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Queues with resequencing a survey and recent results
Theory, Methods d Applicatimu, Vol. 30, No. 8, pp. 5447-5456, 1997 Proc. 2nd World Congress ofNonlinear Analysts 0 1997 Elsevier Science Ltd Rinted in Great Britain. All rightsresewed 0362-546X/97 $17.00 + 0.00
NonlinearAnalysis,
PII: SO362-546X(97)00373-8
QUEUES WITH RESEQUENCING.
A SURVEY AND RECENT RESULTS
BOYAN N. DIMITROV Department Key Words:
Performance
of Science & Mathematics, GM1 Engineering & Management Flint, Michigan 485044898, USA evaluation,
Queuing theory, Modeling, Resequencing protocols, Matrix-analytic methods. 1. INTRODUCTION.
Institute
algorithms,
Optimal policies, ARQ
NOTATIONS
The resequencing problems arise in numerous practical situations, where multiple parallel service apply. They appear in communication system networks, distributed database systems, production flow-networks, in information networks and other multi-server queuing models. The reason: simultaneously served jobs use different physical servers when passing through a given node or link of the network. Different service rates, unequal path lengths, possible errors, varying processing times, etc. may cause variable delays into the node, and mixed completion times. However, if these jobs are completed in order different from the one they arrive at the node, it may destroy the right processing on the next nodes. In order to ensure the right processing through the network, the original order needs to be restored. Thus, an additional queue of already served jobs may be formed before departure from the node where late arrived and early served jobs are kept. It is named here resequencing bufier (RSB). The RSB is emptied when all earlier arrived jobs complete their services . The queue in front of the servers (if any) is named primary bufier (PB). The waiting time and the queue length in PB and RSB are denoted by WPB, WRSB, and WPB, URSB correspondingly. The input flow of jobs, if Poissonian, has intensity X with a possible subscript, whenever different classes of jobs are considered. The service rate on server i is p;. The numeration of servers usually is in the order which makes sure that ~1 2 ~2 2 . . . > pn is fulfilled. A principle scheme of a queuing system with resequencing, and a following next service station, is shown on Fig. 1
g ---+
input
p1 CL2
-
x
:
..:g;;
t
_.-
- - -.-
---PB
Fig.
-
. ..o(J
6 servers
1 Resequenced
tin
output
---+
n next service
MB
queuing system - a fragment,
Historically, just Krishnamoorthi( 1963) noticed heuristically that the disorder in a heterogeneous M/M/~/CO queue would be minimal, if the slower server is used only when the (PB) queue ezceeds mfl, where m is the integer part of the number ~1/~2. Disorder he names the relative number of jobs served (and therefore, departure)
before other jobs who have been arrived earlier. In this connection
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Krishnamoorthi proposed the slower server to pick up for service only jobs waiting at the position m + 1, if any. or otherwise to stay unused. About 11 years later Washburn( 1974) analyzed the total delay in a M/M/n/w queue under “no passing rule” 1i.e. when departures must respect the order of arrivals. Also “zero-service times” were allowed with probability p. The expected waiting time was studied. He mentioned that the mean waiting time penalty implied by no passing rule increases with the traffic level even when the number of servers is infinite. Researchers show low interests to resequencing problems in the 60’s and 70’s. Two related papers of Krishnamoorthi( 1963), and Washburn( 1974) bypassed almost invisible. Publicity of practical realizations of resequenced models came before most theoretical studies to trigger an intensive research. The IBM system network architecture (Gray and McNeill, 1979), and the French public network TRANSPAC (Danet et al. 1976) were designed with multi-parallel communication links named Transmission Groups (briefly, TG). First-in-first-out order is required by the TG protocol. A transmitted packet through a link of a TG cannot be scheduled for further transmission until all packets, started their transmission earlier within the same TG, arrive at the receiving node. Researchers took the model under thorough analysis and further development. More than 40 papers, Ph.D. thesis, technical reports and many conference talks traced a remarkable interest to this kind of problems. However, the mechanism of resequencing processes (not a simple one) requires high mathematical and combinatorial techniques to be analyzed. Here we review first and foremost the journal papers. Theoretical studies reveal numerous interesting stochastic properties. Also variety of other applications results is limited research group.
of queuing models to what we found
2. GENERAL
with resequencing have been found. Our discussion on obtained interesting. Separate attention we give to the results of GM1
RESULTS.
INFINITE
NUMBER
OF SERVERS
Most of the explicit analytic results are obtained for queuing resequencing models with multiple servers, Poisson arrivals, and exponential service times. However, the work of Baccelli et a1.(1984) gives a general and most abstract statement of the resequencing problems as follows: Consider a sequence of objects {&,}FZO who are entering a service system at (an increasing sequence of) instants {a,}~=e, correspondingly. Each &, is delayed within the system by some time D,, n = 0,1, . At the output of the system the objects appear at instants t, = a, + D, which are not necessarily in chronological order any more (i.e. it is possible that t, > tk for k > n ). The objects are then processed by a resequencing algorithm (RA); +,, will receive some service of duration S, and will depart at time d,, . However, this service can only be given in the same order as the original arrival instants; that is, & must be served after &-I and before +,,+I The RA & , (k < n) , h ave been released by the RA. (One delays an object 4, until all of the objects could recognize here the G/G/m system with resequencing and a next service incorporated, as it is shown on Fig. 1. Service time will be D,. The second service must accept the jobs in their original arriving order at the first node). The case of S,, = 0 for all n 2 1 is named “pure resequencing problem”. The non-zero service times are assumed to consider the “effect of an output service time on resequencing delays”. The questions they ask are the following: What all this would cost, and in particular, what is the total effective delay T, = d, - a,? Let A,, = a, - a,_1 be the inter-arrival times. The following Total Delay Equation for the Resequencing Algorithm is obtained:
Tn= ::,:; +[T,-l
-A,,
- Dn]
if D, > T,,-l - A,,, otherwise.
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In the case of {A,,} , {II,,}, and {Sn} independent sequences of i.i.d. random variables it is shown that under the condition E[S,,_l - A,] < 0 , the sequence of delays Yi = D1 , Y, = max [&, (Y&i + &+I - &I, n > 2 converges in distribution to a proper random variable Y. Its conditional limiting probability distribution U(z) = lim,+,, P{Yn < z 1 D, 5 z} is obtained in terms of Laplace-Stieltjes Transform (LST) in cases of Poisson arrivals and hyper-exponential delays. The Winner-Hopf factorization technique is applied. In particular, for N = 1, and S,, = 0 the results of Kamoun et al. (1981) concerning M/M/W resequenced queue are obtained. Areas of application, as Packet-Switching Networks, Distributed Databases, and Pure Mathematical Problems are outlined. Kamoun et al(1981) introduced the concept of Star Tusk (briefly, ST). This is each job which does not wait in RSB. A ST will eventually release the RSB from all jobs started later and served earlier. Then the imbeded ~RsB is the bunch of jobs departing with the ST. The authors found the inter-departure distribution between two consecutive STs as well as the waiting time distribution of jobs which are not STs. Harrus and Plateau(1982) extended these results for M/G/co queue with resequencing, i.e. for general distribution of the service time. Also they found a number of stationary characteristics: the mean number of jobs in the RSB, the p.d.f. of the service time of an ST, as well as the joint distribution of inter-departure time between two consecutive STs, the distribution of the number of jobs in RSB just before departure time of an ST. Finite second moment of the service time is required in order to ensure finite expectations of the listed characteristics. Explicit particular results are obtained for the case of exponential service time and the early known results for M/M/co queue with same kind of resequencing were confirmed. Graphical illustrations show the dependence of the mean stationary characteristics on the variance of the service time.
3. FINITE
NUMBER
OF SERVERS
The picture’ becomes considerably different and reach of opportunities in discussion of finite resequencing queues. It is possible to construct some specific policy of directing the waiting jobs to the available servers of lower rate, in order to reduce, say, the total waiting time, or the proportion of jobs that will experience resequencing, or other important performance measure. The policy which gives optimal performance measure is named optimal policy (OP). The importance of the remark of Krishnamoorthi(l963) was noticed in the work of Lin and Kumar(1984). They proved that in case of resequenced queue with two heterogeneous servers and common PB the OP which minimizes the expected total delay of a job in the system, is of the following threshold type: The faster server will serve the first job brn
the head of the PB whenever it becomes available for
service. The slower server should be utilized if and only if the queue length exceeds a readily computed
threshold value N*.
Lin and Kumar(1984) also calculated the mean total delay for any given threshold N in the resequencing queue. The optimal N* is then determined by the use of an iterative algorithm. Iliadis and Lien(1988) give an extensive discussion of the published in that time studies on resequencing, and consider again the M/M/2 queue. For any given N they introduce in consideration N + 1 fixed position policies (FPP) xi(N), i = l,..., N + 1, which work in the following way: If any job is in PB, the faster server must serve. If the waiting jobs in PB are more than N and the slower server becomes (or is) idle, it takes for service the job from position i. The authors derive explicit expressions for the steady state probability distribution of the triplet (vpg,zi,zs), where xi’s are indicators (busy or not) of the two servers (i = 1,2). Then for each of the two policies xi(N), and TN+1 (N) they derive explicit expressions for the resequencing delay, queue lengths, and the proportion of jobs that will experience resequencing delay. It is proven that xN+i always yields M/M/2
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smaller proportion decision
of resequenced
jobs than ~1, provided that it is true ~1 > 1-12. The polynomial
function fN(d
=
z N+2
+
(N - 1)~’ - 2Nr f N
(1)
determines the policy that yields minimum mean delay in steady state, for z = PI/(P~ + ~2). The OP does not depend neither on the buffer size B (it just needs to be at least N + l), nor on the load of the system, but on the relative service ratio z. Numerical examples show a comparison between performance measures of the two policies xl(N) and r~+r(N), and also how to use the decision rule above in determination of preference of one policy versus the other. Iliadis and Lien(1993) continue study M/M/2/B system under an arbitrary policy xi(N). Using an imbeded Markov chain approach for an extended state process, they obtain that steady state resequencing delay is determined by the survival function
p{wRSB
> t}
=
p(l
lpB)[z(ztS- l)eePzl
+ (1 - .z)$l j=O
- riPj)@$-eUPtt].
Here p = X/(pr +pz) is the traffic load, z is the relative service ratio, pg is the loss probability, and T and S are explicit expressions given in terms of the steady state probabilities of the process. The OP r~*+r(N*) yields smallest fraction of resequenced jobs, if ~1 > ~2. Moreover, the following holds: Let n* be the integer part of the number In(1 - z)/Znz. Then for any given threshold N < n* the optimal FPP is TN++(N), and it minimizes the mean resequencing delay; if N 1 n*, then the optimal FPP is x,. . The OP depends on the PB’s size B only when B < n’. The impact of policies rri(N) on the resequencing delay of a job was numerically illustrated. Ayoun and Rosberg( 1991) also study M/M/2 JB model with the question: are there other FPPs for which the policy 7r,,*(N) is still the optimal one? The authors conduct precise probability analysis to calculate performance characteristics under each of proposed new policies which differ by the rule of dispatching a job from the queue to the slower server when it is getting free. Their logistic analysis and exact calculations reveal a large scale of feasible rerouting policies and discounted times (costs) of resequencing, for which, if the PB size B > n*, then every threshold policy with a given N can be improved by rerouting jobs from position n* to slower server. This is an analytical work and no numerical examples were presented. model, where C classes of jobs form the Sasase et al(1992) analyze the most general Id/M/n/B input, and class-dependent thresholds policies are introduced. The fastest server 1 always serves the job from the head of the queue. Whenever server j becomes idle (j = 2, . . . . n), and servers l,Z,...,j - 1 are busy, server j starts processing a waiting job of class i taken from the head of the queue if and only if the total number of jobs in PB exceeds the given threshold number Nj_1, i; otherwise it stays idle. The order N1, i 5 Nz, i 5 . . . 5 N,,_l, i is assumed. The resequenced jobs wait only those of their own class that have been bypassed during its transmission. (Note, that in this model some class(es) of jobs may not be subject of resequencing.) The authors use steady state balance equations and the method of inclusion-exclusion to derive expressions for the mean queuing delay and the resequencing delay for each class and for the entire system. Also they mention a possibility two jobs of same class to start simultaneously being served on two different servers. This creates a new component in the study of resequencing delay, which has been completely analyzed. A numerical example illustrates the policy on the model g/M/2/B, where jobs of class 2 are not resequenced. Compare to conventional N-threshold policy, it is shown that the class-dependent (Nr,i, NQ) threshold policy is a way to reduce the resequencing delay within some specific classes of jobs. Varma(l99la) demonstrates a new approach in the study of MfMJPJB resequenced system, in order to obtain the probability distribution of URSB. By introducing a variable Z to show which of
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the two busy servers has started servicing earlier, he notices that the infinitesimal matrix of the extended process (vpn, zi, 22, v~s, Z), has a block diagonal structure. This yields a matrix-geometric solution for the steady state probabilities of the corresponding continuous time Markov chain. (The fact is suggested by Neuts(l981), and opened the horizon for more general studies.) Getting the marginal distributions of each component of the extended process, and effectively calculating various performance measures is then a matter of suitable choice of algorithmic manipulations. By making use of the matrix-geometric approach Varma suggests an effective way to determine the size of PB, or that of RSB, which warrants certain minimum percentage of losses, or to reduce over-waiting resequenced jobs. Varma(l99lb) analyses also the M/M/2/ co model under threshold type policy that minimizes the end-to-end delay. He introduces the costper unit time t functionc(t) = YPB(~)+z~(~)+~*(~)+uRSB(~) under discounting rate CY. A policy K* is optimal if it minimizes the cost per policy J(n) = sp eeat c(t)&! in steady state conditions. The author uses specific results of stochastic dynamic programming (based on discrete structure of components of c(t)), to derive that the OP does not depend on the number of jobs present in RSB. He found, that keeping faster server busy whenever there are jobs in PB is a step towards the OP. His conjecture is that the threshold FPP policy 1r(n*) is optimal for the new cost approach too. 4. ARQ PROTOCOLS
Specific resequencing problems arise in communication networks with finite number of servers of independent service rates at each node. Jobs are packets of data to be transmitted. The queue in front of the node will be denoted again by PB. Here the arriving jobs are waiting service at the node. A successful service must be confirmed by the receiver, otherwise it is repeated until confirmed. Additional queue similar to resequenced ones is needed for transmitted but not confirmed jobs. The jobs served by different servers may encounter different delays, and may be subject of errors, but must leave in the original arriving order. Therefore, a RSB for either the non-confirmed, or for early served jobs is required. This situation in communication networks is known as automatic-repeat-request (it ARQ) protocols. It generates new problems similar to the resequencing, but not exactly the same. A pair of nodes (a transmitter and a receiver with known buffer capacities) are considered. Data packets that arrive at the transmitter are assigned consecutive integers to serve as their identifiers (ID). Packets must go from the receiver, for further processing by the network, in their original order (i.e. in FIFO). Data packets are sent from a transmitter to a receiver over a noisy path. An it ARQ is established by adding an error-detecting code (e.g. cyclic redundancy check, CRC) to the information bits, and requiring that packets be retransmitted when errors are reported via a feedback channel. Three basic it ARQ protocols are used: Stop-and-Wait (SW), Go-back-N (GBN), and Selective Repeat (SR). Numerous papers treat several important performance measures associated with it ARQ protocols. A relatively complete review of these can be found in Rosberg and Shacham(l989), and in Yoshimoto et al.( 1993). In SW it ARQ protocols the transmitter sends a packet to the receiver, and waits for an acknowledgment from it. If an negative acknowledgment (NACK) is reported, the transmitter resends the same packet, and sends the next packet if a positive acknowledgment (ACK) is received. It is simple, but inefficient protocol because a lot of idle time is spent for waiting acknowledgment of each transmitted packet. RSB takes the early served jobs. In GBN it ARQ protocol packets are continuously transmitted. Each packet contains N bits. The transmitter does not wait for an acknowledgment but continuously transmits packets. When an NACK is received, the transmitter stops sending a new packet and resends the negatively acknowledged packet and its succeeding ones. At the receiver all packets received after the erroneous one are
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discarded regardless of whether they are error-free or not. At any given time the receiver’s buffer holds all packets that have been received correctly transmitted, but for which at least one packet with lower number has not yet been correctly received. The SR it ARQ protocols have several forms differing in the size of available buffers. The ideal SR it ARQ protocol is assumed to have an infinite receiving buffer (notation SR,). In SR it ARQ protocols packets are transmitted continuously and only those which are detected to be in error are retransmitted. However, some SR protocols may have assigned a r~indow size N. Given the transmission of a packet i, the transmitter must await until after it finishes the transmission of up to N - 1 subsequent packets (new or retransmitted), 1 5 N 5 00. If an ACK arrives at the transmitter during this waiting period, the corresponding packet is released from the transmitter’s buffer and the Nth subsequent transmission is a new packet. If a NACK or no acknowledgment is obtained (timeout), the Nth subsequent transmission is again the packet i. RSB is needed for non-acknowledged jobs at the transmitter, and for reordering at the receiver. Yum and Ngai(1986) consider a close to the M/M/n/cc network model. A packet is directed to the fastest idle link. The resequencing delay equals to the waiting time at RSB of a packet until all early started packets be correctly transmitted. This is the difference between its own service time and the maximum of the all other services currently made. The authors introduce indicator r.v.s X3 = 1 if the jth server is busy, and Xi = 0 otherwise. With 2 = (Xi, . . . . X,) they use a Markov birth-and-death technique to find the joint distribution of the pair (2, vpg), and then to derive the distribution of WR~B and its expected characteristics. The Little’s identity WVRSB)
=EWRSB)W~
-PB)
is then used to determine the average queue size in the RSB for channel-level resequencing (CHREFIFO) case, where one RSB is assigned to all jobs. A virtual circuit (VC-REFIFO) is the alternative. It considers the input X = Xi + . . . + X, of jobs preliminary partitioned into T types. Each type i will wait at RSBi only for completing the order of the same type jobs. The above technique is used to establish the waiting times WRSB(i) and queue lengths ~RSB(~). An overall performance measure of the system given by the sum WPB + S,JCT + WRSB, where SACT is the actual service time of a job, is then numerically studied. It is shown that for exponential service times the resequencing at the end of the network gives larger expected overall delay than the total delay when resequencing is done at each node. Also an uniformly partitioned VC-REFIFO gives proportionally smaller overall delay than CH-REFIFO. Rosberg and Shacham( 1989) analyze an SR it ARQ protocol of random window size N in discrete time. The time unit equals to the transaction of a window group. The highest, and the lowest IDS of packets transmitted within a window uniquely specify (in a combinatorics fashion) the waiting packets at the RSB. Binomial probabilities (with probability p of an erroneous transmission) determine the number of erroneously received packets. At the end of a window some new packets may join the RSB as well as some of the waiting there can be released. There is an infinite input rate. Transition probabilities between the RSB states are found, and the steady state of corresponding Markov chain is proven. The probability generating functions of URSB and of W-B are obtained. In particular &
is the expected length of the RSB queue. The expected
total delay, and the waiting time are measured in terms of the window time and in multiple to l/N part of it (the time for transmission of one packet). The Little’s identity here (1 - P)E(WRSB)=E(V~B) is used again to study the waiting time. Approximation of the exact distributions for p -+ 0, or p -+ 1 are derived. Graphical illustrations show the expected RSB queue behavior with variation of window size N.
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Shacham and Towsley(1991) consider the SR it ARQ protocol in multi-cast environment: a transmitter sends packets to M receivers. Each NACK after the timeout causes retransmission of corresponding packet to all the receivers. Each receiver has its own RSB. The pair transmitter-receiver is treated under the effect of multi-cast transmission. Backward acknowledgments are non-reliable; with probability qi transmitter gets the ACK from 8” receiver. The case of M = 1, q = 1 is the above reported by Rosberg and Shacham. Here the authors derive the probability distribution of the time 2 until a packet is successfully transmitted to all of the M receivers. The distribution of WRss(i) is in terms of a system of recursive equations, which allow an effective moments calculation. The Little’s formula E(vRsB(~J)=& E(WRsB(i)) gives the expected queue size at each receiver. Reservation buffer is named the number of packets that will potentially join the RSB(i) after a window is transmitted. The total resequenced queue is the algebraic sum of the actual and reservation buffers. Qualitative result is that for fixed value of pi’s the expected RSB occupancy E(L/~B(~)) decreases, if considered as a function of M or of pi (as it does E(Z)); and increases, if considered as function of qi or N. Lee and Hu(1992) discuss the SR it ARQ protocol with M receivers and repeated n copies, retransmitted with each NACK packet. Also individual error probabilities pi are presented together with an error probability pc for common wrong transmission of a packet to all receivers. Packets arrive for transmission according to a renewal process with geometric distribution. The WPB is the time spent by a packet at the transmitter’s buffer from its first start until it is confirmed by all receivers. Performance measure is the sum WPB + WRsB(i). Each component is explicitly studied. A discrete time Markov chain technique is applied to derive relationships between suitably introduced random variables (to describe the process). Equations for the equilibrium probability distributions are obtained. The optimal number of copies which minimizes the expected total delay of a packet in the system is subject of a control. It is shown that by sending the optimal number of copies of a packet, the mean overall delay can be substantially reduced. The finite size of PB has a negligible effect on the total delay if this size is grater than the window size. Yoshimoto et a1.(1993) search a large scale of ARC protocols in review and in action. There is a geometric input flow of messages. Each message may have a random length. Then it is partitioned in single packets for transmission according to GBN, or SR it ARQ protocol, until the entire message is transmitted. The messages itself are treated as under SW it ARQ protocol. The primary queue of packets in PB is then similar to McX)/G/l queuing system in discrete time. It is used to derive what time is needed to transmit one message in whole. For instance, GBN it ARQ protocol has generating function of the service time S*(z)=B* (I$&) zN, where B*(z) is the generating function of the number of packets generated by one message, and N is the window size. Corresponding expression is obtained also for SR it ARQ protocol. It is in terms of transmission times of all the involved packets and their waiting times in the RSB. The use of probability generating functions styles the best tradition of queuing theory. Numerical results show that in some sense SR is better than GBN scheme. Also dependence of the throughput on the message length and on the window size is illustrated.
5. THE MAP/MI2
MODELS
possibly traced a new way in resequencing studies. It was seen that the Matrix-analytic and study of performance measures, when explicit equations for the state probabilities are at hands. Almost all of the previous works with numerical component dealt with Poisson input and exponential service times, or were discrete time models. One model, in which a Poisson process superimposed by a deterministic bursty process as system input and normally distributed service times, was recently studded in Plotkin and Varaiya(l993). Varma(l99la)
methods can be helpful in effective computations
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However, in communication engineering packets may arrive in a very bursty and stochastic manner; the inter-arrival times may be dependent. This motivates the use of more general input processes which can be modeled by Markov arrival process (MAP). At GM1 an intensive study of these models for resequenced systems is going. The research group consists of Dr. S. Chakravarthy, Dr. S. Chukova, Dr. B. Dimitrov, and works in conjunction with master and undergraduate students, as S. Lin, T. Dimitrova, S. Sassy and others. Bin and Chakravarthy(l996a) conduct a complete steady state analysis of MAP/M/2 system with non-renewal input, two heterogeneous servers and threshold-type scheduling. Bin and Chakravarthy (1996b) continue their study to find the optimal allocation of messages for the same system in the sense of search of optimal policy, as it was stated previously by Iliadis and Lien(1988, 1993). It is assumed ~1 > ~2, one threshold number N, and policies nl(N), or r~+r(N) are used. By making use of the method of Matrix-Geometric solution, they involve a large scale state space X (~pg,Xl, X2,77), where Xi indicates the state of ith server; n indicates the phase of the MAP, (1 <_ n 5 m). In this way the stationary distribution at arrival, and at arbitrary time is obtained as solution of a matrix equation. It allows to calculate effectively the c.d.f. of residual delay in RSB, the probability of a job to be delayed, to establish all necessary and sufficient conditions for minimizing the resequenced proportion, or expected system delay. A numerically stable algorithm for computing performance measures is developed. It is proven that policy KN+~(N) always yields a smaller fraction of resequenced jobs than rr(N). Also, there exists a polynomial decision &n&on in z = pl/(pr +/Jz) which determines the policy that yields the minimum delay. Its form is given by equation (1)) found by Iliadis and Lien(1988) for the Markovian models. Chakravarthy et a1.(1996) gave another point of view in the analysis of queuing models with resequencing, namely by introducing randomized policy in directing the heading jobs to the idle model with ~1 > ~2 Infinite capacity of RSB. The proservers. They consider the MAP/M/2/B posed randomized (probability) control policy n(p) means that when the new arriving job finds both servers idle, with probability p E [0, l] it is directed to server 1; otherwise it gets server 2. The finite capacity of RSB may cause blocking of servers when RSB (if finite) is full. In addition another randomized policy r@,pB) appears: When the blocked RSB is released, the waiting job from the head of the queue with probability pn E [0, l] is directed to server 1; otherwise it gets server 2. Matrix-Geometric approach is applicable again. It gives the following results: The random vector (VRSB, zr, z2, ~pg, E, n), where E indicates which server serves the job entered earlier (and the order of this description is also important), suits to apply this approach. The infinitesimal operator governing the process is obtained, and an explicit steady state analysis of the process is performed. Algorithmic procedures for calculation of the steady state probabilities of the vector (VRSB, ~1, x2, VpB, &,n) are worked out. Stationary waiting time distribution of an admitted job is then obtained. The system performance measures E(vp~), E(VRSB), PB, the mean sojourn time in the system are effectively found. Numerical examples with correlated and uncorrelated inter-arrival times show some curious effects: There are values of admission probabilities p, and pB which are neither 0, nor 1, which produce optimal performance of that particular model.
6. CONCLUSIONS
The resequencing problems in Queuing Theory have its specific scope of research. Mostly Markovian models have been studded. There is a number of practical system that work under resequencing mechanisms. The it ARQ models offer interesting problems in resequencing which differ from the models considered in “classic setups” of queuing theory. The Matrix-Geometric approach, and MAP models offer new technical, theoretical and practical way in the study of resequencing.
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The search of optimal resequencing policies is an additional and interesting field in the study of resequenced queues. Threshold selection policies, preliminary partitioning for resequencing, group rerouting, windowing, multiple repeated services, non-traditional randomized control policies in resequenced system may reduce considerably the delay, and may improve the total performance of the system. Queues with resequencing will keep attention of researchers active and challenging.
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IBM Syst. J., 18, 263-279 (1979) issue.
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in computer networks. Ph.D. thesis, Dept.
12. ILIADIS I. & LIEN L. Y.-C. Resequencing control for a queuing l+ans. on Communications, 41, No 6, 951-961 (1993).
system
on Software Engineering,
Electr.
with two heterogeneous
13. ILIADIS I. & LIEN L. Y.-C. Resequencing delay for a queuing system with two heterogeneous threshold-type scheduling. IEEE ‘Ikans. on Communications, 36, No 6, 692-702 (1988).
Eng., Columbia
servers.
IEEE
servers under
a
14. KAMOUN F., KLEINROCK L. & MUNTZ R. Queuing analysis of reordering issue in a distributed database concurrency control mechanism. In Proc. Z’d Int. Conf. on Distributed Computing Systems, IEEE Press, 13-23 (1981). 15. KRISHNAMOORTHI (1963).
B. On Poisson
16. LEE T.-H. & HU J.-K. Some selective Pubkcation, ICC’92, 1242-1246 (1992).
queue with two heterogeneous
repeat
it ARQ schemes
servers.
Operations Research, 11, 321-330
for point-to-multipoint
communication.
IEEE
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17. LIN W. & KUMAR P. R. Optimal control of a queuing Automatic Control, 29, No. 8, 696-703 (1984).
of Nonlinear Analysts system with two heterogeneous
servers.
IEEE Tkans. on
18. NEUTS M. F. Matrix=Geometric SoUron in Stochastic Models: An Algorithmw Approach. The Johns Hopkins University Press, Baltimore, MD (1981). 19. NEUTS
M. F. Structured Stochasttc Matrices of M/G/l type and Thew Applwatlons, Marcel Dekker, NY (1989).
20. NEUTS M. F. Modeling based on the Markovian No. 12, 1255-1265 (1981).
Arrival
21. PLOTKIN N. T. & VARAIYA P. P. Performance analysis cation. IEEE ‘Bans. on Communications (1993).
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ATM connections
22. ROSBERG 2. & SHACHAM N. Resequencing delay and buffer occupancy IEEE tins. on Information Theory, 35, No 1, 166-173 (1989).
under
for Gigabit
Speed appli-
the selective-repeat
it ARQ.
23. SASASE I., NISHIO Y. & NAKAMURA H. The effect of message-class dependent threshold-type the delay for the M/M/N queue. IEEE IEEE Publication, ICC’92, 506-510 (1992).
scheduling
24. SHACHAM N. & TOWSLEY D. Resequencing delay and buffer occupancy in selective multiple receivers. IEEE Zkans. on Communications, 39, No 6, 928-936 (1991).
it ARQ with
25. VARMA S. Some problems in queuing systems with resequencing. MD (1987). 26. VARMA S. Optimal allocation Control, 36, No 11, 1288-1293 27. VARMA (199lb).
S. A matrix
of customers (1991a).
geometric
solution
MS. thesis,
Univ.
in a two server queue with resequencing.
to a resequencing
problem.
Perfoknance
repeat
Maryland,
on
College Park,
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Evakation,
12,
103-114
28. WASHBURN
A. A multi-server
29. YOSHIMOTO for go-back-N (1993).
M., TAKINE T., TAKAHASHI Y. & HASEGAWA T. Waiting time and queue length distributions IEEE ‘Zkans. on Communications, 41, No 11, 1687-1693 and selective-repeat it ARQ protocols.
queue with no passing.
30. YUM T.-S. P. & NGAI T.-Y. Resequencing nications, 34, No. 2, 143-149 (1986).
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IEEE ZYans. on Commu-
NonlinearAnalysis,
PII: SO362-546X(97)00373-8
QUEUES WITH RESEQUENCING.
A SURVEY AND RECENT RESULTS
BOYAN N. DIMITROV Department Key Words:
Performance
of Science & Mathematics, GM1 Engineering & Management Flint, Michigan 485044898, USA evaluation,
Queuing theory, Modeling, Resequencing protocols, Matrix-analytic methods. 1. INTRODUCTION.
Institute
algorithms,
Optimal policies, ARQ
NOTATIONS
The resequencing problems arise in numerous practical situations, where multiple parallel service apply. They appear in communication system networks, distributed database systems, production flow-networks, in information networks and other multi-server queuing models. The reason: simultaneously served jobs use different physical servers when passing through a given node or link of the network. Different service rates, unequal path lengths, possible errors, varying processing times, etc. may cause variable delays into the node, and mixed completion times. However, if these jobs are completed in order different from the one they arrive at the node, it may destroy the right processing on the next nodes. In order to ensure the right processing through the network, the original order needs to be restored. Thus, an additional queue of already served jobs may be formed before departure from the node where late arrived and early served jobs are kept. It is named here resequencing bufier (RSB). The RSB is emptied when all earlier arrived jobs complete their services . The queue in front of the servers (if any) is named primary bufier (PB). The waiting time and the queue length in PB and RSB are denoted by WPB, WRSB, and WPB, URSB correspondingly. The input flow of jobs, if Poissonian, has intensity X with a possible subscript, whenever different classes of jobs are considered. The service rate on server i is p;. The numeration of servers usually is in the order which makes sure that ~1 2 ~2 2 . . . > pn is fulfilled. A principle scheme of a queuing system with resequencing, and a following next service station, is shown on Fig. 1
g ---+
input
p1 CL2
-
x
:
..:g;;
t
_.-
- - -.-
---PB
Fig.
-
. ..o(J
6 servers
1 Resequenced
tin
output
---+
n next service
MB
queuing system - a fragment,
Historically, just Krishnamoorthi( 1963) noticed heuristically that the disorder in a heterogeneous M/M/~/CO queue would be minimal, if the slower server is used only when the (PB) queue ezceeds mfl, where m is the integer part of the number ~1/~2. Disorder he names the relative number of jobs served (and therefore, departure)
before other jobs who have been arrived earlier. In this connection
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Krishnamoorthi proposed the slower server to pick up for service only jobs waiting at the position m + 1, if any. or otherwise to stay unused. About 11 years later Washburn( 1974) analyzed the total delay in a M/M/n/w queue under “no passing rule” 1i.e. when departures must respect the order of arrivals. Also “zero-service times” were allowed with probability p. The expected waiting time was studied. He mentioned that the mean waiting time penalty implied by no passing rule increases with the traffic level even when the number of servers is infinite. Researchers show low interests to resequencing problems in the 60’s and 70’s. Two related papers of Krishnamoorthi( 1963), and Washburn( 1974) bypassed almost invisible. Publicity of practical realizations of resequenced models came before most theoretical studies to trigger an intensive research. The IBM system network architecture (Gray and McNeill, 1979), and the French public network TRANSPAC (Danet et al. 1976) were designed with multi-parallel communication links named Transmission Groups (briefly, TG). First-in-first-out order is required by the TG protocol. A transmitted packet through a link of a TG cannot be scheduled for further transmission until all packets, started their transmission earlier within the same TG, arrive at the receiving node. Researchers took the model under thorough analysis and further development. More than 40 papers, Ph.D. thesis, technical reports and many conference talks traced a remarkable interest to this kind of problems. However, the mechanism of resequencing processes (not a simple one) requires high mathematical and combinatorial techniques to be analyzed. Here we review first and foremost the journal papers. Theoretical studies reveal numerous interesting stochastic properties. Also variety of other applications results is limited research group.
of queuing models to what we found
2. GENERAL
with resequencing have been found. Our discussion on obtained interesting. Separate attention we give to the results of GM1
RESULTS.
INFINITE
NUMBER
OF SERVERS
Most of the explicit analytic results are obtained for queuing resequencing models with multiple servers, Poisson arrivals, and exponential service times. However, the work of Baccelli et a1.(1984) gives a general and most abstract statement of the resequencing problems as follows: Consider a sequence of objects {&,}FZO who are entering a service system at (an increasing sequence of) instants {a,}~=e, correspondingly. Each &, is delayed within the system by some time D,, n = 0,1, . At the output of the system the objects appear at instants t, = a, + D, which are not necessarily in chronological order any more (i.e. it is possible that t, > tk for k > n ). The objects are then processed by a resequencing algorithm (RA); +,, will receive some service of duration S, and will depart at time d,, . However, this service can only be given in the same order as the original arrival instants; that is, & must be served after &-I and before +,,+I The RA & , (k < n) , h ave been released by the RA. (One delays an object 4, until all of the objects could recognize here the G/G/m system with resequencing and a next service incorporated, as it is shown on Fig. 1. Service time will be D,. The second service must accept the jobs in their original arriving order at the first node). The case of S,, = 0 for all n 2 1 is named “pure resequencing problem”. The non-zero service times are assumed to consider the “effect of an output service time on resequencing delays”. The questions they ask are the following: What all this would cost, and in particular, what is the total effective delay T, = d, - a,? Let A,, = a, - a,_1 be the inter-arrival times. The following Total Delay Equation for the Resequencing Algorithm is obtained:
Tn= ::,:; +[T,-l
-A,,
- Dn]
if D, > T,,-l - A,,, otherwise.
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In the case of {A,,} , {II,,}, and {Sn} independent sequences of i.i.d. random variables it is shown that under the condition E[S,,_l - A,] < 0 , the sequence of delays Yi = D1 , Y, = max [&, (Y&i + &+I - &I, n > 2 converges in distribution to a proper random variable Y. Its conditional limiting probability distribution U(z) = lim,+,, P{Yn < z 1 D, 5 z} is obtained in terms of Laplace-Stieltjes Transform (LST) in cases of Poisson arrivals and hyper-exponential delays. The Winner-Hopf factorization technique is applied. In particular, for N = 1, and S,, = 0 the results of Kamoun et al. (1981) concerning M/M/W resequenced queue are obtained. Areas of application, as Packet-Switching Networks, Distributed Databases, and Pure Mathematical Problems are outlined. Kamoun et al(1981) introduced the concept of Star Tusk (briefly, ST). This is each job which does not wait in RSB. A ST will eventually release the RSB from all jobs started later and served earlier. Then the imbeded ~RsB is the bunch of jobs departing with the ST. The authors found the inter-departure distribution between two consecutive STs as well as the waiting time distribution of jobs which are not STs. Harrus and Plateau(1982) extended these results for M/G/co queue with resequencing, i.e. for general distribution of the service time. Also they found a number of stationary characteristics: the mean number of jobs in the RSB, the p.d.f. of the service time of an ST, as well as the joint distribution of inter-departure time between two consecutive STs, the distribution of the number of jobs in RSB just before departure time of an ST. Finite second moment of the service time is required in order to ensure finite expectations of the listed characteristics. Explicit particular results are obtained for the case of exponential service time and the early known results for M/M/co queue with same kind of resequencing were confirmed. Graphical illustrations show the dependence of the mean stationary characteristics on the variance of the service time.
3. FINITE
NUMBER
OF SERVERS
The picture’ becomes considerably different and reach of opportunities in discussion of finite resequencing queues. It is possible to construct some specific policy of directing the waiting jobs to the available servers of lower rate, in order to reduce, say, the total waiting time, or the proportion of jobs that will experience resequencing, or other important performance measure. The policy which gives optimal performance measure is named optimal policy (OP). The importance of the remark of Krishnamoorthi(l963) was noticed in the work of Lin and Kumar(1984). They proved that in case of resequenced queue with two heterogeneous servers and common PB the OP which minimizes the expected total delay of a job in the system, is of the following threshold type: The faster server will serve the first job brn
the head of the PB whenever it becomes available for
service. The slower server should be utilized if and only if the queue length exceeds a readily computed
threshold value N*.
Lin and Kumar(1984) also calculated the mean total delay for any given threshold N in the resequencing queue. The optimal N* is then determined by the use of an iterative algorithm. Iliadis and Lien(1988) give an extensive discussion of the published in that time studies on resequencing, and consider again the M/M/2 queue. For any given N they introduce in consideration N + 1 fixed position policies (FPP) xi(N), i = l,..., N + 1, which work in the following way: If any job is in PB, the faster server must serve. If the waiting jobs in PB are more than N and the slower server becomes (or is) idle, it takes for service the job from position i. The authors derive explicit expressions for the steady state probability distribution of the triplet (vpg,zi,zs), where xi’s are indicators (busy or not) of the two servers (i = 1,2). Then for each of the two policies xi(N), and TN+1 (N) they derive explicit expressions for the resequencing delay, queue lengths, and the proportion of jobs that will experience resequencing delay. It is proven that xN+i always yields M/M/2
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smaller proportion decision
of resequenced
jobs than ~1, provided that it is true ~1 > 1-12. The polynomial
function fN(d
=
z N+2
+
(N - 1)~’ - 2Nr f N
(1)
determines the policy that yields minimum mean delay in steady state, for z = PI/(P~ + ~2). The OP does not depend neither on the buffer size B (it just needs to be at least N + l), nor on the load of the system, but on the relative service ratio z. Numerical examples show a comparison between performance measures of the two policies xl(N) and r~+r(N), and also how to use the decision rule above in determination of preference of one policy versus the other. Iliadis and Lien(1993) continue study M/M/2/B system under an arbitrary policy xi(N). Using an imbeded Markov chain approach for an extended state process, they obtain that steady state resequencing delay is determined by the survival function
p{wRSB
> t}
=
p(l
lpB)[z(ztS- l)eePzl
+ (1 - .z)$l j=O
- riPj)@$-eUPtt].
Here p = X/(pr +pz) is the traffic load, z is the relative service ratio, pg is the loss probability, and T and S are explicit expressions given in terms of the steady state probabilities of the process. The OP r~*+r(N*) yields smallest fraction of resequenced jobs, if ~1 > ~2. Moreover, the following holds: Let n* be the integer part of the number In(1 - z)/Znz. Then for any given threshold N < n* the optimal FPP is TN++(N), and it minimizes the mean resequencing delay; if N 1 n*, then the optimal FPP is x,. . The OP depends on the PB’s size B only when B < n’. The impact of policies rri(N) on the resequencing delay of a job was numerically illustrated. Ayoun and Rosberg( 1991) also study M/M/2 JB model with the question: are there other FPPs for which the policy 7r,,*(N) is still the optimal one? The authors conduct precise probability analysis to calculate performance characteristics under each of proposed new policies which differ by the rule of dispatching a job from the queue to the slower server when it is getting free. Their logistic analysis and exact calculations reveal a large scale of feasible rerouting policies and discounted times (costs) of resequencing, for which, if the PB size B > n*, then every threshold policy with a given N can be improved by rerouting jobs from position n* to slower server. This is an analytical work and no numerical examples were presented. model, where C classes of jobs form the Sasase et al(1992) analyze the most general Id/M/n/B input, and class-dependent thresholds policies are introduced. The fastest server 1 always serves the job from the head of the queue. Whenever server j becomes idle (j = 2, . . . . n), and servers l,Z,...,j - 1 are busy, server j starts processing a waiting job of class i taken from the head of the queue if and only if the total number of jobs in PB exceeds the given threshold number Nj_1, i; otherwise it stays idle. The order N1, i 5 Nz, i 5 . . . 5 N,,_l, i is assumed. The resequenced jobs wait only those of their own class that have been bypassed during its transmission. (Note, that in this model some class(es) of jobs may not be subject of resequencing.) The authors use steady state balance equations and the method of inclusion-exclusion to derive expressions for the mean queuing delay and the resequencing delay for each class and for the entire system. Also they mention a possibility two jobs of same class to start simultaneously being served on two different servers. This creates a new component in the study of resequencing delay, which has been completely analyzed. A numerical example illustrates the policy on the model g/M/2/B, where jobs of class 2 are not resequenced. Compare to conventional N-threshold policy, it is shown that the class-dependent (Nr,i, NQ) threshold policy is a way to reduce the resequencing delay within some specific classes of jobs. Varma(l99la) demonstrates a new approach in the study of MfMJPJB resequenced system, in order to obtain the probability distribution of URSB. By introducing a variable Z to show which of
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the two busy servers has started servicing earlier, he notices that the infinitesimal matrix of the extended process (vpn, zi, 22, v~s, Z), has a block diagonal structure. This yields a matrix-geometric solution for the steady state probabilities of the corresponding continuous time Markov chain. (The fact is suggested by Neuts(l981), and opened the horizon for more general studies.) Getting the marginal distributions of each component of the extended process, and effectively calculating various performance measures is then a matter of suitable choice of algorithmic manipulations. By making use of the matrix-geometric approach Varma suggests an effective way to determine the size of PB, or that of RSB, which warrants certain minimum percentage of losses, or to reduce over-waiting resequenced jobs. Varma(l99lb) analyses also the M/M/2/ co model under threshold type policy that minimizes the end-to-end delay. He introduces the costper unit time t functionc(t) = YPB(~)+z~(~)+~*(~)+uRSB(~) under discounting rate CY. A policy K* is optimal if it minimizes the cost per policy J(n) = sp eeat c(t)&! in steady state conditions. The author uses specific results of stochastic dynamic programming (based on discrete structure of components of c(t)), to derive that the OP does not depend on the number of jobs present in RSB. He found, that keeping faster server busy whenever there are jobs in PB is a step towards the OP. His conjecture is that the threshold FPP policy 1r(n*) is optimal for the new cost approach too. 4. ARQ PROTOCOLS
Specific resequencing problems arise in communication networks with finite number of servers of independent service rates at each node. Jobs are packets of data to be transmitted. The queue in front of the node will be denoted again by PB. Here the arriving jobs are waiting service at the node. A successful service must be confirmed by the receiver, otherwise it is repeated until confirmed. Additional queue similar to resequenced ones is needed for transmitted but not confirmed jobs. The jobs served by different servers may encounter different delays, and may be subject of errors, but must leave in the original arriving order. Therefore, a RSB for either the non-confirmed, or for early served jobs is required. This situation in communication networks is known as automatic-repeat-request (it ARQ) protocols. It generates new problems similar to the resequencing, but not exactly the same. A pair of nodes (a transmitter and a receiver with known buffer capacities) are considered. Data packets that arrive at the transmitter are assigned consecutive integers to serve as their identifiers (ID). Packets must go from the receiver, for further processing by the network, in their original order (i.e. in FIFO). Data packets are sent from a transmitter to a receiver over a noisy path. An it ARQ is established by adding an error-detecting code (e.g. cyclic redundancy check, CRC) to the information bits, and requiring that packets be retransmitted when errors are reported via a feedback channel. Three basic it ARQ protocols are used: Stop-and-Wait (SW), Go-back-N (GBN), and Selective Repeat (SR). Numerous papers treat several important performance measures associated with it ARQ protocols. A relatively complete review of these can be found in Rosberg and Shacham(l989), and in Yoshimoto et al.( 1993). In SW it ARQ protocols the transmitter sends a packet to the receiver, and waits for an acknowledgment from it. If an negative acknowledgment (NACK) is reported, the transmitter resends the same packet, and sends the next packet if a positive acknowledgment (ACK) is received. It is simple, but inefficient protocol because a lot of idle time is spent for waiting acknowledgment of each transmitted packet. RSB takes the early served jobs. In GBN it ARQ protocol packets are continuously transmitted. Each packet contains N bits. The transmitter does not wait for an acknowledgment but continuously transmits packets. When an NACK is received, the transmitter stops sending a new packet and resends the negatively acknowledged packet and its succeeding ones. At the receiver all packets received after the erroneous one are
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discarded regardless of whether they are error-free or not. At any given time the receiver’s buffer holds all packets that have been received correctly transmitted, but for which at least one packet with lower number has not yet been correctly received. The SR it ARQ protocols have several forms differing in the size of available buffers. The ideal SR it ARQ protocol is assumed to have an infinite receiving buffer (notation SR,). In SR it ARQ protocols packets are transmitted continuously and only those which are detected to be in error are retransmitted. However, some SR protocols may have assigned a r~indow size N. Given the transmission of a packet i, the transmitter must await until after it finishes the transmission of up to N - 1 subsequent packets (new or retransmitted), 1 5 N 5 00. If an ACK arrives at the transmitter during this waiting period, the corresponding packet is released from the transmitter’s buffer and the Nth subsequent transmission is a new packet. If a NACK or no acknowledgment is obtained (timeout), the Nth subsequent transmission is again the packet i. RSB is needed for non-acknowledged jobs at the transmitter, and for reordering at the receiver. Yum and Ngai(1986) consider a close to the M/M/n/cc network model. A packet is directed to the fastest idle link. The resequencing delay equals to the waiting time at RSB of a packet until all early started packets be correctly transmitted. This is the difference between its own service time and the maximum of the all other services currently made. The authors introduce indicator r.v.s X3 = 1 if the jth server is busy, and Xi = 0 otherwise. With 2 = (Xi, . . . . X,) they use a Markov birth-and-death technique to find the joint distribution of the pair (2, vpg), and then to derive the distribution of WR~B and its expected characteristics. The Little’s identity WVRSB)
=EWRSB)W~
-PB)
is then used to determine the average queue size in the RSB for channel-level resequencing (CHREFIFO) case, where one RSB is assigned to all jobs. A virtual circuit (VC-REFIFO) is the alternative. It considers the input X = Xi + . . . + X, of jobs preliminary partitioned into T types. Each type i will wait at RSBi only for completing the order of the same type jobs. The above technique is used to establish the waiting times WRSB(i) and queue lengths ~RSB(~). An overall performance measure of the system given by the sum WPB + S,JCT + WRSB, where SACT is the actual service time of a job, is then numerically studied. It is shown that for exponential service times the resequencing at the end of the network gives larger expected overall delay than the total delay when resequencing is done at each node. Also an uniformly partitioned VC-REFIFO gives proportionally smaller overall delay than CH-REFIFO. Rosberg and Shacham( 1989) analyze an SR it ARQ protocol of random window size N in discrete time. The time unit equals to the transaction of a window group. The highest, and the lowest IDS of packets transmitted within a window uniquely specify (in a combinatorics fashion) the waiting packets at the RSB. Binomial probabilities (with probability p of an erroneous transmission) determine the number of erroneously received packets. At the end of a window some new packets may join the RSB as well as some of the waiting there can be released. There is an infinite input rate. Transition probabilities between the RSB states are found, and the steady state of corresponding Markov chain is proven. The probability generating functions of URSB and of W-B are obtained. In particular &
is the expected length of the RSB queue. The expected
total delay, and the waiting time are measured in terms of the window time and in multiple to l/N part of it (the time for transmission of one packet). The Little’s identity here (1 - P)E(WRSB)=E(V~B) is used again to study the waiting time. Approximation of the exact distributions for p -+ 0, or p -+ 1 are derived. Graphical illustrations show the expected RSB queue behavior with variation of window size N.
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Shacham and Towsley(1991) consider the SR it ARQ protocol in multi-cast environment: a transmitter sends packets to M receivers. Each NACK after the timeout causes retransmission of corresponding packet to all the receivers. Each receiver has its own RSB. The pair transmitter-receiver is treated under the effect of multi-cast transmission. Backward acknowledgments are non-reliable; with probability qi transmitter gets the ACK from 8” receiver. The case of M = 1, q = 1 is the above reported by Rosberg and Shacham. Here the authors derive the probability distribution of the time 2 until a packet is successfully transmitted to all of the M receivers. The distribution of WRss(i) is in terms of a system of recursive equations, which allow an effective moments calculation. The Little’s formula E(vRsB(~J)=& E(WRsB(i)) gives the expected queue size at each receiver. Reservation buffer is named the number of packets that will potentially join the RSB(i) after a window is transmitted. The total resequenced queue is the algebraic sum of the actual and reservation buffers. Qualitative result is that for fixed value of pi’s the expected RSB occupancy E(L/~B(~)) decreases, if considered as a function of M or of pi (as it does E(Z)); and increases, if considered as function of qi or N. Lee and Hu(1992) discuss the SR it ARQ protocol with M receivers and repeated n copies, retransmitted with each NACK packet. Also individual error probabilities pi are presented together with an error probability pc for common wrong transmission of a packet to all receivers. Packets arrive for transmission according to a renewal process with geometric distribution. The WPB is the time spent by a packet at the transmitter’s buffer from its first start until it is confirmed by all receivers. Performance measure is the sum WPB + WRsB(i). Each component is explicitly studied. A discrete time Markov chain technique is applied to derive relationships between suitably introduced random variables (to describe the process). Equations for the equilibrium probability distributions are obtained. The optimal number of copies which minimizes the expected total delay of a packet in the system is subject of a control. It is shown that by sending the optimal number of copies of a packet, the mean overall delay can be substantially reduced. The finite size of PB has a negligible effect on the total delay if this size is grater than the window size. Yoshimoto et a1.(1993) search a large scale of ARC protocols in review and in action. There is a geometric input flow of messages. Each message may have a random length. Then it is partitioned in single packets for transmission according to GBN, or SR it ARQ protocol, until the entire message is transmitted. The messages itself are treated as under SW it ARQ protocol. The primary queue of packets in PB is then similar to McX)/G/l queuing system in discrete time. It is used to derive what time is needed to transmit one message in whole. For instance, GBN it ARQ protocol has generating function of the service time S*(z)=B* (I$&) zN, where B*(z) is the generating function of the number of packets generated by one message, and N is the window size. Corresponding expression is obtained also for SR it ARQ protocol. It is in terms of transmission times of all the involved packets and their waiting times in the RSB. The use of probability generating functions styles the best tradition of queuing theory. Numerical results show that in some sense SR is better than GBN scheme. Also dependence of the throughput on the message length and on the window size is illustrated.
5. THE MAP/MI2
MODELS
possibly traced a new way in resequencing studies. It was seen that the Matrix-analytic and study of performance measures, when explicit equations for the state probabilities are at hands. Almost all of the previous works with numerical component dealt with Poisson input and exponential service times, or were discrete time models. One model, in which a Poisson process superimposed by a deterministic bursty process as system input and normally distributed service times, was recently studded in Plotkin and Varaiya(l993). Varma(l99la)
methods can be helpful in effective computations
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However, in communication engineering packets may arrive in a very bursty and stochastic manner; the inter-arrival times may be dependent. This motivates the use of more general input processes which can be modeled by Markov arrival process (MAP). At GM1 an intensive study of these models for resequenced systems is going. The research group consists of Dr. S. Chakravarthy, Dr. S. Chukova, Dr. B. Dimitrov, and works in conjunction with master and undergraduate students, as S. Lin, T. Dimitrova, S. Sassy and others. Bin and Chakravarthy(l996a) conduct a complete steady state analysis of MAP/M/2 system with non-renewal input, two heterogeneous servers and threshold-type scheduling. Bin and Chakravarthy (1996b) continue their study to find the optimal allocation of messages for the same system in the sense of search of optimal policy, as it was stated previously by Iliadis and Lien(1988, 1993). It is assumed ~1 > ~2, one threshold number N, and policies nl(N), or r~+r(N) are used. By making use of the method of Matrix-Geometric solution, they involve a large scale state space X (~pg,Xl, X2,77), where Xi indicates the state of ith server; n indicates the phase of the MAP, (1 <_ n 5 m). In this way the stationary distribution at arrival, and at arbitrary time is obtained as solution of a matrix equation. It allows to calculate effectively the c.d.f. of residual delay in RSB, the probability of a job to be delayed, to establish all necessary and sufficient conditions for minimizing the resequenced proportion, or expected system delay. A numerically stable algorithm for computing performance measures is developed. It is proven that policy KN+~(N) always yields a smaller fraction of resequenced jobs than rr(N). Also, there exists a polynomial decision &n&on in z = pl/(pr +/Jz) which determines the policy that yields the minimum delay. Its form is given by equation (1)) found by Iliadis and Lien(1988) for the Markovian models. Chakravarthy et a1.(1996) gave another point of view in the analysis of queuing models with resequencing, namely by introducing randomized policy in directing the heading jobs to the idle model with ~1 > ~2 Infinite capacity of RSB. The proservers. They consider the MAP/M/2/B posed randomized (probability) control policy n(p) means that when the new arriving job finds both servers idle, with probability p E [0, l] it is directed to server 1; otherwise it gets server 2. The finite capacity of RSB may cause blocking of servers when RSB (if finite) is full. In addition another randomized policy r@,pB) appears: When the blocked RSB is released, the waiting job from the head of the queue with probability pn E [0, l] is directed to server 1; otherwise it gets server 2. Matrix-Geometric approach is applicable again. It gives the following results: The random vector (VRSB, zr, z2, ~pg, E, n), where E indicates which server serves the job entered earlier (and the order of this description is also important), suits to apply this approach. The infinitesimal operator governing the process is obtained, and an explicit steady state analysis of the process is performed. Algorithmic procedures for calculation of the steady state probabilities of the vector (VRSB, ~1, x2, VpB, &,n) are worked out. Stationary waiting time distribution of an admitted job is then obtained. The system performance measures E(vp~), E(VRSB), PB, the mean sojourn time in the system are effectively found. Numerical examples with correlated and uncorrelated inter-arrival times show some curious effects: There are values of admission probabilities p, and pB which are neither 0, nor 1, which produce optimal performance of that particular model.
6. CONCLUSIONS
The resequencing problems in Queuing Theory have its specific scope of research. Mostly Markovian models have been studded. There is a number of practical system that work under resequencing mechanisms. The it ARQ models offer interesting problems in resequencing which differ from the models considered in “classic setups” of queuing theory. The Matrix-Geometric approach, and MAP models offer new technical, theoretical and practical way in the study of resequencing.
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The search of optimal resequencing policies is an additional and interesting field in the study of resequenced queues. Threshold selection policies, preliminary partitioning for resequencing, group rerouting, windowing, multiple repeated services, non-traditional randomized control policies in resequenced system may reduce considerably the delay, and may improve the total performance of the system. Queues with resequencing will keep attention of researchers active and challenging.
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