IEEE TRANSACTIONS ON COMPUTERS, VOL. 42, NO. 1, JANUARY 1993
83
Performance Evaluation of a Threshold
Policy for Scheduling Readers and Writers
Alexander Thomasian and Victor F. Nicola, Member, IEEE
Abstract-This paper is concerned with the analysis of a threshold policy for scheduling “readers” and “writers” in a multiserver
system and a comparison of its performance with the FCFS
policy. A writer must be processed singly, thus using all servers
in the system, while readers can be processed concurrently, each
using one server. Higher throughput and better response time can
be achieved by increasing the degree of reader concurrency; this
improvement is more significant when the reader processing time
is much longer than the writer processing time. A high degree
of concurrency cannot be sustained with the FCFS policy, due to
the service order imposed on arrivals to the system, in addition
to the inefficiency resulting from the need to frequently empty
the system from readers before processing a writer. On the other
hand, high levels of reader concurrency can be achieved with
the Threshold Fastest Emptying (TFE) policy, which empties
the system from readers (if any) to serve writers only when a
threshold li on the number of writers in the system is reached. If
there are no readers, the system starts processing writers (if any),
even if the threshold is not reached. The TFE policy is analyzed,
in a system with writer arrivals and an infinite backlog of readers
(reader-saturated system), using a Markovian model as well as
a vacationing server model to yield closed form expressions for
the mean writer response time and the reader throughput. The
maximum throughput achieved by the TFE policy is shown to
be an increasing function of I<, which even at I< = 1 exceeds
the maximum throughput achieved by FCFS. In a system with
reader and writer arrivals (nonsaturated system), simulation is
used to study and compare the mean response time for the FCFS
and the TFE policies.
Index Terms- Concurrency control, full capacity disciplines,
locking, Markov chain model, mutual exclusion, parallel processing, performance evaluation, queueing analysis, readers and
writers, threshold scheduling, vacationing server model.
I. INTRODUCTION
T
HE “readers and writers” problem is a classical problem
in operating system theory and practice (see e.g., [6]),
in which writers are mutually exclusive with respect to each
other and readers, while readers can be processed concurrently.
A variant of this problem occurs in transaction processing
systems, where shared or exclusive locks are required before
a database object can be accessed. In this case shared and
exclusive lock requests correspond to readers and writers, respectively. Other variants of the problem occur in manipulating
replicated data in a distributed database environment (see [lo]
and references therein). A special case is that of replicating
data on multiple disks, which is referred to as disk mirroring
Manuscript reeceived December 23, 1990; revised September 3, 1991 and
March 6 , 1992.
The authors are with IBM T. J. Watson Research Center, Yorktown Heights,
NY 10598.
IEEE Log Number 9202618.
(shadowing). Read and write accesses to disk correspond to
readers and writers, respectively.
In a multiserver system with a mixed stream of reader and
writer arrivals, system performance is improved when readers
are processed at a high degree of concurrency. The FCFS
policy for a system with both reader and writer arrivals results
in a severe reduction in the potential degree of concurrency in
processing readers due to the service order imposed on arrivals
to the system,’ in addition to the inefficiency resulting from
frequent switchovers to empty the system from readers before
processing a writer. This inefficiency is more significant when
the processing time of readers is much longer than that of
writers, so that readers contribute heavily to system load.
A higher degree of concurrency for readers is achieved
with the Exhaustive Service (ES) policy, which gives priority
to readers (resp. writers) over writers (resp. readers) in the
queue while other readers (resp. writers) are being processed.
In other words, the processing of writers (if any) resumes only
when the system empties from readers, and vice versa. Readers
accumulated during writer processing can be processed at a
higher degree of concurrency. It is obvious that this policy
is unfair to writers (resp. readers), which arrive during the
processing of readers (resp. writers). Although, as shown in
this paper, ES attains the highest maximum throughput, other
scheduling policies may result in a better mean response time
at reasonably low arrival rates.
An effective scheduling policy which yields a higher maximum throughput than FCFS and a better mean response time
than ES (at reasonably low arrival rates) is the Threshold
Fastest Emptying (TFE) policy [SI. This paper is concerned
with the analysis and evaluation of this policy in comparison
with the FCFS and ES policies.
The TFE policy, with a threshold K for writers, operates as
follows: an empty system starts processing readers or writers
(whichever comes first). Once the system starts processing
writers, it continues to process all writers before switching to
process readers (if any). However, the processing of readers
is disrupted when the number of writers in the system reaches
the threshold K . At that point, no new readers are admitted
to service and all active readers are processed to completion.
Then the system starts processing writers. However, if there
are no readers, the system may start processing writers (if any),
even if the threshold is not reached. Notice that, in the limit
as K -+ m, the TFE policy corresponds to the ES policy.
Consider a hypothetical (worst-case) situation with arrivals alternating
between readers and writers. With a FCFS scheduler only one reader will be
processed in each read epoch, resulting in very low processing concurrency.
0018-9340/93$03.00 0 1993 IEEE
84
There have been several studies of this problem (see e.g.,
[1]-[5], [lo], [13), which differ in the following main aspects:
The policy for scheduling readers and writers, e.g.,
FCFS, TFE, etc.
Preemptive or nonpreemptive policies (for writers with
respect to readers).
Sequential or parallel writing (e.g., when updating the
copies of a replicated database).
Synchronous or nonsynchronous writing (e.g., in a replicated database, synchronous writing implies that readers
cannot start accessing the copies of the database unless
the writing of all copies has been completed. The
nonsynchronous policy does not have this requirement).
the context of a replicated database, the analysis of a
system with FCFS scheduling of readers and writers using a
matrix-geometric approach appears in [lo]. Poisson arrivals,
nonpreemptive writers, synchronous and nonsynchronous writing policies are considered. The stability of a queueing system
with FCFS scheduling of readers and writers is also studied
in [4], but this system is different from others (including [5])
in that there are separate arrival streams for each copy of the
replicated database. A FCFS policy for scheduling readers and
writers under rather general assumptions is also considered
in [13], where it is shown that the stability condition is
sensitive to the interarrival time distribution for writers and
the processing time distribution for readers, but depends on
only the mean processing time for writers.
The TFE policy in a system with an infinite backlog
of readers, and nonpreemptive and synchronous writers is
considered in [5].This system empties from readers to process
writers only when the threshold K is reached. Two problems
are considered: the first with (different) rewards for reader
and writer completions, and the second with a reward for
reader completion and a cost for writer waiting time. Using
a Markovian decision process formulation, it was determined
that the TFE policy maximizes the discounted and long run
average reward in both problems. While discounted criteria
are of interest in some applications, it is common to consider
long run average criteria in performance related studies.
A two-server system with Poisson arrivals for two job
classes, which correspond to readers and writers, is analyzed in
[111 for five different scheduling disciplines to obtain response
time distributions and the maximum throughput for each discipline. In [ll], a full capacity discipline is defined as “a discipline which will cause an infinite expected waiting time only if
every other discipline also causes an infinite expected waiting
time for the same arrival stream.” The maximum throughput
attained by a full capacity discipline, for a given arrival stream,
is called the system capacity. It turns out in [ll], that (preemptive) disciplines which preclude any server from idling (as long
as there are jobs in the queue) are full capacity. However, full
capacity disciplines may not necessarily utilize all servers [121,
[15]. A full capacity discipline may result in a higher mean
response time than others at low arrival rates. It is interesting
to note that ES is a nonpreemptive full-capacity discipline.
Similar to the studies mentioned above, in this paper we also
consider a multiserver system processing (concurrent) readers
IEEE TRANSACTIONS ON COMPUTERS, VOL. 42, NO. 1, JANUARY 1993
and (exclusive) writers. As we have qualitatively argued
earlier, the TFE policy results in performance improvements
over the FCFS and ES policies. Our goal is to quantify these
improvements for varying system and scheduling parameters;
this is accomplished by means of analysis and/or simulation.
Such parametric studies are very useful for gaining insight into
the behavior and relative performance of different policies for
scheduling readers and writers.
First, we consider a system with an open stream of writer
arrivals and an infinite backlog of readers (a reader-saturated
system). Our approach provides a queueing analysis of the
system, which complements the control theoretic approach in
[5]. Furthermore, the throughput analysis in this system is
essential for obtaining the maximum throughput in a system
with an open stream of readers and writer arrivals (a nonsaturated system), which is also considered in this paper. Using a
renewal theoretic argument and a vacationing server model, we
derive closed form expressions for the reader throughput and
the mean writer response time in the reader-saturated system.
Alternatively, we also present a continuous-time Markov chain
model representing the queueing behavior of the system.
The Markov chain analysis provides an interesting technique,
involving appropriate manipulations of balance equations, to
yield (in this particular case) closed form expressions for
performance measures of interest. It is shown that, for large
but finite threshold values ( K ) , the TFE policy attains a
throughput arbitrarily close to system capacity. However, K
should remain finite; otherwise, no writers will be processed.
The nonsaturated system, i.e., with both reader and writer
arrivals, is of much practical interest. Therefore, also in this
system, we consider the performance of the TFE policy and
compare it with that of the FCFS and the ES policies. We
derive closed form expressions for the maximum throughput
attainable with the FCFS, ES, and TFE policies. It is shown
that the maximum throughput attained by the TFE policy is
an increasing function of K , which, even for K = 1, is
higher than that attained by the FCFS policy. When K is
allowed to be sufficiently large, such that reader processing is
never disrupted, the TFE policy becomes equivalent to the ES
policy, which attains the system capacity. We also investigate
the effect of system parameters on the maximum throughput
attainable by different policies. The queueing analysis of the
nonsaturated system is of much interest; however, it is a
nontrivial open problem which we do not consider in this
paper. Instead, we use simulation to study the response time
characteristic in a nonsaturated system. While the mean writer
response time increases with K , the mean reader response
time typically decreases with K . Notice that the TFE policy
provides the flexibility (control) of choosing an optimal K ,
which minimizes an appropriate function (e.g., a weighted
sum) of reader and writer response times.
The paper is organized as follows. In Section I1 we give a
formal description of the system model and introduce some
notation. The analysis and behavior of the reader-saturated
system is considered in Section 111. (The corresponding Markovian analysis is presented in Appendix A.) Analysis and
discussion of the reader throughput is given in Section 111-A.
The mean writer response time is obtained in Section 111-E. In
THOMASIAN AND NICOLA: THRESHOLD POLICY FOR SCHEDULING READERS AND WRITERS
Section 111-C we study the behavior of the reader-saturated
system. For the nonsaturated system, in Section IV-A we
derive expressions for the maximum throughput for different
policies and compare them numerically in Section IV-B. In
Section IV-C we use simulation to compare the mean response
time in the nonsaturated system for the FCFS, ES, and TFE
policies. Conclusions appear in Section V.
85
the Markov chain specifying the behavior of the system with
the TFE policy is shown in Fig. 1. Appropriate manipulations
with the state equilibrium equations for this Markov chain
yield closed form expressions for performance measures of
interest. These derivations are rather lengthy and therefore are
given in Appendix A.
In fact the system can also be analyzed for a general
distribution of writer processing times. The reader throughput
11. THE SYSTEMMODEL
depends on the threshold K of the TFE policy and therefore
Readers and writers are processed on a system with M is denoted by y ( K ) , which can be derived using a renewal
servers. Writers arrive according to a Poisson process at a property of the system operation, as will be discussed in
rate A. Each writer is processed singly in the system, and is Section 111-A. In Section 111-B, the mean number of writers
assumed to utilize all M servers during its entire processing in the system is determined by analyzing an appropriately
time. However, readers can be processed concurrently in the modified vacationing server model [8]. It is worthwhile to
system (each on one server). Therefore, at most M readers can mention that this analysis is not valid for estimating the
be processed concurrently at any given time. The processing distribution of the writer response time, because the vacation
time for writers (resp. readers) is given by the random variable time depends on the arrival process [8]. However, the steadyX (resp. p) with a distribution G ( t ) (resp. H ( t ) ) . The ith state probabilities of the Markov chain model (in Appendix
moment for the writer (resp. reader) processing time is denoted A) can be used to compute the distribution of the writer
with the first moment 3 = 1 / p (resp.ij = l / u ) . response time. In Section 111-C we investigate the behavior
by 2 [resp.
For the sake of mathematical tractability, the reader processing of the reader-saturated system.
time is assumed to be exponentially distributed. The writer
processing time is assumed to be generally distributed. There- A . Reader Throughpul
fore, in a replicated database, we can model sequential, parallel
;The reader-saturated system operates in three phases: (I)
or any other synchronous writing strategy (Le., strategies that
require exclusive access to all replicas in the system during Processing readers. (11) Emptying the system from readers.
(111) Processing writers. The system undergoes renewal cycles
the write operation).
In a reader-saturated system, there is an infinite backlog each beginning with Phase I. Therefore, y ( K ) can be exof readers; this case is equivalent to a closed system with pressed as the ratio of the mean number of readers completed
M readers. Performance measures of interest in the reader- per cycle and the mean cycle length [14], which are derived
saturated system are the reader throughput, denoted by y,and in the following.
1) Phase I . The mean duration of this period (TI) equals
the mean writer response time, denoted by R,. The analysis
the expected time for K writers to arrive, Le., TI =
of the reader-saturated system with the TFE policy appears in
K/X. During this phase, readers are processed at the
Section I11 and Appendix A.
maximum degree of concurrency ( M ) . Therefore, the
In the nonsaturated system, the reader (resp. writer) arrival
mean number of readers completed during this phase
process is Poisson with a rate y (resp. A); therefore, the
is N I = ( K / X ) x M u , when the reader processing
overall arrival rate is A = X y. The fraction of writers and
time is exponentially distributed. For a general reader
readers in the arrival stream is assumed a constant (determined
processing time distribution, the mean number of readers
only by the workload) given by f, = X/A and f, = y/A,
processed in Phase I can be expressed as follows
respectively. A performance measure of interest in this case is
the maximum throughput attainable by the system for different
policies, which is denoted by ALZ,"", A:,
and ,:A
::
for
the FCFS, ES, and TFE (with a finite threshold K ) policies,
respectively. However, it is important to note that only some
of the forementioned policies may attain the system capacity,
which is denoted by A,,,. Other measures of interest are the
where H,(t) is the n-fold convolution of H ( t ) . The
writer, reader, and overall mean response times, denoted by
sumx,",l H,(t) is the mean number of readers completed by one server in the interval (0, t ) [14], which in
R,, R, and R(= f, x R,
f,. x R,), respectively. The
stability conditions (maximum throughputs) for the FCFS and
the case of the exponential distribution equals vt.
the TFE policies are derived in Section IV-A. In Sections IV2 ) Phase ZZ. In this phase, the system is being emptied
B and IV-C we use simulation to perform parametric studies
from the M readers; therefore, NIr = M . The duration
on the performance of the nonsaturated system with different
of this-phase is distributed according to: Z,,,(M)
=
policies.
max(Z1,2 2 ,. . . , ZM),
where 2, is the residual processing time of the ith reader currently active in the system.
111. ANALYSIS OF THE TFE POLICY
The mean duration of Phase I1 is TII = Zm,(M),
IN THE READER-SATURATED
SYSTEM
where Z,,,(M)
is the mean of Zmax(M).Due to
the memoryless property of the exponential distribution
We first consider the analysis of the reader-saturated system
under Markovian assumptions. The state transition diagram for
= Y,,,(M) = H M / u , where
we have: Z,,,(M)
GI,
+
+
a6
IEEE TRANSACTIONS ON COMPUTERS, VOL. 42, NO. 1, JANUARY 1993
I
2v
I
...
Fig. 1. The state transition diagram for the Markov chain model.
ymax(M)is the mean of the maximum of M reader
processing times and H M = CjZ1
M l/j.
this becomes tantamount to maximizing the system throughput
A ( K ) = X y ( K ) , K < co. However, the following
3) Phase IZI. No readers are processed in this phase; there- arguments will also hold when we associate arbitrary positive
fore, N I I I = 0. This phase corresponds to a (writer) rewards with reader and writer completions. At K = 00, the
busy period starting with K+ J writers, where J denotes system processes readers only and writers are never processed;
the (random) number of writer arrivals during Phase I1 in this case, the system throughput is given by Mu. The
(with a mean J). Since writers arrive according to a maximum throughput attainable with a sufficiently large, but
Poisson process J = XTII. When the reader processing finite, K (i.e., K -, 00) is given by X (1 - pw)Mv,where
time is exponentially distributed, then J = X x H M / u . the second term corresponds to the value of Ymax in (3.2). It
The mean busy period (started by a single writer), follows that, an infinite K will maximize throughput only if
denoted by 6, corresponds to that of an M/G/1 queueing Mv > X (1- pw)Mv,Le., X ( 1 - M v / p ) < 0 or Mv > p.
system [14], hence 6 = ? / ( l - p,), where p, = A? is In other words, when the system capacity to process readers
the writer utilization. Since Phase I11 starts with K J (Mv)exceeds the system capacity to process writers ( p ) ,
writers, it follows that TIII
= ( K J ) x 6.
the system throughput (or reward) is maximized by shutting
off writers completely. Otherwise, the maximum throughput
The reader throughput, y ( K ) , is therefore given by
X (1 - p,)Mv can be approached as closely as desired by
setting K to a sufficiently large, but finite, value. Notice that
writers will eventually get processed as long as K remains
"
(3.1) finite; however, increasing K results in a detrimental effect
Notice that the second equality in (3.1) is true only when the on R, (this will be discussed in Sections 111-C and IV-C).
It is interesting to note that a preemptive resume policy for
reader processing time is exponential. The same expression
can be obtained by substituting for p ( 0 , M ) from (A.2) into writers (over readers) is equivalent to a preemptive threshold
(A.3) for the reader throughput in Appendix A. It follows policy with K = 1, which achieves the maximum reader
as given by (3.2). The throughput is maxfrom (3.1) that y ( K ) increases with K and that the efficiency throughput (rmax)
resulting from uninterrupted processing for readers in Phase I imized because there is no wasted work when readers are
dominates the inefficiency caused by a longer Phase 111. In the preempted. In this case, a preemptive threshold policy with
K > 1 is not beneficial, since it introduces unnecessary writer
limit as K -, 00, y ( K ) approaches its maximum
delays without further increasing the reader throughput.
The utilization of the system is given by: psys = p,
Ymax 1(1 - p w ) M v .
(34
y ( K ) / ( M v ) ,where r ( K ) is given by (3.1). The first and
The expression for the maximum reader throughput is intu- second terms correspond to system utilization due to writers
itively appealing; the maximum reader throughput equals the and readers, respectively. When K >> ( X / V ) H M ,then full
fraction of time the system is available to readers multiplied system utilization is approached. When K << X/u then
by Mu. It turns out (see the discussion below) that the above p s y s E pw
(1 - p w ) / H M .
expression is an asymptote (upper bound) approached for very
large, but finite, values of K.
Let us consider the problem of maximizing the reward for B. Analysis of the Vacationing Server Model
The writers can be viewed as the primary customers of an
reader and writer completions in the long run. When the
rewards for reader and writer completions are set to unity, M/G/1 queueing system with vacations. The completion of
+
+
+
+
+
+
+
+
THOMASIAN AND NICOLA THRESHOLD POLICY FOR SCHEDULING READERS AND WRITERS
a (writer) busy period (Phase 111) is the beginning of a server
vacation. The duration of the vacation is determined by the
time it takes for K writer arrivals and the time to empty the
system from readers (Phases I and 11). Notice that the duration
of the vacation depends on the writer arrival process.
The number of writers in the system is given by [7] fi, =
N&ffGfI
Nu,,, where f i A f f G f l is the number of customers
(writers) in an identical M/G/1 queue (with the same arrival
rate and service time distribution) without vacations. fiuac denotes the number of writer arrivals during the residual lifetime
of a vacation. The z-transform for N A f / G l l is expressed as [8]
87
K ( K - l ) ( m + q2v2 + 2 K ( m + 1)vX + 2 x 2
( m 113
+
(3.10)
+
(3.3)
where v(.) denotes the Laplace transform of the probability
density function of the writer processing time. Let a(.) (resp.
,L?(z)) denote the z-transform for the number of customers
arriving during a vacation (resp. the residual lifetime of a
vacation), then the z-transform for Nu,, is [7]
1- a(.)
p ( z ) = a’( 1)(1 - 2 )
(3.4)
The first term in (A.8) and (3.8) corresponds to the mean
number of writers in a FCFS M/G/1 queueing system. When
there are no readers in the system, we have an M/G/1 system
with a threshold policy [9]; in this case, the additional term in
(3.8) is given by ( K - 1)/2, which follows by setting v = 30
in (3.8). The mean response time for writers is obtained from
Little’s law, thus R,; = N,/X.
It is interesting to note that, using different analyses, distinct
closed-form expressions [(3.8) and (A.8)] are obtained for
the same quantity N,. It remains to be shown algebraically
that these two expressions constitute a combinatorial identity.
However, this has been verified only numerically.
C. The Behavior of the Reader-Saturated System
This section deals with a parametric study of the behavior
of the reader-saturated system. We first study the effect of
varying the threshold K on the reader throughput ( y ( K ) )and
the vean response time of both readers (R,) and writers (&).
We then consider the effect of varying the writer arrival rate
As discussed in Section 111-A, the number of customers (A) on R, and y ( K ) .
In Fig. 2 we plot the normalized throughput for readers
(writers) afriving during a vacation is K
j , where K is
fixed and J is the number of arrivals during Phase 11. 3 is the y n o r m ( K )= y ( K ) / ( M v ( l - p w ) ) versus K , when the system
number of Poisson arrivals during an interval determined by capacity for readers Mv = 10, for = X = 0.5, p, = 1 and
the maximum of M random variables, each- corresponding to different values of M . It is observed that a system with a small
the residual processing time of a reader ( 2 ) .Therefore, the number of fast servers results in a higher y n o r m ( K )[due to
the term Hhf in the denominator of (3.1)]. As K increases
probability mass function for 3 is
ynorm( K ) approaches one, but this happens much faster for
smaller values of M . An intuitive explanation for this is that
the system is more efficient (in terms of being emptied from
readers) for smaller values of M and higher values of v.
Fig. 3 shows the mean response time for readers (R,),
- H ( z ) ) d z is the distribution of 2.
where F ( t ) = AY s‘(1
O
writers (R,) and the overall mean response time ( R =
a ( z ) is the product of the z-transform of P r [ S = j ] and the j , x R, f r x R,) versus K , for X = 0.5, ,u = 1, v = 0.1,
z-transform of a step of size K (Le., z K ) . For an exponential and M = 10. Note that f U . = X / ( X + y ( K ) ) and f , = 1- f,,
reader processing time distribution F ( t ) = H ( t ) = 1 - e-”t, so that the fraction of completed readers (and writers) varies
then
with K . It is observed that as K increases, R, decreases and
M-1
R, increases almost linearly. The overall mean response time
2K
a ( z )= M u
( R )is minimized at some value of K . The issue of minimizing
I ) ( m 1)” X ( 1 - z )
m=O
the overall mean response time is also meaningful in the case
(3.7) of a nonsaturated system [16].
The mean number of writers in the system can be obtained
In Fig. 4 we plot the mean writer response time (R,) versus
from + ( z ) in (3.5) as
the arrival rate for writers ( A ) for different values of K . The
parameters used are 11 = 1, and 7) = 0.1, and M = 10.
R, is affected by the time to attain the threshold, the time
to empty the system from readers, and the time to reach the
where d l ) ( l )and a ( 2 ) ( 1are
) given as follows
head of the queue. For very low writer arrival rates, the writer
response time is dominated by the time to attain the threshold,
M-l
M
so
that, initially, the mean writer response time decreases very
a(l)(l)= (-1y
rapidly with increasing A. For intermediate values of A, R, is
m=n
(3.9) dominated by the reader emptying time, which is independent
It follows that, the z-transform for
fi,
is
+
+
(Mi+
’
+
88
IEEE TRANSACTIONS ON COMPUTERS, VOL. 42, NO. 1, JANUARY 1993
M=l
h=0.5
p= 1
M v=10
I
I
I
I
I
BO
40
0
I
I
120
THRESHOLD VALUE (K)
Fig. 2. The normalized reader throughput versus the threshold value in a reader-saturated system.
I
20
10
30
40
THRESHOLD VALUE (K)
Fig. 3. Reader, writer, and overall mean response times versus the threshold value in a reader-saturated system.
of A. As X increases further, the time to reach the head of the
queue dominates and we see the usual sudden increase in R,
as X approaches the system capacity for writers ( p ) . R, is
larger for higher values of K , simply because it takes longer
to attain the threshold.
In Fig. 5 we use the same parameter values as in Fig. 4 to
plot the reader throughput (-y(K))versus X for different values
of K . The reader throughput is a monotonically decreasing
function in X [see (3.1)]. The inefficiency of small threshold
values is exhibited by the fact that the reader throughput drops
rapidly for increasing values of X when K is small, but this
is not so for higher values of K . The readers are shut-off
completely as pw + 1, since the TFE policy prioritizes the
processing of writers.
Iv.
STUDY O F THE
NONSATURATED
SYSTEM
In a nonsaturated system readers as well as writers arrive
randomly at the system with a total rate A. The fraction
of writers (resp. readers) in the arrival stream is a constant
f , (resp. f r = 1 - f w ) . The arrival rate for writers (resp.
readers) is therefore X = A f , (resp. y = A f r ) . In Section IV-
THOMASIAN AND NICOLA: THRESHOLD POLICY FOR SCHEDULING READERS AND WRITERS
L
-
K=64
/A = 1.0
u = 0.1
M = 10
0.4
0.2
0.6
WRITER ARRIVAL RATE
Fig. 4.
a9
0.8
1 .o
(A)
Mean writer response time versus the writer arrival rate in a reader-saturated system, for different values of Ii
/A = 1.0
u = 0.1
M = 10
I
0.2
0.4
0.6
WRITER ARRIVAL RATE
0.8
1.o
(A)
Fig. 5. Reader throughput versus the writer arrival rate in a reader-saturated system, for different values of I<.
A, we first obtain expressions for the maximum throughput
that can be sustained in a nonsaturated system with the
following policies: FCFS, ES, and TFE for a given threshold
K . In Section IV-B we carry out a parametric study to
understand the effect of various parameters on the maximum
throughput attained by different policies. In Section IV-C we
use simulation to study the response time characteristic of the
nonsaturated system under different policies.
It is important to note that the stability limit for writers
may not be the same as that for readers; this is particularly
true for the TFE policy with a finite threshold value. In this
case, once the system is switched to writers, priority is given to
processing all writers in the system before switching to readers.
Therefore, while the system remains stable for writers (as long
as pw < l), the fraction of time available for processing
readers may not be enough to maintain a stable queue for
readers. The maximum throughput of the system, obtained in
Section IV-A, is the stability limit for readers.
A. Analytic Expression for Maximum Throughput
In this section we determine the maximum throughput that
can be sustained by the system with different policies. First,
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IEEE TRANSACTIONS ON COMPUTERS, VOL. 42, NO. 1, JANUARY 1993
let us consider the system with the FCFS policy. In this case,
the stability limit can be derived analytically by viewing the
system (at saturation) as an M/G/l server in which customer
arrivals correspond to writer arrivals. However, in this system,
a customer requires servicing one writer and a random number
of readers (those readers arriving between two consecutive
writers). The probability of i readers arriving between two
consecutive writers is given by (1 - f r ) f ; , in which case
the (conditional) expected service time of a customer, s(i),
is given by
(TFE) and the subscript (max) for the sake of readability
(note that A ( K ) was also used in Section 111 to denote the
total throughput in a reader-saturated system). The fraction
of readers and writers in the arrival stream is fi. and fw,
respectively. The system is stable for writers as long as the
total arrival rate A is less than p / f w , i.e., the stability limit for
writers is independent of K . However, given that the system
is stable for writers, the stability limit for readers depends
on K . This stability limit can be determined by substituting
X = A ( K )x f w and y(K) = A ( K )x fT into (3.1). We obtain
the following quadratic equation in A ( K )
By unconditioning on i , and with some manipulations, we
obtain the (unconditional) expected service time of a customer,
Customers arrive to the system at the same rate as writers;
therefore, for stability, we must have A f w s < 1. It follows
that the maximum throughput,
is given by
Mv
i=l
1
-l
(4.1)
.
The stability limit in (4.1) can also be derived using a matrixgeometric approach similar to that followed in [lo].
The maximum throughput that can be attained with the ES
policy corresponds to that of the TFE policy for sufficiently
large K (such that the processing of readers is never disrupted). The maximum throughput for readers (ymax)follows
(3.2) for the reader-saturated system. The maximum throughput for a specified fraction of readers in the arrival stream is
therefore determined from ymax/AEF= f r . Therefore,
-1
=
(5+ &)
fr9
=
( +)..
fwZ
-’
.
A(K)-K=0.
(4.3)
It can be shown that the above equation has one real positive
root, which is the stability limit for readers and the overall
arrival stream.
For M > 1, the TFE policy outperforms the FCFS policy,
even for the smallest threshold, K = 1. This is because the
TFE policy utilizes the system more efficiently than the FCFS
policy, by completing the “busy period” due to writers before
switching back to readers, Le., the TFE -policy. reduces the need
to frequently empty the system from readers before processing
a writer. Furthermore, there are more readers to process when
a writer’s busy period is completed.
It can be shown that A ( K ) is an increasing function2 of K
and that
5 A(1). It follows that’::A:
5 A ( K ) ,for
all K 2 1. Equation (4.3) can be rewritten as A A Z ( K )
( B K - C ) A ( K )- K = 0, where A = ( f w / v ) ( f w / p
f r H M / ( M V ) ) ,B = f w / p f r / ( M v ) ,and C = f w / ~ . The
positive root of the quadratic equation is
AE:FS
+
+
+
-BK
+ C + [ ( B K - C)’ + 4AK] 1/2
A(K) =
2A
.
(4.4)
+
(4.2)
The above expression is also true for general distributions of
writer and reader processing times; therefore, we have replaced
1/11by Z and 1 / u by 9 in the second expression. A n intuitive
explanation of A:& follows from the fact that the system
behaves as a single-server (resp. multiserver) queueing system
in processing writers (resp. readers). With the ES policy, the
system does not idle as long as there are customers. It follows
that ES is a full capacity discipline and, at the maximum
throughput, the sum of system utilizations in these two modes
is equal to one. Note that, for the ES policy, the queue length
for both readers and writers grows infinitely at the same
stability limit in (4.2). It follows from (4.1) and (4.2) that
RE
’:
<
for all combinations of f w , p, and Mu.
Equality holds in the extreme cases when the system processes
only readers or only writers and when M = 1.
The maximum throughput for the TFE policy with a finite
threshold K is denoted by AZCF(K). In what follows, we
use A ( K ) instead of AZ:F(K), by eliding the superscript
.... .-.
We show that d A ( K ) / d K > 0 or A(K 1) > A ( K ) for any
K . This leads to the requirement of showing that A / B > C ,
which is true as long as M > 1. To show that’::A
:
5 A(1)
we need to show that substituting A:“,“’
from (4.1) for
A(1) in (4.3) will result in an expression which is zero or
negative [because A’$:;
is positive and therefore larger than
the negative root of the equation, but should be smaller than
A( l)]. Therefore, the inequality to be proven is as follows
+
($+
-
fw
- 1 5 0.
(4.5)
The proof of the above inequality is given in Appendix B.
This establishes that AiC,“’ 5 A ( K ) , K 2 1. The equality
holds in the following trivial cases: i) M = 1 (a single
server), regardless of the value of fw, ii) there are only writers
2The situation is slightly different here because previously (Section 111-A)
the writer throughput was specified and y ( K ) was shown to be an increasing
function in K , while at this point the fraction of readers and writers is
specified.
~
THOMASIAN AND NICOLA: THRESHOLD POLICY FOR SCHEDULING READERS AND WRITERS
jL=
0
0.2
91
FCFS
_ _ _ _ _ K=
1
-K--
1
0.4
0.6
0.8
1.a
WRITER FREQUENCY (fw)
Fig. 6. li~$/s,
.1(1) and
as functions of
(fw =
1) or only readers (f, = l),regardless of the value of
M . Numerical results also show that A(K)/A:FFs 2 1 and
is an increasing function of M for all values of K (see Fig. 7).
Consider the effect of using faster servers for processing
readers, while fixing the system capacity for processing readers
and writers, M u and p, respectively. It can be easily shown
[by referring to (4.1) and (3.1)] that the maximum throughput
attained by the FCFS and the TFE (with a finite K ) policies
increases with decreasing values of M , while M u remains
fixed. In fact, increasing u results in a decrease in the emptying
time for readers. On the other hand, the maximum throughput
attained by the ES policy is insensitive to M , as long as M u
is constant.
The maximum system utilization attained by different policies can be obtained directly from their maximum throughput.
Full system utilization can be achieved by the ES policy,
while the maximum utilization for the FCFS and the TFE
(with a threshold K ) policies are given by pEzFs = (fw%
f,9/M)ALzFS, and PK~F(K)
= (fw% f,Y/M)A(K),
respectively. A measure for the effective degree of concurrency of a given policy can be obtained from its maximum
throughput. For example, for the TFE policy with a threshold
K , it isgiven by f,A(K)jj/(l-f,A(K)lc). For the ES policy,
this expression yields M , which is the maximum degree of
concurrency.
+
+
B. Comparison of Maximum Throughput for Different Policies
In this section we compare the maximum throughput for
different scheduling policies and varying system parameters.
Our main goal is to determine when the TFE policy results
in a significant improvement in performance compared to the
FCFS policy.
Consider the nonsaturated system with any policy (FCFS
or TFE). Observe that, with no writers in the arrival stream
fu,
for different values of v ('Wv
> p).
(Le., f w = 0,) the maximum throughput is given by the reader
= Mu. Also note that, with
processing capacity, Le., A,,,
no readers in the arrival stream (i.e., fw = l), the maximum
throughput is given by the writers processing capacity, i.e.,
Amax = p. Therefore, A,,, is invariant at the extreme points
fw = 0 and fw = 1 and is given by p and M u , respectively.
In Fig. 6 we consider the system with the FCFS and the TFE
policies, with p = 1, M u = 5 and M u = 10. For a given M u
we obtain a family of curves by varying K . Observe that, even
for K = 1, A ( K ) is higher than AEZFs and that it increases
with K . In fact, the curves for FCFS and TFE with K + 00
form a lower and an upper envelope, respectively, containing
curves for all finite values of K . The improvement with K
is more significant for lower (resp. higher) but not extreme
values of f w , if M u > p (resp. if M u < p). Fig. 7 shows the
relative improvement in the maximum throughput of the TFE
policy with respect to the maximum throughput of the FCFS
policy (A(K)/(A:FFs - 1)) as a function of M for different
values of K . For M = 1 there is no improvement, since in
this case the system is fully utilized also while being emptied
from the sole reader. The improvement in the maximum
throughput increases with M and K when the readers have
a longer processing time than writers, but this improvement is
insignificant when the readers are not much longer than writers
(see Fig. 7, for u = 1.0). It is interesting to note that as M +
FCFS
03, A ( K )
p / f w and Amax
( f w / ~- f w l n f w / ~ ) - l .
It follows that, (A(K)/(Ai:Fs - 1)) + -(p/u)lnf,.
In
other words, the relative improvement in throughput increases
with the fraction of readers in the arrival stream and with
the mean reader processing time (relative to the mean writer
processing time); in this circumstance, the potential for higher
concurrency is better exploited by the TFE policy. For the set
of parameters used in Fig. 7 ( p = 1, u = 0.01, fw = O.l), this
relative improvement approximately equals 230.
+
+
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IEEE TRANSACTIONS ON COMPUTERS, VOL. 42, NO. 1, JANUARY 1993
u=O.Ol
////
st-
1
p= 1.o
Iw=O.l
1 0’
104
NUMBER OF READERS (M)
Fig. 7. Relative improvement in the maximum throughput, (.\(
C. Comparison of Response Time for Different Policies
For the FCFS and ES policies, the stability limit is the
same for readers and writers. However, for the TFE policy
with a finite threshold, the system first becomes unstable for
readers (since the writers are implicitly prioritized), while it
remains stable for writers as long as p, = Afw/p < 1.
In Section IV-B, it is shown that, as the load increases, the
system becomes unstable for a discipline with low maximum
throughput (stability limit for readers), while it remains stable
for a discipline with a higher maximum throughput. Therefore,
at heavy loads, the maximum throughput is an appropriate
measure for comparing different policies. At light loads that
can be sustained for most disciplines, the (readers, writers,
or overall) mean response time is an appropriate measure for
comparing different policies.
The queueing analysis of the nonsaturated system with the
FCFS policy has been considered in [lo] using a numerical
matrix-geometric approach. The queueing analysis for the TFE
policy is a nontrivial problem, which is not considered in
this paper. Instead, we use simulation to compare the reader
and writer mean response times with the FCFS, TFE, and ES
policies. In all simulation runs the reader and writer processing
times are exponentially distributed with means jj = 1/v = 10
and 5 = l / p = 1, respectively. The number of servers
M = 10. The batch means method was used to estimate
the mean response times with confidence intervals within 5%
of the mean at 90% confidence level (with few exceptions,
usually close to the stability limit).
In Figs. 8 and 9 we plot the reader and writer mean response
times, R, and R,, respectively, versus K , for different overall
arrival rates (A) and for f, = 0.1. We first consider the effect
of K on R, (see Fig. 8). We note, that for small values of
K , the system is unstable for values of A greater than A ( K )
Ii)/.\K:T.’
- 1).
[as given by (4.4)]. Furthermore, it is observed that R, drops
with increasing K ; this drop is similar to what was observed
in Fig. 3 for a reader-saturated system. However, unlike the
reader-saturated system, with further increase in K , R, starts
increasing. This increase can be explained by the fact that
beyond a certain K the efficiency gained in processing readers
is offset by the delay encountered by readers arriving during
the much longer writer busy periods. For sufficiently large
values of K , the system empties from readers and switches
to process writers before reaching the threshold. In this case,
R, reaches its asymptote and does not change with increasing
K . Also note that, for such large values of K , the TFE policy
has become equivalent to the ES policy. In Fig. 9, R, is
an increasing function of K , but for sufficiently large K
reaches an asymptotic value (for the same reason mentioned
above).
We next compare the relative performance of the FCFS,
ES, and TFE policies by plotting R, R,, and R, versus A
for the following set of parameters: fw = 0.1, M = 10,
v = 0.1, and p = 1. In Fig. 10 we plot the mean reader
response time for the FCFS policy and the TFE policy with
K = 1, 4, 16, 64, 256, 1024 and ES. The FCFS policy
has the worst performance, since the response time increases
= 0.395. For the
rapidly and the system saturates at A,,,
given system parameters, the mean response times for the
TFE policy with K = 1024 and for the ES policy are
indistinguishable. Note that high values of K may result in
a degraded response time, for writers as well as readers, so
that the overall mean response time is also degraded. The
stability limit for readers (and the overall stability limit)
is determined by the threshold K [see (4.4)] as follows:
A ( l ) = 0.605, A(4) = 0.779, A(16) = 0.914, A(64) = 0.975,
A(256) = 0.993, A(1024) = 0.998, and
= 1.0. It
THOMASIAN AND NICOLA: THRESHOLD POLICY FOR SCHEDULING READERS AND WRITERS
f
IJJ
93
= 0.1
M=10
v-0.1
5:
W
z
IW
v)
Z
0
n
U
E
0
2
N
0
1
10
100
THRESHOLD VALUE
1000
(K)
Fig. 8. Mean reader response time versus the threshold value
f w = 0.1
M=10
v=o. 1
p=l.O
1
A=0.2
1
10
100
1 000
THRESHOLD VALUE (K)
Fig. 9. Mean writer response time versus the threshold value
follows that the mean response time curves, for, say, K = 256
and K = 64, must intersect before the stability limit for
K = 64.
Fig. 11 gives the mean response time for writers. The system
remains stable for writers beyond the stability limit for readers,
up to a maximum arrival rate of p/f,. The mean writer
response time is composed of the following components, each
of which dominates at some range as the total arrival rate A
increases:
1) At low arrival rate: the time to empty the system from
all readers is less than the time to attain the threshold.
Therefore, it is most dominant and increases with A.
2) At low and intermediate arrival rates: the time to attain
the threshold value for writers is less than the time to
empty the system from all readers. This time decreases
with increasing A.
3) At intermediate to high arrival rates: the time to empty
the system from active readers is most dominant. This
time is independent of A.
4) At high arrival rate: the time to reach the head of the
queue is most dominant. This causes the sudden increase
94
IEEE TRANSACTIONS ON COMPUTERS, VOL. 42, NO. 1, JANUARY 1993
8
gz
F
W
v)
;s
!
LL
0.2
0.6
0.4
0.8
(A)
ARRIVAL RATE
Fig. 10. Mean reader response time versus the arrival rate A, for
8
f
tCEB
fw
fu,
= 0.1
= 0.1
M=10
v=o.1
p= 1.o
I
2
I
I
I
6
4
ARRIVAL RATE
I
I
B
1
10
(A)
Fig, 11. Mean writer response time versus the arrival rate A, for f w = 0.1.
in response time as A approaches the stability limit
4 f W .
To summarize, when fw is small (say O.l), increasing A
primarily results in an increase in the reader arrival rate, such
that writer delay is affected mostly by interference by readers.
As A is increased, the number of readers encountered by
writers increases; on the other hand, the time to attain the
threshold decreases. These two effects yield a maximum for
Rw.
Finally, we investigate the effect of varying f,, for the set
of parameters A = 0.2, M = 10, u = 0.1, p = 10, when
different policies are in effect. The mean response times R,
and Rw are given in Figs. 12 and 13, respectively. When
f, + 0, the arrival of writers is very infrequent, therefore
R, can be determined directly by analyzing the corresponding
M / M / c system for readers (c = M is the number of servers).
R, is determined by the time to empty the system from
the readers preceding the writer (resp. the currently active
readers) when the FCFS policy (resp. TFE with K = 1) is
in effect. Since the state probabilities for the M / M / c system
THOMASIAN AND NICOLA: THRESHOLD POLICY FOR SCHEDULING READERS AND WRITERS
95
h=0.2
M=10
u=o.1
p= 1.0
0
0.2
0.4
0.6
WRITER FREQUENCY
0.8
1 .o
(fw)
Fig. 12. Mean reader response time versus f,,
are known, then R, in both cases can be easily determined
from the corresponding pure death process. When K > 1
and since writer arrivals are infrequent, then a writer arriving
during a busy period for readers has to wait for the residual
busy period of M f M f c. We next consider the other extreme
when f w -+ 1. In this case, we have an M f M f 1 queueing
system for both readers and writers. The waiting time for
readers with the FCFS policy is the same as writers. With
the ES and TFE (with K 2 1) policies, readers arriving
during a writer busy period are delayed until the system is
emptied from writers; therefore, the mean waiting time for
readers (which have to wait) equals the mean residual busy
period for writers. It is interesting to note the deterioration
of the reader and writer mean response times for the FCFS
policy at intermediate frequencies. This can be explained
by the fact that the system is operating closer to a worst
case scenario of alternating arrivals of readers and writers,
thus significantly reducing the effective degree of reader
concurrency. The reader and overall mean response times for
the TFE and ES policies are less sensitive to f U , , such that
there is only a small increase in response time at intermediate
frequencies.
system with a general distribution for the writer processing
time.
From the analysis of the reader-saturated system, it is shown
that the reader throughput is an increasing function of the
threshold value K , which can be chosen sufficiently large to
attain a throughput arbitrarily close to the maximum. The
same analysis also yields the maximum throughput A( K )
for the TFE policy (with a threshold K ) in a nonsaturated
system. It was shown that h ( K ) is an increasing function of
K and that for all A’ 2 1 it is greater than the maximum
throughput attained by the FCFS policy. This increase in
maximum throughput is more significant when readers have
a longer processing time relative to writers, and for smaller
fractions of writers in the arrival stream. When K is large
enough such that the processing of readers is never disrupted,
the TFE policy becomes equivalent to the ES (exhaustive
service) policy, which is a full capacity discipline (as defined
in [ll]).
The mean response time characteristic for the TFE and
FCFS policies was investigated using simulation. Although
the ES policy yields the maximum throughput, at high utilizations it results in a poor response time for writers (resp.
readers) at a high fraction of readers (resp. writers) in the
arrival stream. The TFE policy obviously outperforms the
V. CONCLUSIONS
FCFS policy, particularly when the maximum throughput
We have considered a multisever queueing system in for the FCFS policy is approached. The TFE policy is also
which readers are processed concurrently and writers are adaptive in that an appropriate value for K can be used
processed one at a time. We have analyzed the system to minimize the reader, writer, or overall mean response
with the TFE (threshold fastest emptying) policy in the time.
It would be interesting to compare the performance of
special case when the reader queue is saturated. Under the
assumption of exponential distributions for reader and writer different scheduling policies for readers and writers other
processing times, we have obtained performance measures than those considered in this paper. Also of interest is the
of interest from the corresponding Markov chain. The development of analytic solutions for a nonsaturated system
MfGI1 queue with server vacations is used to model the with the TFE policy.
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IEEE TRANSACTIONS ON COMPUTERS, VOL. 42, NO. 1, JANUARY 1993
h=0.2
M=10
u=O.l
L
0
I
I
I
0.2
I
1
0.4
I
I
0.6
WRITER FREQUENCY
,
:
E
I
1.o
(fw)
Fig. 13. Mean writer response time versus
APPENDIXA
MARKOVIAN
ANALYSIS
OF THE TFE
1
0.8
fu.
l / j . Hence,
POLICY
Assuming that the processing times of readers and writers
are exponentially distributed, the behavior of the readersaturated system (see Section 111) can be modeled by the
continuous-time Markov chain shown in Fig. 1. In this Appendix we derive closed form expressions for performance
measures of interest by appropriate manipulations of balance
equations in the Markov chain model. In Fig. 1, a state ( i , j )
represents the system with i writers and j readers in service,
i 2 0, 0 I j _< M (j’
= 0 implies that a writer is in service).
The (steady-state) probability of being in state (i,j ) , is denoted
by p ( i , j ) . We define p, as the probability of j readers in
service, 0 5 j 5 M .
The ergodicity condition follows from p ( 0 , M) > 0. For finite
values of K and M , the system stability is determined only
by the writer utilization, Le., A / p < 1. The reader throughput,
y is given by
with p ( 0 , M) from (A.2). The mean response time of a reader
follows from Little’s formula, R, = M / y . The utilization of
the system due to readers (0,) is given by
Note that p,. < 1 - pw because the system is not fully utilized
while
being emptied from readers. Define the quantities Nj =
Let p, be the fraction of time the system is servicing writers
(writer utilization). It is equal to p o . Therefore, po = A/p. C ~ l i p ( i , j )0 , 5 j 5 M. N j / p , can be interpreted as
The probabilities p ( i , M ) , 1 5 i 5 K - 1, are given by the average number of writers in the system, given that j
the balance equations, p ( i , M ) = p(O,M), 1 5 i 5 K - 1. readers are in service. The (unconditional) average number of
Nj.
Balance equations at the set of states (i,j),i 2 K, yield the writers in the system can be expressed by N , = CEO
In
the
following
we
derive
relations
for
the
quantities
N
j,
following for p,, 1 5 j 5 M ,
0 5 j 5 M , leading to a closed form expression for N,.
From balance equations at the states ( i , O ) and (i, M), for
1 5 i 5 K - 1, we have
M
p ( 0 , M) can now be obtained from the normalizing equation,
cEopj =
( K $ H M ) p ( O , M ) = 1, with HM =
p+
+
A x p ( i , j )= pp(i
j=O
+ l,O),
i 2 K.
THOMASIAN AND NICOLA: THRESHOLD POLICY FOR SCHEDULING READERS AND WRITERS
91
Multiplying the above balance equations by i and taking the
sum for all i 2 1, we get
x
No = - ( N u
P
+ 1).
From the balance equation at the state ( i , O ) , for i
have
(A
2 1, we
+ P M i , 0) =
PP(i
+
1.O),
Xp(i - 1,O) + p p ( i
Xp(i - 1,O) PP(i
+
+ l,O),
+ 1,O)
+up(i, 1).
i
= 1,
2 5 i 5 K - 1,
i 2 K.
5 (1 - :)
From the balance equation at the state ( K
and 1 5 j 5 M , we have
+ i , j ) , for i
PROOF OF THE
AZ2Fs
5 A ( K ) , for
In this Appendix we prove that
all K , Le., the maximum throughput attained by the TFE
policy is greater or equal to that attained by the FCFS policy.
The equality holds for M = 1 and at the extremes, when
fr = 1 or fr = 0. It was shown in Section IV-A that
A ( K 1) > A ( K ) ,for all K 2 1. Therefore, it is sufficient
to show that
5 R ( 1 ) ,which is equivalent to showing
that the Inequality (4.5) holds. For convenience, we use the
following notation: x = 1,y =
r = fr. Substituting for
from (5.1), with :ome manipulations, the Inequality
(4.5) can be rewritten as follows:
+
Multiplying the above balance equation by i and taking the
sum for all i 2 1, we get
N1 =
APPENDIXB
INEQUALITY
(A.4)
ALCFs
i,
2 1
+ ( j + l)vp(K + i , j + l),
i 21.
Multiplying the above equation by i and taking the sum for all
i 2 1, we get the equation shown at the bottom of this page
where we made use of N M =
i p ( i ,M ) x z o ( K
i ) p ( i , M ) .The probabilities p ( K , j ) , 1 5 j I: M , in (A.6),
can be determined from the balance equations at the states
( K , j ) , 1 I: j 5 M ,
czyl
+
+
Noting that x 2 0 and 0 5 r 5 1, the term containing x is
always positive, therefore, it can be omitted from the r.h.s. of
the above inequality. Furthermore, since y 2 0, the inequality
can be reduced to
After simple manipulations we get
A closed form expression for N , is obtained by substituting
from (A.l), (A.2) and (A.4) to (A.7) into N , =
Nj.
Finally, we have
,:E
Pw
+
N , = - yHM
1-Pw
To prove the Inequality (B.l), we use induction on M . It is
simple to show that it holds for M = 1 and for M = 2. Now,
assuming that it holds for arbitrary M , we prove that it holds
for M + 1. After some manipulations, the Inequality (B.l), for
M 1, can be written as follows
+
IEEE TRANSACTIONS ON COMPUTERS, VOL. 42, NO. 1, JANUARY 1993
98
Using the induction hypothesis for M [Inequality (B.l)] and
after some manipulations, it remains only to show that
Denote by I ( r ) the r.h.s. of the Inequality (B.3). Then replace
i in the denominators of the second and third summations by
M 1 and M , respectively. It follows that
+
.
M
K. Omahen and L. Schrage, “A queueing analysis of a multiprocessor system with shared memory,” in Proc. Symp. Comput.-Commun.
Networks and Teletraffic, Polytechnic Institute of Brooklyn, NY,Apr.
1972, pp. 77-88.
K. J. Omahen, “Capacity bounds for multi-resource queues,” J. ACM,
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i=l
The r.h.s. of the above inequality is greater than or equal to
1, which completes the proof.
ACKNOWLEDGMENT
The authors acknowledge fruitful discussions with Prof. Y.
Viniotis from the ECE Dept. at North Carolina State University
and Dr. R. Nelson from IBM Research. The proof in Appendix
B is due to Dr. R. Gail from IBM Research. Comments and
criticisms of anonymous referees have helped improve the
quality of presentation.
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Alexander Thomasian received the B.S. degree in
electrical engineering from the University of Tehran,
Iran, (with honours) and the M.Sc. and Ph.D. degrees in computer science from the University of
California at Los Angeles.
He has been a member of research staff in the
Systems Analysis Department at the IBM T. J.
Watson Research Center since 1985. He was a
faculty member at Case Western University and the
University of Southern California and a senior staff
scientist at Burroughs (now Unisys) Corporation.
He has taught tutorials on high-performance architectures for transaction
and query processing and has been an Adjunct Professor at the Computer
Science Department at Columbia University. His current research is in the
area of performance analysis and design of parallel and distributed systems
and especially disk arrays. His prior research has been concerned with
the applications of queueing network models in performance evaluation of
computer and communication systems and techniques to analyze the effect
of database concurrency control on the performance of transaction processing
systems.
Dr. Thomasian has served on the program committees of the ACM
Sigmetrics, IEEE Data Engineering, and Distributed Computing Conferences.
He is a member of the IEEE Computer Society and the Association for
Computing Machinery.
Victor F. Nicola (M’89) received the Ph.D. degree
in computer science from Duke University, North
Carolina, the B.S. and the M.S. degrees in electrical
engineering from Cairo University, Egypt, and Eindhoven University of Technology, The Netherlands,
respectively.
From 1979 he held scientific and research staff
positions at Eindhoven University and Duke University. Since 1987, he has been a Research Staff
Member at the IBM Thomas J. Watson Research
Center. His research interests include performance
and reliability modeling of computer systems, queueing theory, fault-tolerance
and simulation.
Dr. Nicola is a member of the IEEE Computer Society.