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IRJET- On the Analysis of Shortest Queue with Catastrophes and Restricted Capacities
1.
International Research Journal
of Engineering and Technology (IRJET) e-ISSN: 2395-0056 Volume: 06 Issue: 06 | June 2019 www.irjet.net p-ISSN: 2395-0072 © 2019, IRJET | Impact Factor value: 7.211 | ISO 9001:2008 Certified Journal | Page 2014 On the Analysis of Shortest Queue with Catastrophes and Restricted Capacities A.M.K. Tarabia1, A.H. El-Baz2, A.A. Aziz3 1Professor of Mathematical Statistics, Dept. of Mathematics, Damietta Faculty of Science, New Damietta, Egypt. 2Assistant Professor of Statistics, Dept. of Mathematics, Damietta Faculty of Science, New Damietta, Egypt. 3M.Sc. School Basic Sciences – Department of Mathematical Sciences –Statistics Division - New Damietta, Egypt. ---------------------------------------------------------------------***---------------------------------------------------------------------- Abstract - In this paper we present a catastrophe on a system consisting of two parallel queues with jockeying and limited capacities. The capacity of each queue is restricted to L including that was being served. Customers arrive according to a Poisson process and on arrival; they join the shortest feasible queue. Where we study the impact of catastrophe on the system and derive some important performance measures. Key Words: Steady state solution, Shortest queue, Jockeying, Catastrophes, Single server queue. 1. INTRODUCTION System in this paper consisting of two parallel servers with different service rates With a jockeying system, Jobs arrive according to a Poisson .Jockeying can be described as the movement of a waiting customer from one queue to another of shorter length or which appears to be moving faster, etc. in anticipation of a shorter delay. On arrival a job joins the shortest queue as and in case both queues have equal lengths, he joins the first queue with probability α and the second one with probability β . Haight [14] originally introduced the problem Kingman [16] and Flatto and McKean [11] treatedthesymmetric problem by using a generating function analysis. They showed that the generating function for the equilibrium distributionof the lengths of the two queues is a meromorphic function. Then by the decomposition of the generating function into partial fractions, it follows that the equilibrium probabilities can be represented by an infinite sum of product form solutions. However, the decomposition leads to cumbersome formulae for theequilibriumprobabilitiesandthemethoddoes notseemto be generalizable to the asymmetric problem. Cohen and Boxma [6] andFayolleandIasnogorodski[12]studiedthe asymmetric shortest queue problem and showed that the analysis of the asymmetric shortest queue problem can be reduced to a simultaneous boundary value problem in two unknowns. The resulting boundary value problem however,isnotof a standard type and further research remains to be done here.For the numerical approach, seeAden etal.[1–3]showedforthesymmetric shortest queue problemthat the steady state distribution of the queue lengths of the two queues can be found in an elementaryway directly from the equilibrium equations. They showed the compensation procedure can be easily extendedto the asymmetric shortest queue problem. Moreover, they extended the compensation method to the case oftwo identical Erlange-k servers and shortest expected delay routing. Conolly [8] discussed the finite waiting room version of the shortest queue problemand showed that this problem can be solved efficiently, essentiallybydimensionreduction.Zhaoand Grassman [21] obtained explicitsolutions for the equilibrium probabilities, the expected customers, and the expected waiting time of a customerin the system. Van Houtum et al. [19] study a production system consisting of a group ofparallel machinesproducing multiple job types. On arrival, a job joins the shortest queue among all queues capable of serving that job. They formulate anumerical approach to determine the upper and lower bounds for the mean waiting time. For further overviewof solution methods of the performance analysis of parallel and distributed system see [5] Recently, Yao and Knessl [20] considered two parallel M/M/1 queues. Tarabia [17,18] study a system consisting of two parallel queueswithjockeyinganddifferentserver’s rates. In recent times, a new class of queuing systems with Catastrophes has been introduced. These catastrophes may occur indiscriminately, killing all customers and temporarily disrupting the service facilityuntil newcustomersarrive.Thedisasters may come from outside the system or from another service station. have been investigated by Boucherie and Boxma [7], Jain and Sigman [15] and Dudin and Nishimura [10]. The notion of catastrophes occurring at random, leading to annihilation of all the customers there and the momentary inactivation of the service facilities until a new arrival of customersisnot uncommon in many practical situations. The catastrophes may come either from outside the system or from another service station. Comprehensive treatment of queueing models with catastrophes can be found in Gelenbe and Pujolle [13], Chao et al. [9] and Artalejo [4] a new class of queueing systems with catastrophes. Queueing models with catastrophes and breakdown are
2.
International Research Journal
of Engineering and Technology (IRJET) e-ISSN: 2395-0056 Volume: 06 Issue: 06 | June 2019 www.irjet.net p-ISSN: 2395-0072 © 2019, IRJET | Impact Factor value: 7.211 | ISO 9001:2008 Certified Journal | Page 2015 extensively studied as mentioned above, no work has been found in the literature which studies queueing systems taking together the above mentioned features. Based on this observation, In this paper we have been studying parallel lines with jockeying, limited capabilities and disasters. The results of this paper are organized as follows: In Section 2, we describe the model and we formulate its difference equations. Section 3, the solution for the stationary distribution of these equations is given in simple matrix form.InSection4, we present some important performance measures our model are given. In Section 5, we compute the mean of queue. in Section 6, some other numerical results are presented . 2. Model Description In this model the two queues in parallel with jockeying and the capabilities are constrained with Catastrophes and the customer access rate is the Poisson process with the arrival rate . The queueing system consists of twoparallel serverswith different rates 1 and 2 , respectively. Where the client organizes the next to the short queue, if the two queues are equal it chooses the first column with probability or second column with probability of , where 1 . With disasters occurring on this system as an independent Poisson process with parameter v and inactivate the server upon arrival as shown in Figure (1). Chart-1: Shows the catastrophes of the system The capacity of each queue is restricted to L including the one being served. The momenttheserver becomesidleandthereisa customer waiting in the other queue, the customer immediatelyfollowingthecustomerwhoisreceivingserviceatthatcounter is transferred to the idle server’s queue. Let N1 and N2 be, respectively, the number of customers in each queue. Since all distributions are exponential, the stochastic process (N1, N2)isa Markovchainwithstatespace{0,1,2,…,L}x{0,1,2,…,L}andwe can write ),Pr( 21, jNiNP ji for the steadystateprobability.Fromtheabove assumptions,theprobability jiP, satisfies the following system of difference equations: 1 1,0 2 0,1 0,0( ) 0P P P (1) 0,0 1 1,1 2 0,1 0P P P (2) 0,0 2 1,1 1 1,0 0P P P (3) 0,1 1,0 1 2,1 2 1,2 1 2 1,1( ) ( ) 0P P P P P (4) 1,1 1 2,2 1 1,3 2 1,3 1 2 1,2( ) 0P P P P P (5) L,LL-1,LK,L1,L L,L-1L-1,L-1 n,nn-1,nK,n1,n N,n-1 L,kn,k L,1n,1 k
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of Engineering and Technology (IRJET) e-ISSN: 2395-0056 Volume: 06 Issue: 06 | June 2019 www.irjet.net p-ISSN: 2395-0072 © 2019, IRJET | Impact Factor value: 7.211 | ISO 9001:2008 Certified Journal | Page 2016 1,1 2 2,2 1 3,1 2 3,1 1 2 2,1( ) 0P P P P P (6) 1,2 2,1 1 3,2 2 2,3 1 2 2,2( ) ( ) 0P P P t P P (7) 1 2 1, 1 1 2, 1 2 1,( ) 0, 3 1n n nP P P n L (8) 2 ,2 1 1,1 2 1,1 1 2 ,1 0 3 1n n n nP P P P n L (9) 1, 1 1, 2 , 1 1 2 , 0 2 2k n k n k n k nP P P P k n (10) , 1 1 1, 2 , 1 1 2 , 0, 2 2n k n k n k n kP P P P k n (11) 2, 1, 1 1 , 2 1, 1 1 2 1, 0n n n n n n n n n nP P P P P (12) 1, 1 , 2 1 1, 1 2 , 1 2 , 1( ) 0n n n n n n n n n nP P P P P (13) 1, , 1 1 1,1 2 ,, 1 1 2 , 0n n n n n n n n nP P P P P (14) 1 2, 1 2 1, 0L LP P (15) 2 ,2 1 2 ,1 0L LP P (16) 1, 1 1, 1 2 , 0, 2 2k L k L k LP P P k L (17) , 1 2 , 1 1 2 , 0, 2 2L k L k L kP P P k L (18) 2, 1, 1 1 , 1 2 1, 0L L L L L L L LP P P P (19) , 2 1, 1 2 , 1 2 , 1 0L L L L L L L LP P P P (20) 1, , 1 1 2 , 0L L L L L LP P P (21) Define the column vector kP of 12 k elements as ,,...,3,2,1,),,,...,,,,( ,1,,12,,21,,1 LkPPPPPPPP t kkkkkkkkkkk and .,...,3,2,1,),,,( 2,0,11,0 LkPPPPP t kkk Also, let 1 2 1 2 1 2 1 2 1 2 1 2 , , .and Ignoring the Eqs. (1)–(4) the above system (5)–(7) can be rewritten in the following block-matrix form: 1 1 2 2 3 3 0,P P C P Where
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of Engineering and Technology (IRJET) e-ISSN: 2395-0056 Volume: 06 Issue: 06 | June 2019 www.irjet.net p-ISSN: 2395-0072 © 2019, IRJET | Impact Factor value: 7.211 | ISO 9001:2008 Certified Journal | Page 2017 000 00010 00001 )1( )1(0 0)1( , 0 12 32 1 21 Cand In general, the above system (5)-(21) can be rewritten as : 1 1 1 1 0, 2,3,4,..., 1,k k k k k kP P C P k L (22) 1 1 0,L L L LP P k L (23) And the normalizing equation : ,1 0 0 L i L j ijP where (2 1) (2 1)( ) , 1,2,..., 1,k ij k ka k L with 2 1,2 1 2 ,2 1,k k k ka a and , 0i ja otherwise : , (2 1) (2 1)( ) , 2,3,... ,k i j k kb k L with : ,,...,3,2,,, 212,22112,3232,1222,12 Lkbbbb kkkkkkkk 22,...,3,2,1),1(, kib ii 42,...,3,2,1,,2 kib ii ,2,...,3,2,1,, 222,2112,12 kibb iiii ,),1(12,12 Lkb kk ,112,12 LLb and , 0i jb otherwise. Finally ,,...,4,3,)( )12()32(, LkcC kkjik with: ,12,21,1 cc ,,...,4,3,, 232,32122,32 Lkcc kkkk ,2,...,3,2,, 212,1212,2 kicc iiii and , 0,i jc otherwise.
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of Engineering and Technology (IRJET) e-ISSN: 2395-0056 Volume: 06 Issue: 06 | June 2019 www.irjet.net p-ISSN: 2395-0072 © 2019, IRJET | Impact Factor value: 7.211 | ISO 9001:2008 Certified Journal | Page 2018 3. Theoretical Results : In this section, we formulate the solution for the stationary distribution of the system (1)–(21). Clearly,usingtheEqs. (22) and (23) the stationary distribution of the given Markov process has a matrix solution and can be obtainedrecursivelyby solving the Eq. (22). More specifically, we have first to prove the following lemma which helps us in obtaining the stationary solution of the above system. Lemma 1.The inverses of Lkk ,...,3,2, are exist and their determinants can be computing by 1 11 2 1 1 ,,...,3,2,)det( k i ii kk k Lked ed s where 1,...,3,2,)1(),1( 1 1 1 ki d dd i i 1,...,3,2,)1(),1( 1 2 1 ki e ee i i )1( s for Lk and 1s for Lk Proof. We prove the lemma by calculating the inverse 1 k . The idea of method for evaluating the determinant of k is to reduce the given matrix to upper triangular matrix form by elementary row operations, we obtain : Det(Bk)= 1 2 1 1 21 3 3 22 12 21 11 000......0 00 .0 .0 .00 ...000 00..000 00...00 kk k ed s e e d e d e d Where ,1,...,3,2,)1(),1( 1 1 1 ki d dd i i ,1,...,3,2,)1(),1( 1 2 1 ki e ee i i )1( s for Lk and 1s for ,Lk
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of Engineering and Technology (IRJET) e-ISSN: 2395-0056 Volume: 06 Issue: 06 | June 2019 www.irjet.net p-ISSN: 2395-0072 © 2019, IRJET | Impact Factor value: 7.211 | ISO 9001:2008 Certified Journal | Page 2019 simply we have 1 11 2 1 1 .)det( k i ii kk k ed ed s Clearly.det 0)( k for any value of , then k is invertible for Lk ,...,3,2 . This completes the proof . Theorem 1. Let kP be the unique positive solution of the system (1)–(21), then LkpRRRP kkkk k k ,...,3,2,1,...)1( 1112211 1 (24) )22( )()2())(( 21 210,021211 0,1 P P (25) )22( )()2())(( 21 120,012212 1,0 P P (26) 1 2 1 2 1 2 1 2 1 2 1 2 1 2 0,0 1,1 1 2 1 2 1 2 1 2 1 2 1 2 3 ( 3 ) ( ) (2 2 ) ( 2 ) ( ) (2 2 ) P P With (27) 2 1 1 2 2 2 1 2 1 2 2 2 1 1 1 2 2 2 1 2 1 2 1 2 1 2 1 2 0,0 (1 ) (2 2 1) ( ) ( ) ( )(1 ) ( 2 1) ( 1) , (1 ) (2 2 1) ( ) (2 ) ( ) (2 ) ( )(1 ) ( 3 ) ( 3 1) ( 1) L L P 2 1 1 2 1 2 1 2 2 2 1 1 1 2 1 2 1 2 1 2 1 2 1 (3 2 ) (1 ) (1 ) (2 2 ) (1 ) (2 2)(1 ) , 1 (3 2 ) (1 ) (2 ) (1 ) (2 ) ( )( 3 1) (1 ) (2 2)(1 ) L L (28) 1 1 2 1 2 1 2 1 0 1 2 1 2 1 2 1 2 1 2 1 1 1 , , , , 1, , , [ ] , 1, 2,...,3, 2. L L k L k k k R A R B and R B C R A k L L Proof. From Lemma 1, we have proved that matrix k is invertible. Now using Eq. (23) we obtain 1 1, ,L L L LP R P k L (29) where 1 LLR Hence 221 ,...,, RRR LL can be solved recursively by the following formula:
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of Engineering and Technology (IRJET) e-ISSN: 2395-0056 Volume: 06 Issue: 06 | June 2019 www.irjet.net p-ISSN: 2395-0072 © 2019, IRJET | Impact Factor value: 7.211 | ISO 9001:2008 Certified Journal | Page 2020 2,3,...,2,1, 1 11 LLkRCR kkkkk Furthermore, from Eq. (22) we get 1,...,3,2,11 LkPRP kkkk (30) Hence 1321 ,...,,, PPPP LLL can be obtained recursively using (29) and (30) in terms of 1,1P , we get ,,...,3,2,1,...)1( 1112211 1 LkPRRRP kkkk k k (31) Now by simple algebraic manipulations, Eqs.(25)–(27) can be obtained by solving the system (1)–(3) in terms of 0,0P . Finally to complete our proof, let us assume that ,21 NNN be the numberofcustomersinthesystemandlet )( kNPg rk . Now, by induction on k we needtoshowthat Lkgg k k 2,...,4,3,)( 2 3 .Thefollowingcasesarise:forthesimplecase, using Eq. (4), we get .23 gg . Similarly, working on Eqs.(5) and (6),we obtain ),)(())(( 1,33,12,2211,12,11,221 PPPPPP 4 3 2 2 4 2 2 (1 ) ( ) g g g g g g In general for ,2rk , replacing n by rrrr ),...,32(),22(),12( in the Eqs. (8)–(14) respectively and add, we have )()( )()( 122,1 2 2 ,1221 2 2 12,21 1 ,12 1 12,, 1 1 ,2 1 1 12,21 rr i iir i iri r i iir r i irirr r i iir r i ri PPP PPPPPP 1,...,4,3,)1( 12212 Lrggg rrr Working likewise for the case ,12 rk by replacing n by 1),...,22(),12(,2 rrrr one can easily establish the following relation: 1,...,4,3,)1( 21222 Lrggg rrr In general, we can write Lkggg kkk 2,...,4,3,2,)1( 21 (32) Clearly, the case for Lk 2 can be easily obtained usingEqs.(19)and(20).Nowlettherelation ,)( 2 3 gg k k istruefor some km (say). On making use of Eq. (30), we get 2 3 1 1 1 2 2(1 ) (1 )( ) ( )( ) ( )m m m m m mg g g g g Hencethe relation ,)( 2 3 gg k k is true for all .2,...,4,3 Lk Using the normalizing equation ,1 2 0 L k kg we get 2 3 0 1 2 3 ( ) 1 L k k g g g (33) Using the facts 1,120,11,010,00 ,, PgPPgPg and using Eqs. (25)–(27)into the last one, the theorem gets fully established. 4. Particular case Furthermore, another interesting result can be obtained for finite waiting space queueing system by putting : 21, and , 2 we get : )12(2)(2,)(,)( 0,0 2 1,10,01,00,00,1 PPPPPP
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of Engineering and Technology (IRJET) e-ISSN: 2395-0056 Volume: 06 Issue: 06 | June 2019 www.irjet.net p-ISSN: 2395-0072 © 2019, IRJET | Impact Factor value: 7.211 | ISO 9001:2008 Certified Journal | Page 2021 2 2 2 2 2 2 2 2 0,0 (1 )(1 2 ) ( )(2 4 )(1 ) , 0, 1, (1 )(1 2( )) ( )(2( ) )(1 ) (1 2 ) (2 2)(1 )(3 4 ) , 0, 1, (3 2 ) (2 2)(1 )(2(1 ) ) 2( ) 2 ( 2 1), 1 2 , L L n n n L g n L P n L (34) 5. Mean of the queue We can define the mean of the total of costumers N is the system as: 2 , 0 0 0 ( ) ( ) L L L n i j n i j E N ng i j p 2 1 2 2 2 1 1,1 2 2 ( ) (1 ) (2 1)( ) ( ) (1 ) L L L P L E N (35) 6. Numerical results In this section we not try to provide a numerical analysis of the impact of Catastrophes on some of the important performance measures of this system. The computational results obtained by employing the above technique are discussed through tables and graphs. For differentvaluesofparameters 1, , , , and 2 ,weconductedseveral calculationson 0,0P withthesame parameters according to values, as shown in Tables. Table 1 Effect of on ( )E N , this table has been generated using : 1 20.4, 0.6, 0.4, 0.6, 25L Table -1: Effect of on ( )E N 0.25 0.50 0.75 ( )E N ( )E N ( )E N 0.1 0.671019 1.513310 3.424730 0.25 0.652010 1.314780 2.663630 0.50 0.643295 0.996633 1.844020 0.75 0.530110 0.754861 1.346920 0.95 0.444996 0.612408 1.080010
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of Engineering and Technology (IRJET) e-ISSN: 2395-0056 Volume: 06 Issue: 06 | June 2019 www.irjet.net p-ISSN: 2395-0072 © 2019, IRJET | Impact Factor value: 7.211 | ISO 9001:2008 Certified Journal | Page 2022 Chart -2: Effect of v on E(N) For the sake of clarity, We compared between our new model, and the Tarabia model [17] in terms of the λ effect on E(N)when = 0. As shown below: Chart -3 Comparison between our new model, and the Tarabia model [17] in terms of the λ effect on E(N)when 0 Table -2: Effect of 1 on Performance measures, this table has been generated using : 2 0.6, 0.1, 0.1, 0.4, 0.6, 25L Table -2: Effect of 1 ( )E N1,1P0,1P1,0P0,0P1 0.5833330.0306120.0449910.2606680.7653060.1 0.4160540.0195050.0492800.1440310.8278790.25 0.3428730.0138860.0507090.1014180.8549650.4 0.2919910.0102230.0513480.0731880.8736060.6 0.2690120.0085420.0514760.0606730.8819490.75 0.2483210.0070100.0514650.0494770.8894090.95 With increase in the value of 1 increases 0,0P and 0,1P while ( )E N , 1,0P and 1,1P decreases.
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of Engineering and Technology (IRJET) e-ISSN: 2395-0056 Volume: 06 Issue: 06 | June 2019 www.irjet.net p-ISSN: 2395-0072 © 2019, IRJET | Impact Factor value: 7.211 | ISO 9001:2008 Certified Journal | Page 2023 Table -3: Effect of 2 on Performance measures, this table has been generated using : 1 0.4, 0.1, 0.1, 0.4, 0.6, 25L Table -3: Effect of 2 ( )E N1,1P0,1P1,0P0,0P2 33.623900.4539220.4836380.0925840.6011350.1 14.247300.2381330.2234450.1138790.6468940.25 8.9988900.1767050.1482820.1302690.6754870.4 5.8997800.1365380.1016230.1464980.6987810.6 4.6170700.1178300.0815670.1556890.7098600.75 3.5255100.1001340.0640300.1652230.7197660.95 With increase in the value of 2 increases 0,0P and 1,0P while ( )E N , 0,1P and 1,1P decreases. 7. CONCLUSIONS In the previous analysis, a two queues in parallel with jockeying, and restricted capabilities and Catastrophes is considered to obtain the steady state probabilities of the system size are also studied. Finally, some important performance measures have been obtained from the steady state probabilities, Through the tables and diagrams we note the effect of Catastrophes on this system. REFERENCES [1] I.J.B.F. Adan, J. Wessels, W.H.M. Zijm, "Analysis of the symmetric shortest queueing problem", Stoch. Models, vol.6,1990, pp. 691–713. [2] I.J.B.F. Adan, J. Wessels, W.H.M. Zijm, "Analysis of the asymmetric shortest queueing problem with threshold jockeying", Stoch. Models, vol. 7, issue. 41, 1991, pp. 615–628. [3] I.J.B.F. Adan, G.J. Van Houtum, J. Van Der Wal, "Upper and lower bounds for the waiting time in the symmetric shortest queue system", Ann. Oper. Res, vol. 48, 1994, pp. 197–217. [4] J. R. Artalejo, G-Networks: "A versatile approach for work removal in queueing networks,EuropeanJournal ofOperations Research", Vol.126, issue. 2, 2000, pp. 233-249. [5] O.J. Boxma, G. Koole, Z. Liu, "Queueing-theoretic solution methods for models of parallel anddistributedsystems",Report No. BSR9425, CWI, Amsterdam, 1994. [6] J.W. Cohen, O.J. Boxma, "Boundary Value Problems in Queueing System Analysis", North-Holland, Amsterdam, 1983. [7] R. J. Boucherie and O. J. Boxma, "The workload in the M/G/1 queue with work removal", Probability in Engineering and Informational Sciences, Vol. 10, 1996, pp. 261-277. [8] B.W. Conolly, "The autostrada queueing problems", J. Appl. Prob, vol. 21, 1984, pp. 394–403. [9] Choa A, Zheng Y, "Transient analysis of immigration-birth-death process with total catastrophes",Prob.Eng.Inform.Sci., vol. 17, 2003, pp. 83-106. [10] A. N. Dudin and S. Nishimura, "A BMAP/SM/1 queueing system with Markovian arrival input of disasters",J.Appl.Prob., Vol.36, 1999, pp. 868-881. [11] L. Flatto, H.P. McKean, "Two queues in parallel", Comm. Pure. Appl. Math., vol. 30, 1977, pp. 255, 263. [12] G. Fayolle, R. Iasnogorodski, "Two couple processors: the reduction to a Riemann–Hilbert problem", Z. Wahrsch. Verw.Gebiete, vol. 47, 1979, pp. 325–351. [13] E. Gelenbe and G. Pujolle, "Introduction to Queueing Networks", (2nd edition), John Wiley and Sons, Chichester, 1998. [14] F.A. Haight, "Two queues in parallel", Biometrika, vol. 45, 1958, pp. 401–410.
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International Research Journal
of Engineering and Technology (IRJET) e-ISSN: 2395-0056 Volume: 06 Issue: 06 | June 2019 www.irjet.net p-ISSN: 2395-0072 © 2019, IRJET | Impact Factor value: 7.211 | ISO 9001:2008 Certified Journal | Page 2024 [15] G. Jain and K. Sigman, "A Pollaczek-Khinchine formula for M/G/1 queues with disasters",J.Appl.Prob.,Vol.33,1996,pp. 1191-1200. [16] J.F.C. Kingman, "Two similar queues in parallel", Ann. Math. Stat., vol. 32, 1961, pp. 1314–1323. [17] A.M.K. Tarabia, " Analysis of two queues in parallel with jockeying andrestrictedcapacities",J.Appl.Math.Model.,vol.32, 2008, pp. 802–810 . [18] A.M.K. Tarabia, "Transient analysis of two queues in parallel with jockeying", J.Appl. Math. Model., 2008,p. 802. [19] G.J. Van Houtum, I.J.B.F. Adan, J. Wessels, W.H.M. Zijm, "Performance analysis of parallel identical machines with a generalized shortest queue arrival mechanism", OR Spektrum, vol. 23 ,2001, pp. 411–427. [20] H. Yao, C. Knessl, "On the infinite server shortest queue problem: Non-symmetric case", Queueing Syst., vol.52 , issue.2, 2006, pp. 157–177. [21] Y. Zhao, W.K. Grassman, "The shortest queue model withjockeying",Nav.Res.Logist.,vol.37,issue.5,1990,pp.773–787.
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