A vacation model for the non-saturated Readers and Writers system with a threshold policy

Performance Evaluation 50 (2002) 233–244

A vacation model for the non-saturated Readers andWriters system with a threshold policy

Eric Xua, Attahiru Sule Alfab,∗a Cogency Semiconductor Inc., 362 Terry Fox Drive, Kanata, Ont., Canada K2K 2P5

b Department of Industrial and Manufacturing Systems Engineering, University of Windsor,401 Sunset Avenue, Windsor, Ont., Canada N9B 3P4

Received 8 January 2001; received in revised form 14 March 2002

Abstract

The non-saturated Writers and Readers system with threshold service policy is a general situation of the classical Writersand Readers problem. It considers the case where both Writers and Readers have thresholds for beginning of service. Thispaper sets up two vacation models for the Writers and Readers, respectively. Since the dependency between Writers andReaders is approximated, this model system is an approximation. The numerical results compared with the simulation resultsshow that this vacation model system approximates the non-saturated Writers and Readers problem very well.© 2002 Elsevier Science B.V. All rights reserved.

Keywords:Threshold; Vacation models; Decomposition; Approximation

1. Introduction

The Writers and Readers problem is a queuing system which has two kinds of customers and multipleservers. The two kinds of customers are called “Writers” and “Readers”. The service policies betweenWriters and Readers are different. When the servers attend to Readers they perform as a multiple serversystem, thus each server could serve one Reader separately and simultaneously, i.e. the Readers could beprocessed concurrently. On the other hand, when the servers attend to Writers they perform like a singleserver system, i.e. the Writers are mutually exclusive with respect to each other and all the servers serveone Writer together as a single server. The Writers and Readers problem is a classical problem whichoccurs in operating systems. It can also be considered as a general class of two-queue problems. Manyreal-life problems could be considered as special cases of the Writers and Readers problem, either directlyor after simple modifications. Some examples are the distributed database management systems, sharedmemory systems, telecommunication networks, manufacturing systems and other transaction processingsystems as well.

∗ Corresponding author. Tel.:+1-519-253-3000x3917; fax:+1-519-971-3667.E-mail addresses:[email protected] (E. Xu), [email protected] (A.S. Alfa).

0166-5316/02/$ – see front matter © 2002 Elsevier Science B.V. All rights reserved.PII: S0166-5316(02)00084-6

234 E. Xu, A.S. Alfa / Performance Evaluation 50 (2002) 233–244

The non-saturated Writers and Readers system with threshold service policy is a general situation ofthe Writers and Readers problems. It considers the case where both Writers and Readers have thresholdsfor beginning of service. By choosing different values for the parameters, the special cases of a Writersand Readers system could be obtained. For example, by choosing the number of servers to be 1, we canhave a two-queue, single server system with threshold for service. Furthermore, by choosing the valueof the thresholds to be 1, we can have a simple two-queue, single server system. By setting the thresholdvalue for Readers to be 1, increasing the Reader arrival rate to be infinite and considering the Writersto have priority over Readers for service, then we can have the Reader-saturated Writers and Readerssystem, and so on.

Under threshold service policy, the servers start to serve one kind of customer only from an idle stateor after they finish serving other kinds of customers, when the number of this kind of customers inthe waiting queue reaches a certain number, i.e. the threshold. Otherwise, the servers cannot start toserve this class of customers. The threshold policy can be applied on one or both classes of customers.The threshold policy applied to Writers and Readers problems makes these kinds of problems moreflexible and more meaningful. It allows the servers to stay in an idling state for a certain period ratherthan starting to serve the new kind of customer immediately, once they have finished serving the previouskind. By changing the value of the thresholds, the system could be optimized to seek the goals of themost efficient, the highest throughput, the least cost, least time-consuming or the best economical results,depending on the system users.

A major study in this area related to this paper was done by Thomasian and Nicola[1], which focusedon the Reader-saturated, Writers and Readers system. Their work was based on the situation of Readersaturation, so the Writers always have priority over the Readers. Otherwise, once the servers start to servethe Readers, there are always enough Readers in the queue waiting for service—hence, saturation, andthere is no chance for Writers to be served. Since the priority service discipline was applied in this Writersand Readers system, an analytical model could be established mainly focussing on the performance ofthe Writers. The Readers have an effect on the system only when the priority is limited, i.e. the Writershave only non-preemptive priority over the Readers. Thus, once the Writers reach the threshold to startservice, there is a lock-up procedure to terminate the service for the Readers which were on going. So theanalytical model of the Reader-saturated Writers and Readers system in paper[1] could be understoodas a single server queuing system with threshold service policy and warm-up time or lock-up timefor start of service. This well-known locking mechanism is extensively used in distributed databasesmanagement, shared memory systems, manufacturing systems and other transaction processing systems.The Reader-saturated queuing structure assumption and the priority service policy tremendously reducethe difficulty for analytical modeling and avoid the dilemma of the two infinite queue systems. Langarisand Moutzoukis[2] studied the batch arrival Writers and Readers queuing system with retrial Writers.In their work, Readers have non-preemptive priorities over Writers. Not servers, but Writers will takevacations when they arrive if they find the system is busy. It does not matter if they are busy on servingReaders or Writers. Retrial was used to describe this situation in their paper. Furthermore, batch arrival,multi-class of Readers, single class retrial Writers and infinite servers with the server’s vacations wereconsidered. The stability conditions and the generating functions of the state probability of the Writersand LSTs of the virtual queuing time for both Readers and Writers were obtained. The infinite numberof servers Readers and Writers queuing system with Readers preemptive priority over Writers was firststudied by Kulkarni and Puryear[3], based on an analysis of M/G/∞ system. They derived the meanqueuing times for both Writers and Readers. The ordinary Writers and Readers system without priorities

E. Xu, A.S. Alfa / Performance Evaluation 50 (2002) 233–244 235

was studied by Courcoubetis et al.[4] and Baccelli et al.[5]. They obtained the stability conditions. Thethreshold policy was discussed by Kella[6] in a M/G/1 queue with server vacation.

Thomasian and Nicola[1] discussed the Reader-saturated Writers and Readers system. They usedboth analytical and simulation methods to study the system. As a natural extension to their study, weconsider how to analytically model the Writers and Readers non-saturated system. Specifically, we want todetermine what the results will be when both Writers and Readers have thresholds for services. Substantialdifficulties exist in the computation of steady state distribution. Thus, it is difficult to compute the queuelength, waiting time and other queuing features of this system, since this is a two-queue system and bothare infinite queues without possessing any special structure for us to take advantage of. The aim of thispaper is to build up an analytical model system for this very general situation and find a way to solve thesekinds of Writers and Readers problems analytically, rather than with a simulation method. Throughputstudies, and efficiency studies of the non-saturated Writers and Readers system using this model system,are important for real-life applications. However, they are next steps, not the main subjects in this paper,and thus are left for future works.

In this paper, the very strict assumption of the saturation of the Readers is taken off from the Writ-ers and Readers system. The Writers and Readers are treated equally, i.e. there is no priority betweenthe Writers and Readers. This gives the system a more realistic fit to real-life practice, and moreover,the threshold service policy is applied to Readers as well. The approximated analytical model systemof the non-saturated Writers and Readers problem is the main progress this paper has achieved (Fig. 1).

Fig. 1. Non-saturated Writers and Readers system with double thresholds.


2. The non-saturated Writers and Readers system

Consider a Writers and Readers system withM servers working on two queues, the Writer queue andthe Reader queue. Both Writer queue and Reader queue have a threshold,kw andkr, respectively, forstarting the service. The service for both queues is based on an exhaustive service discipline. There is nopriority between these two queues. Thus, when both queues are below their threshold values, the serverswill keep idling and not perform any service until the number of customers in one of the two queuesreaches that queue’s threshold. The servers will start to serve the queue which reaches the threshold first.Once the servers are operating on one queue, the service will continue until that queue becomes empty. Tounderstand the system clearly, assume the number of Readers in the Readers queue reaches its thresholdkr first, then the server will start to serve Readers and keep on serving Readers until the Reader queue isempty. At this moment, the servers will check the number of Writers in the Writer queue. If the numberof Writers in the Writer queue has reached or exceeded the thresholdkw, the servers will start to servethe Writers. Otherwise, if the number of Writers in the Writer queue is less thankw, then the servers willbe idle again waiting for one of the two queues to reach their threshold values. When the servers attendto the Writers allM servers will work together, as a “single” server system. However, when the serversattend to the Readers they will work separately and concurrently as a multiple server system, each servercan serve one Reader. There are thresholds applied in this Writers and Readers system on the Writerqueue and Reader queue, respectively. This could help the system to allocate the service time on the twoqueues, in real-life applications to achieve high throughput and low cost by adjusting the values of thethresholdskw andkr, and even the value of the number of serversM as well. However, how to decide theoptimal values of thekw, kr andM is not the topic here. The aim of this paper is to model this Writersand Readers system theoretically and to develop an algorithm to solve it analytically. Then the behaviorof this system will be more obvious and clear and the system could be measured and controlled.

We consider both Writers and Readers have Poisson arrivals with the arrival rates ofλw andλr, andexponential service with the service rates ofµw andµr.

Let Ω represent the state space, where

Ω = ((i, j, k) : i = 0, 1, . . . , kw − 1; j = 0, 1, . . . , kr − 1; k = 0)⋃((i, j, k) : i = 0, 1, . . . ,∞; j = 0, 1, . . . ,∞; k = 1, 2; i + j ≥ 1),

wherei is the number of Writers in the system,j the number of Readers in the system andk the status ofthe servers:

k = 0: the servers are idling.k = 1: the servers are serving Writers.k = 2: the servers are serving Readers.

According to the state space on above, the structure of the Markov chain of this Writers and Readerssystem is very clear. The generator matrixQcould be easily written down. However, since both the Writerqueue and Reader queue are infinite queues, the matrix geometric method could not be applied directlyto solve this problem. There are two ways we could solve the problem. One is by truncating and the otheris using a vacation model idea. Both are approximations. We chose the vacation approach because it ismore realistic.


2.1. Vacation model system

We focus on Writers or Readers separately, and set up two vacation models for both the Writersand Readers, respectively, i.e. decompose this Writers and Readers system into two separate systems:the Writers system and the Readers system. However, both models are not solvable directly due toinsufficient information. By considering the dependency between the two models, the analytical resultscould be obtained by solving the two models together.

The vacation method can be used to approximately capture the dependency between the Writers andReaders. Thus, two vacation models are set up for the Writers and Readers, respectively, and the problemis solved analytically.

3. System modeling

3.1. Model for Writers

Consider a single server queuing system, where Writers are the only kind of customers served by asingle server. The server can serve the Writers or go on vacation. The server starts to serve the Writersonly when the number of Writers in the waiting queue reaches the thresholdkw with exhaustive servicepolicy applied, otherwise the server will be idling or will take a vacation.

Let kw be the threshold for satisfying the service condition to start serving Writers. Consider a Markovchain described by the state space∆ = (i, j), i ≥ 0, j = 0, 1, wherei is the number of Writers in thesystem, andj represents the states of the servers:

j = 0 represents the service condition is not established or the server is on vacation.j = 1 represents the system in service state, i.e. the server is busy serving Writers.

So in this Writers system, the server has two states: available for service or on vacation. The vaca-tion period distribution is unknown, but the necessary information could be obtained from the Readerssystem.

Assumptions

1. The vacation process is independent and therefore does not depend on the Writers arrival process andservice process.

2. Steady state exists for this system.3. In the system steady state, the system has probabilityP 1

w of being on vacation or not available forservice and probabilityP 0

w of being available for service, thusP 1w + P 0

w = 1. P 1w andP 0

w could becomputed from the Readers system.

The resulting Markov chain of this system has the following generator matrixQw:

Qw =[

A00 A01

A10 A11

],


where

A00 =

−λw λw

−λw λ

−λw λ

. . .. . .

−λw P 1wλw

−λw P 1wλw

−λw P 1wλw

. . .. . .

,

A01 =

. . .

P 0wλw

P 0wλw

P 0wλw

. . .

, A10 =

µw

,

A11 =

−(λw + µw) λw

µw −(λw + µw) λw

µw −(λw + µw) λw

. . .. . .

. . .

µw −(λw + µw) µw



. . .. . .

.

The special structure of this matrix makes the steady state equation solvable.Let πw be the invariant vector,πw = (xw, yw):

xw = xw0 , xw

1 , xw0 , . . ., yw = yw

0 , yw1 , yw

0 , . . ., πwe = 1.

So,πw satisfies the steady state equation:

πwQw = 0,

i.e.

(xw, yw)

[A00 A01

A10 A11

]= 0.


Let

ρw = λw

µw.

Solving this equation, we obtain the analytical results of the steady state vector:

yw1 = ρw(1 − ρw)/

(kw + P 1

w

P 0w

), xw

0 = xw1 = · · · = xw

kw−1 = (1 − ρw)/

(kw + P 1

w

P 0w

),

xwkw+i = (P 1

w)i+1(1 − ρw)/

(kw + P 1

w

P 0w

), i ≥ 0,

ywi = ρw(1 − ρi

w)/

(kw + P 1

w

P 0w

), 1 ≤ i ≤ kw,

ywk+i = ρwyw

k+i−1 + (P 1w)iρw(1 − ρw)/

(kw + P 1

w

P 0w

), i ≥ 1.

For detail computations, see[7].

3.2. Model for Readers

The vacation model for Readers could be set up in the same way as that of Writers.Consider a multiple server queuing system in which Readers are the only kind of customers served by

theM servers. The servers can serve the Readers or go on vacation. The servers start to serve the Readersonly when the number of Readers in the waiting queue reaches the thresholdkr with exhaustive servicepolicy applied, otherwise the servers will be idling or take a vacation. When the servers take vacationsthey will behave as a single server.

Let kr represents the threshold for serving Readers andM is the number of servers.Consider a Markov chain described by the state space∆ = (i, j), 1 ≥ 0, j = 0, 1, wherei is the

number of Readers in the system andj represents the states of the servers:

j = 0 represents the service condition is not yet established or the server is on vacation.j = 1 represents the system in service state, i.e. the server is busy serving Readers.

So in this Readers system, the server has two states: available for service or on vacation. The vaca-tion period distribution is unknown, but the necessary information could be obtained from the Writerssystem.

Assumptions

1. The vacation process is independent and does not depend on the Readers arrival process and serviceprocess.

2. Steady state of this system exists.3. In steady state, the servers have the probabilityP 1

r of being on vacation or not serving yet and theprobabilityP 0

r of being available for service, thusP 1r + P 0

r = 1. P 1r andP 0

r could be computed fromthe Writers system.


The resulting Markov chain of this system has the following generator matrixQr:

Qr =[

B00 B01

B10 B11

],

where

B00 =

−λr λr

−λr λr

−λr λr

. . .. . .

−λr P 1r λr

−λr P 1r λr

−λr P 1r λr

. . .. . .

,

B01 =

. . .

P 0r λr

P 0r λr

P 0r λr

. . .

, B10 =

µr

,

B11 =

−(λr + µr) λr

2µr −(λr + 2µr) λr

3µr −(λr + 3µr) λr

. . .. . .

. . .

(M − 1)µr −(λr + (M − 1)µr) λr

Mµr −(λr + Mµr) λr

Mµr −(λr + Mµr) λr

. . .. . .

.

Let π r be the invariant vector,πr = (xr, yr).

xr = xr0, x

r1, x

r0, . . ., yr = yr

0, yr1, y

r0, . . ., πre = 1.

So we have the steady state equation as

πrQr = 0, (xr, yr)

[B00 B01

B10 B11

]= 0.


Let

ρr = λr

Mµr.

Solving the equations, we obtain the analytical results of the steady state vector as

1. yr1 = 1/

1

Mρr

(kr + P 1

r

P 0r

)+ 1

M(1 − ρr)

M∑

i=1

M − i

i!

i∑j=1

(j − 1)!(Mρr)i−j +

(kr + P 1

r

P 0r

) .

2. xr0 = xr

1 = · · · = xrkr−1 = yr

1

Mρr, xr

kr+i = (P 1r )i+1 yr

1

Mρr, i ≥ 0,

yri = 1

i!

i∑j=1

(j − 1)!(Mρr)i−j yr

1, 1 ≤ i ≤ M, yri = ρry

ri−1 + yr

1

M, M + 1 ≤ i ≤ kr,

yrkr+i = ρry

rkr+i + (P 1

r )i−1yr1

M, i ≥ 1.

For detail computations, see[7].

3.3. Alternative method for computing steady state parameters

From the Writer model, we could calculateP 1r andP 0

r for the Readers model. SinceP 1r is the probability

of the server being on vacation for the Readers model, it is thus the probability that the server is busyserving Writers, so

P 0r =

∞∑i=0

xwi , P 1

r =∞∑i=1

ywi and P 0

r + P 1r = 1,

(kw + P 1

w

P 0w

)1

ρwyw

1 +(

kw + P 1w

P 0w

)1

1 − ρwyw

1 = 1.

So we obtain

P 0r = 1 − ρw, P 1

r = ρw.

From the Readers model, we can calculateP 1w andP 0

w for the Writers model:

P 0w =

∞∑i=0

xri , P 1

w =∞∑i=1

yri and P 0

w + P 1w = 1,

yr1

Mρr

(kr + P 1

r

P 0r

)+ 1

M(1 − ρr)

[M∑i=1

(M − i)yri +

(kr + P 1

r

P 0r

)1

1 − ρryr

1

]= 1.

So we obtain:

P 0w =

(k + P 1

r

P 0r

)yr

1

ρr, P 1

w = 1

1 − ρr

[M∑i=1

M − i

Myr

i + yr1

M

(kr + P 1

r

P 0r

)].


We can choose the initial value forP 1r andP 0

r , then analyze the Writers model to obtainP 1w andP 0

w,then repeat the analysis for the Readers model and Writers model alternately untilP 1

w, P 0w andP 1

r , P 0r

converge to stable values. However,P 1r andP 0

r are decided by onlyρw, so theP 1w, P 0

w andP 1r , P 0

r couldbe obtained directly. There is no overlap between the busy period for Writers and the busy period for theReaders. However, there may be an overlap between the servers on vacation period for Writers modeland the servers on vacation period for Readers model, e.g. the servers are in idle state, neither servingthe Writers nor the Readers, so the following constraint as the stability condition must be applied:

P 1w + P 1

r ≤ 1.

The general stability condition forM = 1 isρw + ρr < 1.

4. Comparison of numerical results

The two vacation models have successfully described the Writers and Readers system. The only ap-proximation in the system is the dependence between the Writers and Readers. Numerical results arecompared between the analytical calculations and the simulation results in various situations by choosingthe corresponding parameters. The simulation project concentrates on monitoring the mean of the Writersqueue and Readers queue and the probability of busy period, so the comparative analysis is based onthese two major results.

4.1. The probability of total busy period

In the Writers and Readers system, the server busy period is the total busy period of serving Writers andReaders. So, the probability of the server being busy is the sum of the probability of the servers servingWriters and the probability of the servers serving Readers. The comparison shows extreme agreementbetween calculated results from two separate vacation models and simulated results in all the cases. Theapproximation of our vacation model system is the approximation of the dependence between the two vaca-tion models. The match of the total busy period of the whole system between the calculated and simulatedresults shows that the vacation models approach the Writers and Readers system very well (seeTables 1–4).

4.2. The mean of queue length at low and medium traffic

The comparison shows generally good agreement for the mean of the queue length for both Writersand Readers queues between calculated and simulated results, in the case of low and medium traffic, i.e.when the system is in a stable state, especially whenP 1

w + P 1r ≤ 0.6.

Table 1λw = λr = 1/4, µw = µr = 2, kw = kr = 8 by changingM

Writers queue Readers queue Busy period,P 1w + P 1

r

λw µw kw qw λr µr kr qr M

Analytical 1/4 2 8 3.5993 1/4 2 8 3.5993 1 0.2500Simulation 1/4 2 8 3.5698 1/4 2 8 3.5932 1 0.2468Analytical 1/4 2 8 3.5426 1/4 2 8 3.4316 8 0.1669Simulation 1/4 2 8 3.5057 1/4 2 8 3.4669 8 0.1668


Table 2λw = λr = 1/4, µw = 2, µr = 1, kw = 5, kr = 8 by changingM


r


Analytical 1/4 2 5 2.2262 1/4 1 8 3.6648 1 0.3750Simulation 1/4 2 5 2.1406 1/4 1 8 3.6396 1 0.3692Analytical 1/4 2 5 2.0815 1/4 1 8 3.3244 4 0.2197Simulation 1/4 2 5 2.0152 1/4 1 8 3.2955 4 0.2206Analytical 1/4 2 5 2.0733 1/4 1 8 3.2824 8 0.2087Simulation 1/4 2 5 2.0280 1/4 1 8 3.2419 8 0.2101

Table 3λw = λr = 1/4, µw = 1, µr = 1, kw = 5, kr = 8 by changingM


r



Table 4λw = 1/4, λr = 1/2, µw = 2, µr = 1, kw = 5, kr = 8 by changingM


r



When the system approaches an unstable state, especially whenP 1w +P 1

r > 0.6, the mean of the queuelength for both Writers and Readers queues becomes inaccurate (seeTables 1–4).

This paper presents a first stage study of the analytical modeling of the general non-saturated Writersand Readers systems. The computations of the system characteristics are still limited to the steady statedistribution of the number of customers. However, this distribution is often the most important one inobtaining the other related analytical results. Further research works could go in several directions. Little’slaw could be applied, combined with further analysis to obtain the formulas to compute the other queuingfeatures, such as mean waiting time, busy period, etc.


5. Conclusion

The Writers and Readers systems have extensive applications. The analytical model is very importantfor applying these systems in real-life applications efficiently. But, because of the mentioned difficultieswith the analytical model and computations, simulation techniques are still the main method of analyzingmost real-life problems faced by manufacturing engineers, systems engineers and telecommunication re-searchers. This paper has made progress in the solvable analytical modeling method. Even though the twovacation model system still has to approximate the dependence between the Writers and Readers systems,it describes the system clearly, simply and in a way that is easily understood and utilized. The mathematicalassumptions are rational and conform with real-life problems. The numerical results were compared withthe simulation outputs in various data values, the results match each other reasonably well. It can be con-cluded that this approximated analytical method is an effective method for dealing with the non-saturatedWriters and Readers problems. However, the solvable analytical models without approximation of thegeneral non-saturated Writers and Readers system is still an open question and challenge to researchers.

References

[1] A. Thomasian, V.F. Nicola, Performance evaluation of a threshold policy for scheduling Readers and Writers, IEEE Trans.Comput. 42 (1) (January 1993).

[2] C. Langaris, E. Moutzoukis, A batch arrival reader–writer queue with retrial writers, Commun. Statist. Stochast. Models13 (3) (1997) 523–545.

[3] V.G. Kulkarni, L.C. Puryear, A reader–writer queue with reader preference, Queuing Syst. 15 (1994) 81–97.[4] C.A. Courcoubetis, M.I. Reiman, B. Simon, Stability of a queueing system with concurrent service and locking, SIAM J.

Comput. 16 (1987) 169–178.[5] F. Baccelli, C.A. Courcoubetis, M.I. Reiman, Construction of the stationary regime of queues with locking, Stochast. Proc.

Appl. 26 (1987) 257–265.[6] O. Kella, The threshold policy in the M/G/1 queue with server vacations, Nava Res. Logistics 36 (1989) 111–123.[7] E. Xu, Analysis of the non-saturated readers and writers system with a threshold policy, M.Sc. Thesis, University of Manitoba,

2000.

Eric Xu received his B.S. degree in applied mathematics from Ocean University of Qingdao, Qingdao,China, in 1985, the diploma of graduate studies in mathematical statistics and probability from BeijingUniversity, Beijing, China, in 1987 and M.S. degree in industrial engineering from the University ofManitoba, Winnipeg, Canada, in 2000. From 1987 to 1993, he was a research scientist in the Instituteof Marine Economics, Qingdao, China. After then, he was with ADGA Consulting Group, as a systemengineer between 1999 and 1998 and joined Nortel Networks as a capacity and performance analystin 2000. Since March 2001, he has been with Cogency Semiconductor Inc. as a network protocol andalgorithm designer, all in Ottawa, Canada.

Attahiru Sule Alfa received his B.E. degree from the Ahmadu Bello University, Nigeria, in 1971, M.Sc.degree from the University of Manitoba, Winnipeg, Canada, in 1974, and Ph.D. degree from the Universityof New South Wales, Australia, in 1980. He is a Professor of Industrial and Manufacturing SystemsEngineering (with cross appointments in Mathematics and Statistics, and also Electrical and ComputerEngineering) and an Associate Vice-President of Research at the University of Windsor. His researchinterest is in the area of operations research, with focus in the applications of queuing and network theories totelecommunication systems, especially mobile wireless communications. He also has interests in networkrestoration problems. He has contributed to the development and applications of the matrix-analytic methodfor stochastic models, especially in the area of discrete time queuing systems.

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A vacation model for the non-saturated Readers and Writers system with a threshold policy