
+ +
(17)
Similar operations on (11) and (12) yield
+
+
(18)
Next, from equations (13) and (14) we obtain.
(19)
And from (15), we get.
(20)
Next, we make use of equations (19) and (20) into
equations (17) and (18), simplify and get.
+ (21)
+
+
+
+ (22)
Now, we re-write equations (21) and (22) as:
_
= (23)
(24)
Re-writing equations (23) and (24) in matrix form as:
= ,
(25)
Where:
,
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Volume 23, 2024
,
Solving (25) simultaneously for and
, we obtain:
,
(26)
.
(27)
Next, we use (26) and (27) into (19), (20), simplify and
obtain:
, (28)
, (29)
Where: ,
,
.
Now, at z=1, the generating functions found above
yield:
(30)
(31)
(32)
(33)
It may be noted that Q is the only unknown which
remains to be determined. For this purpose, we will use
the following normalizing condition:
. (34)
On simplifying, (5.18) yields:
(35)
If we use the value of Q from (35) into the results
(30) to (33) we would be able to determine all the
above steady state probabilities in explicit form. Also,
using the value of Q from (35) into the main results
(26) to (29), all the steady state probability generating
functions of the queue length are found in explicit
form.
6 Some Special Cases
6.1 Queue
We substitute , in the system state
probabilities found above in (30) to (33) and (35).
Therefore, we obtain:
(36)
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Volume 23, 2024
(37)
(38)
(39)
(40)
6.2 Queue
Substituting and in the system state
probabilities, we obtain:
(41)
(42)
(43)
(44)
(45)
6.3 M Queue
Putting in the system state probabilities, we get:
(46)
(47)
(48)
(49)
(50)
6.4 Queue: Only
Exponential Service with Both Types of
Vacation
Letting and in the system state
probabilities found above, we get:
(51)
(52)
(53)
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Volume 23, 2024
(54)
(55)
6.5 Queue: Only Deterministic
Service with Both Types of Vacation
We substitute and in the main results
obtained above and obtain.
(56)
(57)
(58)
(59)
(60)
6.6 Queue: Only Exponential
Service with No Server Vacation
We substitute , and to obtain.
(61)
(62)
(63)
(64)
(65)
6.7 Queue: Only Deterministic Service
with No Vacation
Putting , and in the above
results we get:
(66)
(67)
(68)
(69)
(70)
The results derived in special cases 6 and 7 are
known results in queueing theory literature.
Acknowledgment:
The author is sincerely thankful to all reviewers for
their valuable comments and suggestions in revising
the paper in the final form.
References
[1] Doshi, B, Queueing systems with vacations-a
survey, Queueing Systems, No. 1, pp.29-66,
1986.
[2] Gaver, D. P., A waiting line with interrupted
service including priorities, Journal of Royal
Statistical Society, B 24, pp 3-90, 1962.
[3] Fuhrman, S. W., A note on the M/G/1 queue
with server vacations, Operations Research, No.
32, pp.1368-1373, 1984.
[4] Choi, B. D. and Park. K. K., The M/G/1 queue
with Bernoulli schedule, Queueing Systems 7,
219-228, 1990.
[5] Levy, Y. and Yechiali, U., An M/M/s queue
with servers ‘vacations, INFOR. 14(2) pp. 145-
153, 1976.
[6] Servi, L. D., D/G/1 queues with vacation,
Operations Research, No. pp.34619-629, 1986.
[7] Takagi, H., Time-dependent process of
M/G/1vacation models with exhaustive service,
Journal of Applied Probability, No. 29, pp. 418-
429, 1992.
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Volume 23, 2024
[8] Tegham, L. J., On a decomposition result for a
class of vacation queueing systems, Journal of
Applied Probability, No. 27, pp. 227-231, 1990.
[9] Madan, K. C., On a single server queue with
server’s choice for exponential or deterministic
vacations, AIP Conf. Proc. vol. 1975, 020002
(2018), https://doi.org/10.1063/1.5042170.
Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
The author contributed in the present research, at all
stages from the formulation of the problem to the
final findings and solution.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
No funding was received for conducting this study.
Conflict of Interest
The author has no conflicts of interest to declare.
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(Attribution 4.0 International, CC BY 4.0)
This article is published under the terms of the
Creative Commons Attribution License 4.0
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104
Volume 23, 2024