service rule in the top priority queue. The system M, M\lf"b,. M\l. Consider a single ... (8). " ( V ^ ^ l ' ^ ^ + A A-l,n + X2Qm,n-l + vlQm+b ,n. Note that when a = 1 ...
BULL. AUSTRAL. MATH. SOC. VOL.
60K24, 60J27
33 (1986), 237-243.
A PREEMPTIVE PRIORITY QUEUE WITH A GENERAL BULK SERVICE RULE R. SlVASAMY
In this paper a single server preemptive priority queueing system, consisting of two types of units, with unlimited Poisson inputs and exponential service time distributions, is studied.
The higher
priority units are served in batches according to a general bulk service rule and they have preemptive priority over lower priority units.
Steady state queue length distributions, stability
condition and the mean queue lengths are obtained.
Introduction Queues with more than one type of input and with different priority rules have been solved under various assumptions.
Heathcote [2] has
studied a preemptive priority queueing problem with two priorities. Sivasamy [4] has analysed a non-preemptive priority queue by introducing general bulk service rule in the lower priority queue.
In this paper,
we apply similar ideas as in Heathcote [2] and Sivasamy [4] to study a preemptive priority queueing problem by introducing the general bulk service rule in the top priority queue.
The s y s t e m
M, M\lf"b,
M\l
Consider a single server facility in which two independent Poisson classes of units, to be called type-1 units (top priority or higher priority units) and type-2 units (lower priority or nonpriority units), arrive at rates
X
and
A
respectively and form separate queues.
Received 11 June 1985. Copyright Clearance Centre, Inc. Serial-fee code: $A2.00 + 0.00. 237
0004-9727/85
Let
238
R.
Sivasamy
the service times of the type-1 and type-2 units be independently exponentially d i s t r i b u t e d random variables with means respectively.
p.
and
u
The type-1 units have preemptive p r i o r i t y over the type-2
u n i t s and i t operates as follows: As soon as a type-1 unit arrives, the server breaks his serving to a type-2 unit if any, and remains idle.
Then the idle server opens
service for type-1 units when the queue length reaches size If,
at a service epoch, the server finds
where
' q'
'a'
units.
type-1 units in the system
1 < q < a-1 , no matter how many type-2 units are present, he
waits u n t i l there accumulates batch of
'a'
'a'
type-1 u n i t s , whereupon he removes the
units for service; if he finds
a
or more but at most
he takes them a l l in the batch and if he finds more than picks a batch of
b
wait in the queue.
'£>'
units at random for service, while other units If none of the type-1 units i s present at an epoch, the
server s t a r t s servicing the type-2 units.
Under t h i s preemptive p r i o r i t y
d i s c i p l i n e , a type-2 unit may be preempted any number of times. we propose to study
b ,
u n i t s , he
M, M\M'
Hence
, M\ 1 , preemptive p r i o r i t y queueing
system. This queueing system, in a real l i f e s i t u a t i o n , may be observed in the following taxidriver cum repairman problem.
The server i s a taxi
driver who i s also attending to the service of automobiles if he i s free
from taxi driving.
While he is at service as a repairman, if there
is
a demand for his t a x i , then he abandons the automobile service but waits till
a minimum of
'a'
customers accumulate for the taxi service and
takes up the taxi driving with this batch of
'a'
customers.
Then at the
subsequent service epochs, the server attends these two different
jobs, as
indicated in the description of the queueing system under study.
Steady state probabilities and equations P
:
The steady s t a t e probability that there are type-1 units and
n (n > 0)
type-2 units present in the system.
Obviously the server will be at r e s t when and when
(m = 0, n = 0 )
m(0 < m < a-1)
(1 < m < a-1, n > 0)
and he will be busy with the type-2 units
(m = 0, n > 1) .
A preemptive
Q : m,n
priority
queue
239
The steady s t a t e probability that the server i s busy with a batch of type-1 units and there are
m type-1 units
(excluding
the type-1 units included in the batch under service) and type-2 units wait in the respective queues. all
n
I t i s non-zero for
m,n > 0 .
The steady state probability equations of the queueing model under study are then given by:
(2)
-o .
Now the mean number of type-1 units in the system may be obtained by differentiating the joint probability generating function with respect to functions
F (y)
x
and setting and
H (y) .
a-1
(28)
x =y = 1
or from the partial generating
By either method we obtain
finally
°°
L , = 7 m H (1) + V m F (1) =
J^_ [
X
i6
- i-e [ yi a-e)
. a(Q-i)
2
+
ae a + 1 d-e) - e 2 d - e a ) 1 ffd) (1_6)2
I f we compare t h i s r e s u l t w i t h t h e mean queue l e n g t h of q u e u e i n g system d i s c u s s e d i n Medhi is
$(x,y)
c l e a r t h a t t h e mean number
the preemptive p r i o r i t y
J o M \m'
[3] o r i n ,Gohain and Borthakur
\ 1
[J], i t
(28) , i s n o t changed by t h e i m p o s i t i o n of
discipline.
S i m i l a r l y t h e mean number of type-2 u n i t s i n t h e system i s given by
a
~
I
References [/]
R. Gohain and A. Borthakur, "On difference equation technique applied to bulk service queue with one server", Pure Appl. Math. Sci. 10 (1979) , 63-69.
[2]
C. R. Heathcote, "The time dependent problem for a queue with preemptive priorities", Oper. Res.
7 (1959), 670-680.
A preemptive p r i o r i t y queue [3]
J. Medhi, "Waiting time distribution in a poisson queue with general bulk service r u l e " , Managment Sci.
[4]
243
21 (1975), 777-782.
R. Sivasamy, "A non preemptive p r i o r i t y queueing model with thresholds for the acceptance of non-priority units into service", Pure Appl. Math. Sci.
Lecturer in S t a t i s t i c s , Annamalai University, Annamalai Nagar - 608 002, Tamil Nadu, India.
(to appear).
