An interpolation approximation for expected wait in a time-limited polling system

Computers & Operations Research 27 (2000) 353}366

An interpolation approximation for expected wait in a time-limited polling system Martin J. Fischer!, Carl M. Harris",*, Jennifer Xie# !Mitretek Systems, McLean, VA 22101, USA "Department of Systems Engineering & Operations Research, George Mason University, 4400 University Drive, Fairfax, VA 22030-4444, USA #United States Postal Service, Washington, DC 20060, USA Received 1 October 1998; received in revised form 1 March 1999; accepted 1 March 1999

Abstract We consider the problem of determining the expected waiting time of arrivals at a particular queueing station in a multi-system con"guration with a single server operating under a time-limited polling discipline. At each station, we assume Poisson arrivals and a general service distribution, which does not have to be the same at each station. A simple and accurate interpolation approximation is developed for this mean and compared with known and simulation results. Scope and purpose Computer/telecommunication systems frequently use polling schemes to alternate service to multi-queued customers in an equitable manner. But traditional polling rules like exhaustive and gated service have limited usefulness in these systems. As a result, interest in studying the behavior of time-limited disciplines has arisen. In practice, the complicated combination of non-exponential service and/or "xed-length orientation times makes the necessary analysis quite di$cult. In this paper, we develop and test relatively simple formulas for estimating expected customer waiting time in polling systems wherein the server moves to the next queue if the number of customers at the present queue drops to zero or the "xed-time limit is reached, whichever occurs "rst. The derivation of the formulas uses ideas from the theory of functional interpolation. ( 2000 Elsevier Science Ltd. All rights reserved. Keywords: Queues; Telecommunications; Probability applications

* Corresponding author. Tel.: #703-993-1687; fax: #703-993-1521. E-mail address: [email protected] (C.M. Harris) 0305-0548/00/$ - see front matter ( 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 3 0 5 - 0 5 4 8 ( 9 9 ) 0 0 0 5 6 - 8

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1. Introduction In this paper, we analyze the situation where a single server works on several queueing systems in cyclic order and extend results presented by Xie et al. [17], where we studied a loop system that operates under a "xed-time-limited loop system (FTLS) discipline. The FTLS discipline is one where the server visits a speci"c queue for a "xed amount of time, serves any customers present there, and does not serve any other queue during this period, even if there are no customers at the currently visited queue. A modi"cation of this rule which utilizes the server better is one where the server moves to the next queue if the number of customers at the present queue drops to zero or the "xed time limit (i.e., orientation or on-time) is up, whichever occurs "rst. We call this rule a time-limited polling system (TLPS). In Xie et al. [17] we have discussed the applicability of these latter rules to various problems in telecommunications and robotic systems. Because the time epochs at which service stops at a given queue do not form a renewal sequence, standard methods of analysis break down. However, we have found that it is possible to derive exact or near exact limits for the quantities of interest in either light or heavy tra$c. These limiting cases can then be linked together using an exponential interpolation to estimate the expected workload present at a particular queue or sub-system. When this is combined with results found in Xie [16], the (approximate) expected waiting time is found. In light tra$c, the TLPS rule closely approximates the exhaustive service polling scheme (EPS) since we always require that the time allocated on any trip be at least equal to a service time. Under the EPS rule, the server remains oriented (i.e., serves) at a given queue until there are no customers present, at which time it would move to the next queue. Several results for the EPS are available; see Fuhrmann [1]; and Boxma and Groenendijk [2], for example. On the other hand, in heavy tra$c, the TLPS rule approaches the FTLS rule, and the results developed in Xie et al. [17], and Fischer [3,4] can be used. Thus, one appealing target for approximating TLPS is to match up well with the actual limiting solutions. That is, the asymptotic approximation should be very good when the parameters of the model are close to their limits. In this paper, we combine the EPS rule for light tra$c with the FTLS for heavy tra$c and interpolate between them for our waiting-time approximation of the TLPS. If another data point has been determined via simulation or other means, our approach allows the easy introduction of a further parameter that ensures that the approximation and the additional data point agree and that the accuracy of the approximation is increased (see Table 1 below). Interpolations combining light and heavy tra$c limits are a natural approach and yield approximations that are often much better than using either limit by itself. This fact has been demonstrated empirically in Burman and Smith [5], Reiman and Simon [13], Whitt [15], and Fleming and Simon [6]. In Reiman and Simon, the authors utilized a heavy tra$c limit, a light tra$c limit, and the "rst derivative of the light tra$c limit with respect to the arrival rate to develop a quadratic approximation for the quantities of interest. In Whitt [15], an approximation for the steady-state expected workload in a GI/G/1 queue was developed using a polynomial interpolation method, where the coe$cients and the exponents depended on the light and heavy tra$c limits, as well as their "rst derivatives. Chen and Whitt [7] developed heavy-tra$c limit theorems for systems with interruptions. Co!man et al. [8] and Leung [12] considered TLPS, but required that both service and ontimes be exponentially distributed. However, Leung pointed out the mathematical di$culties

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encountered for the alternative case where the on-periods are assumed to be constant. Leung and Eisenberg [11] had previously examined the case of constant on- and o!-times for only the single-system format with vacations, and their analysis required the solution to a functional equation, which they were unable to solve in closed form. By approximating the probability of 0 in the system by a weighted sum of Laguerre functions with unknown coe$cients, an approximation was developed by them and tested. The unknown coe$cients in Leung and Eisenberg's estimate were determined by solving a potentially large set of linear equations. Furthermore, the coe$cients of this linear system are functions of the subjectively chosen time-scaling factor and are derived from the recurrence formulas for Laguerre polynomials. In our work, we develop a signi"cantly numerically simpler approximation that does not require any complicated numerical procedures in approximating system performance. Later in the paper, we compare our approximation to the results developed by Leung and Eisenberg, in addition to providing expanded thoughts on our relative methods. It su$ces to say for now that our entire method can typically be completed with a pencil, paper and hand-held calculator (or small spreadsheet). In fact, closed-form, parsimonious solutions are often absolutely necessary for complex modeling problems, as, for example, in the multi-station case we present in Section 4, and in problems of system and network optimization. In addition, because of the exponential nature of our underlying functional approximation, the desirable properties of continuity and convexity are automatic when using our approximation. Our work on this problem is presented in the following four sections. To begin with, in Section 2, we develop some basic results for the full multi-queue TLPS and then examine more completely what happens through the eyes of one particular queue only. From the perspective of the single, particular queueing sub-system we have chosen, the server is oriented toward it for at most a "xed period of time or until the number of customers present falls to zero, whichever happens "rst. At the point that the server leaves to work elsewhere, it looks to the single queue as if the server is going on a vacation of random duration, where the length of the vacation will depend on the walktime between the queues and the load at each one of them. Subsequently, in Section 3, we develop an approximation for the average load in the singlequeue case, which, when combined with the results from Section 2, completes the derivation of the expected waiting time from the single-station perspective. Because of the structure of the TLPS discipline, studying a single system in isolation is su$cient. So, in Section 4, we extend the results of Section 3 to the full multi-queue system by properly characterizing the vacation time and conclude with a few remarks in Section 5. Numerical results are presented throughout the paper.

2. Basic results for TLPS: modeling from the view of one sub-system Assume that there are N queueing sub-systems being serviced by a single server under the time-limited polling system (TLPS) discipline. Under such a discipline, the server is oriented toward system i for at most ¹ time units during a given cycle. If the number of customers drops to i zero during the orientation time or the ¹ time units are up, the server moves to sub-system i#1. i We assume that the arrival process of customers at sub-system i is Poisson with rate j , i"1, 2,2, N, and that service time is general with mean b and second moment b(2). All these i i i

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random variables are assumed to be independent. The tra$c intensity for system i is o "j b , i i i i"1, 2,2, N. An important point to note is that if there is no walk-time for the server to go from sub-system i to sub-system i#1 and a preemptive resume discipline is assumed, then expected-value results for the TLPS discipline can be analyzed using classical results from M/G/1. This fact follows because the server is always working if there is someone at one of the queueing systems and no work is created or destroyed. Thus, we assume that there is a period of time required to walk from i to i#1, say r , which may be random. After leaving sub-system N, the server goes to sub-system 1. i We are interested in analyzing the behavior of customers at a particular sub-system. In reality, we can get some important information about the behavior of the entire system by considering what happens at a single station where the server is oriented toward this sub-system for at most T time units and then goes away for a random period of time Z. The random variable Z not only depends on the length of service time at the sub-system of interest but also at all other queues. However, we initially develop results for the case of one queue with an independently drawn vacation time, Z. For a more in-depth discussion of these relationships, see Xie [16]. We develop our results for a general service-time distribution and Z assumed random. When we apply these results to numerical examples for the general TLPS, we consider cases where the service distribution is either exponential or constant and assume that Z is constant. De"ne the following random variables (we suppress the subscript indicating the sub-system of interest): C is the length of a cycle, namely, the time between successive visits by the server to a particular sub-system } that is, the sum of the time the server is oriented toward this sub-system plus a vacation, ¹@ is the length of time the server is oriented toward the sub-system on a particular cycle (¹@)¹ and E[¹@](¹), and Z is the length of the vacation. Then we must have C"¹@#Z and E[C]"E[¹@]#E[Z].

(1)

(Remember that the random variables ¹@ and Z are not independent for the full TLPS problem.) The load that arrives at the sub-system during a cycle must be processed (on average) during the server's on-time period (viz., ¹@) in order for the steady-state distributions of interest to exist. So (1/b)E[¹@]"jE[C] or E[¹@]"jbE[C]"oE[C].

(2)

Combining Eqs. (1) and (2), it follows that E[C]"E[Z]/(1!o)

(3)

E[¹@]"oE[Z]/(1!o).

(4)

and

Since E[¹@](¹, we see from Eq. (4) that o(¹/(¹#E[Z]) is the de"ning requirement on tra$c intensity for equilibrium to exist. To continue, we de"ne the following steady-state random variables:

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D is the total steady-state load present at the sub-system of interest, D is the steady-state load in 2 the queue, = is the waiting time in the system, = is the waiting time in the queue, and S is the 2 3 residual service time of the customer in service at a randomly observed time, given that the server is working. From renewal theory and Little's formula, we have E[S ]"b(2)/2b and E[D ]"oE[= ]. 3 2 2 If we assume that a customer waits at most one vacation time, then

(5)

E[=]+E[= ]#b#a@E[Z], (6) 2 where a@ is the probability that the vacation preempts the service of the test customer whose waiting time we are analyzing. Eq. (6) states that the expected system wait of the test customer is the waiting time in the queue plus its service time plus a vacation time if its service is preempted by a vacation. Under preemptive-resume discipline, the second term of the right-hand side of (6) is just the average service time. The probability a@ is made up of two independent events: (1) the test customer is the last one served in a cycle; and (2) a vacation interrupts service. Because the arrivals are Poisson, the probability of the "rst of these events is (1!ejE*C+)/(jE[C]) (see Cooper, [9]), and we henceforth denote the probability of this second event by a. Eq. (6) lacks an additional term to account for the fact that the test customer's waiting may extend beyond T and hence correspond to several vacations. For the examples considered in this paper, we have b;¹ and service distributions assumed to be either exponential or constant, and the contribution of the possible additional term to the test customer's waiting time is minimal. For those cases where the contribution of several vacations (i.e., b not ;¹), one can include the additional factor in (6) and adjust the following analysis appropriately. Now, using Eq. (3), Eq. (6) becomes 1!e~jE*C+ jE[C] E[=]"E[= ]#b#a 2 jE[C] E[Z] "E[= ]#b#a(1!o)(1!ejE*C+)/j. (7) 2 Since E[= ]"E[D ]/o, we need to determine the expected load in the system to resolve (7). 2 2 Begin by de"ning I as an indicator random variable to denote the orientation of the server, such that I"1 if service is being performed on customers from the sub-system of interest and I"0 if the server is on vacation relative to that sub-system. Then E[D]"E[D ]#PrM='0, I"1NE[S ]#PrM='0, I"0NE[S ]. (8) 2 3 3 Xie [16] has shown that PrM='0, I"1N"o and that PrM='0, I"0N"aE[Z]/E[C]" a(1!o). As a result, Eq. (8) becomes E[D]"E[D ]#E[S ][o#a(1!o)], 2 3 and combining Eqs. (5), (7), and (9), we get oE[=]"E[D]#(E[S]!E[S ])[o#a(1!o)]!aE[S](1!o)e~jE*C+. 3

(9)

(10)

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Therefore, the expected waiting time for a customer depends on two unknowns: the average load in the system and the probability that a vacation interrupts service, a. If the service is exponential, then E[S]"E[S ] and a only contributes to the expected wait through the last term in Eq. (10). If 3 the service time S is constant and T is an integer multiple of b, then a"0 and E[S]!E[S ]"E[S]/2. Later we examine the contribution that the "nal term in Eq. (10) has on 3 a customer's expected waiting time; but, "rst, we develop an interpolation formula for E[D] that combines light- and heavy-tra$c results.

3. An approximation for E[D] From Eq. (10) we see that the expected waiting time at a single queue operating under a TLPS rule depends on the average load E[D]. So we now derive an interpolation estimate for E[D] that depends on light- and heavy-tra$c approximations. During a typical cycle, the server works at the subject sub-system for at most T time units and then goes away for a random length of time Z. Such a system has been analyzed under an assumption of heavy tra$c by Fischer [3,4] using a di!usion pproximation. The case where both T and Z are constant was considered by Fischer [3], while Fischer [4] examined the situation where both could instead be random. For the general TLPS discipline, if the walk times are in fact constant and all sub-systems are heavily loaded, the values of T and Z would be constant and the TLPS would approach the FTPS studied in Fischer [3]. From Fischer [3], we have the following heavy-tra$c di!usion approximation for the expected load, E)[D]: E)[D]"jb(2)/[2(¹H!o)],

(11)

where ¹H"¹/(¹#E[Z]) and o(¹H. Some years after Fischer's original proposal (Fischer [3,4]) to use di!usion for approximating systems with service interruptions, Kella and Whitt [10] further re"ned the use of di!usion to approximate single-station queues where the server goes away for periods of time. This work was extended for use in open queueing networks by Chen and Whitt [7]. Fischer [4] reported on a favorable comparison of the di!usion approximation E)[D] given by Eq. (11) with the results of some simulations; Xie [16] continued that investigation by considering more extensive simulation comparisons, all of which showed that E)[D] is an excellent approximation to E[D] as o approaches ¹H. Furthermore, and most importantly, Xie [16] also showed that E)[D] is a lower bound for E[D] and that its maximum error in approximating E[D] is E[Z] ) ¹/[2(¹#E[Z])]. Thus, it follows that the relative contribution of this error goes to 0 as o approaches ¹H since the width of the approximation interval is independent of o. In addition, it is then straightforward to show that the worst possible error in using E)[D] is (¹#E[Z])/8, which is again independent of o. As a result of all of this, we use E)[D] as the heavy-tra$c anchor for our approximation, since we cannot do any better in the limit as o approaches ¹H with the results of either Kella and Whitt [10] or Chen and Whitt [7]. As we shall see shortly, our approximation does extremely well, thus further supporting the use of Eq. (11) as the heavy-tra$c anchor. For the light-tra$c case, the results from the exhaustive service discipline can be used. For that discipline, the server remains oriented toward a given sub-system until there are no more customers

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present. Using the results of Fuhrmann [1], Boxma and Groenendijk [2], or Tagagi [14], we have the light-tra$c approximation for expected load in the queue E-[D ] as 2 E-[D ]"oMjb(2)/[2(1!o)]#E[Z2]/2E[Z]N. (12) 2 Eq. (9) relates the expected load in the system to that in the queue; using this result, we have the following light-tra$c approximation for the expected load E-[D] in the system: E-[D]"E-[D ]#E[S ][o#a(1!o)]. 2 3 But, in light tra$c the probability a that a vacation interrupts service is (essentially) zero, so it follows from (12) that E-[D]+E-[D ]#E[S ]o 2 3 "oMjb(2)/[2(1!o)]#E[Z2]/2E[Z]#E[S ]N. (13) 3 We shall now introduce an exponential approximation for E[D] (see Eq. (18)). It contains the two unknowns, b and b , that are determined by the boundary conditions and Eqs. (11) and (13). 1 2 When the load is light, E-[D] would be the approximation to use, and its value is zero when o"0. On the other hand, E)[D] would be used to approximate for high loads, and its value goes to in"nity as oP¹H. We need to use the values at these limits to determine b and b , so we would 1 2 like them to be well de"ned at these boundaries. Thus we normalize E-[D] and E)[D] by multiplying each by the constant 2(¹H!o)/o, which simultaneously removes the zero in the denominator of Eq. (11) with the term 2(¹H!o) and allows Eq. (13) to have a well-de"ned, non-zero limit as oP0. As a result, we have the adjusted approximations E)[D(o)]"jb(2)/o"b(2)/b"b

(14)

E-[D(o)]"[ob/(1!o)#z#b](¹H!o),

(15)

and

where b is de"ned by the ratio of the second service moment to its mean and similarly z"E[Z2]/E[Z]. In Eqs. (14) and (15), we have notationally changed D to D(o) to denote that it is a function of o. Now, from Eqs. (14) and (15), we have E)[D(¹H)]"b

(16)

E-[D(0)]"(z#b)¹H.

(17)

and

We are now ready to combine our results so far, and consider the simple exponential interpolation approximation for E[D] of the form f (o)"b o exp(ob )/[2(¹H!o)], 1 2

(18)

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where the constants b , i"1, 2, are determined by the cases where o"0 and o"¹H. It should be i noted that this functional form is quite di!erent from the polynomial approach to interpolation typically found in queueing. Depending on the degree of the polynomial used, such a process would require the determination of many constants, thereby complicating the approximation. In the end, we choose the form given by Eq. (18) largely because it requires the calculation of only two constants, and, as we shall see later, leads to very accurate estimates. From Eqs. (17) and (18) we have b "(z#b)¹H 1

(19)

and b "ln(b/b )/¹H. (20) 2 1 In Xie [16], several polynomial approximations were, in fact, examined using these light and heavy tra$c limits as well as their derivatives. None were as simple, consistent, and accurate as the exponential approximation given by Eq. (18). Using Eqs. (10) and (18) we get E[=]+b exp(ob )/[2(¹H!o)]#ME[S]!E[S ]NM1#a(1!1/o)N!aE[S](1/o!1)e~jE*C+, 1 2 3 (21) where we still have the unknown a. Xie (1992) showed for FTLS that o)a)min(1, o/¹H). But we can have a"0 for TLPS; we therefore suggest using a linear interpolation between 0 and min(1, o/¹H) for a as o varies between 0 and ¹H. In Table 1, we o!er an examination of the impact of this approximation for a on E[=]; the results are given in the column labeled E[=], a'0. More broadly, Tables 1}3 display some comparative results for the system where the server works on the class for at most ¹ and then goes on vacation for a constant length of time. For the exponential service case, Leung and Eisenberg [11] presented a comparison of their approximation with a simulation; see Fig. 1 of their paper. Our Table 1 compares our approximation with theirs, as well as indicating the contribution that the last term of Eq. (21) has on the expected waiting time. We got the Leung and Eisenberg simulation numbers directly o! their Fig. 1, since no table was presented for these numbers by them. This "gure is a curve with the simulation results indicated by the symbol `#a on the curve; we could only thus approximate any values from their experiments (our estimates are found in the L&E column of our Table 1). In our Table 1, the service distribution is assumed to be exponential with a mean of 1. The vacation time is a constant 6, while ¹"5. The second column of the table provides the results from the Leung and Eisenberg approximation, as compared to our approximation in the third column, for which we have set a"0. The factor to which we refer in column four is the term aE[S](1/o!1)e~jE*C+ from Eq. (21). Column "ve presents the results for the cases when this extra term is included. The "nal column is for E[=]+b exp(b ob3)/[2(¹H!o)], 1 2 where b is found by using a data point from Leung and Eisenberg and determining its value such 3 that there is exact agreement with Leung and Eisenberg at o"0.3 (gave b "0.9365). 3

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Table 1 Mean waiting time comparisons o

L&E

E[=], a"0

Factor

E[=], a'0

E[=], b 3

0.10 0.20 0.30 0.40 0.45

3.80 4.20 5.30 8.00 11.75

3.807 4.361 5.452 8.173 11.740

0.139 0.116 0.053 0.015 0.006

3.668 4.245 5.398 8.158 11.734

3.737 4.252 5.300 7.944 11.419

Table 2 Comparisons with simulation for constant and exponential service (¹H"1/3) o

0.125 0.200 0.251 0.286 0.309

Exponential service

Constant service

Approx.

Simulation Approx.

Simulation

7.403 9.899 14.420 23.382 43.288

7.313 9.754 14.391 23.217 44.000

6.235 7.117 9.187 13.213 22.470

5.907 6.807 8.874 13.212 23.123

Table 3 Comparisons with simulation for constant and exponential service (¹H"2/3) o

0.100 0.200 0.500 0.542 0.625

Exponential service

Constant service

Approx.

Simulation Approx.

Simulation

3.626 3.877 7.415 9.393 25.303

3.654 3.859 7.061 9.089 23.547

3.569 3.647 4.934 5.799 13.860

3.367 3.428 4.742 5.698 13.585

Several points can be made from Table 1: 1. Our approximation does very well when compared with Leung and Eisenberg. Since their approximation required the solution of a set of linear equations, the approximation given above has appeal because of its simplicity. 2. Using E[=]+b exp(ob )/[2(¹H!o)] 1 2

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by itself (i.e., a"0) does quite well, and introducing the additional factor aE[S](1/o!1)e~jE*C+ does not substantially improve the approximation. 3. If one has known data, then introducing an intermediate anchor (b ) allows one to sharpen our 3 approximation even more. Given these observations, we propose the following approximation for the expected waiting time: E[=]+b exp(ob )/[2(¹H!o)]#ME[S]!E[S ]NM1#a(1!1/o)N. 1 2 3

(23)

If the service time S is constant and ¹ an integer multiple of the expected service time, then E[S]!E[S ]"E[S]/2, a"0 and thus 3 E[=]+b exp(ob )/[2(¹H!o)]#E[S]/2. 1 2

(24)

If, however, the service is exponential, then E[S]!E[S ]"0 and we have 3 E[=]+b exp(ob )/[2(¹H!o)]. 1 2 For other service distributions, we suggest using Eq. (23) with E[S ]"b(2)/2b and a as a linear 3 interpolation between 0 and min(1, o/¹H) as o varies between 0 and ¹H. Tables 2 and 3 continue with a comparison of our approximation for E[=] to results obtained from a carefully designed simulation experiment. Again, we are considering a system where the orientation of the server to the class has a maximum length of time and then the server goes away for a "xed amount of time. For each table, two cases are examined, namely, when service times are exponentially distributed and when they are constant. For both cases, the mean service time is assumed to be one (i.e., b"1). In Table 2, the maximum on-time is 5 time units and the vacation time is assumed to be a constant 10 time units, whereas Table 4 has the reverse (¹"10 and E[Z]"5). For both tables, the approximations closely follow the simulation results and are certainly within any reasonable con"dence interval of the simulation.

Table 4 Comparisons for a two sub-system con"guration o ,o 1 2

0.1,0.1 0.2,0.1 0.3,0.1 0.4,0.1 0.2,0.2 0.3,0.2 0.4,0.2 0.4,0.3

System 1

System 2

Approx.

Simulation Approx.

Simulation

4.352 5.432 8.053 21.332 6.581 11.557 R R

4.352 5.324 7.893 21.349 6.461 11.644 R R

4.373 4.971 5.679 6.455 6.461 7.952 9.754 44.000

4.352 4.929 5.751 6.982 6.581 8.457 9.899 43.288

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4. The multi-system problem We have so far developed an approximation for the expected waiting time for a single-queueing sub-system under the TLPS discipline where the server goes on vacations for random lengths of time Z. In this section, we extend the results to the case where there is more than one sub-system. To do so, we have developed an expression for the expected length of time E[Z] that the server is oriented toward the other sub-systems. But we also need the second moment of Z for the light-tra$c portion of our approximation; however, we have not been able to develop an expression for it in the multi-system case. We have thus assumed Z to be constant here, so that E[Z2]"E[Z]2. However, our comparisons with simulation results suggest that this assumption has little or no e!ect on the quality of the approximation. This time we also start from a lone sub-system, but now one whose vacation time Z depends on the workload at the other systems and the walk between them. We recall that the arrival process at each sub-system is Poisson with rate ji, i"1, 2,2, N, the service distribution is general with mean bi and second moment b(2) i , and the maximum service on-time is ¹i, i"1, 2,2, N. We assume further that the system of interest is sub-system 1 and that the total walk time during a cycle is the constant r. The tra$c intensity at sub-system i is thus oi"jibi. We assume that the steady-state existence conditions are satis"ed for each system. Then if ¹@i is the length of time that the server is actually oriented to system i, then the average length of a cycle is given by E[C]"E[¹@1]#E[¹@2]#2#E[¹@N]#r. For steady state to exist at each system i"1, 2,2, N, we must have E[¹@i]"oiE[C](¹i

(25)

and so E[C]"r/(1!o), where o"o1#o2#2#oN. From Eq. (25), we thus also have E[¹@i]"oir/(1!o). The approximation presented in the prior section contains three unknowns, E[Z], E[Z2], and the upper limit ¹H on sub-system 1's tra$c intensity for its steady state to exist. For ¹H, we have ¹H"¹1/(¹1#r#q)

(26)

where q is the expected length of time sub-systems 2, 3,2, N use the server if sub-system 1 uses it for its full length of time, ¹1. Following the analysis presented at the beginning of this section, we have q"(r#¹1)(o2#o3#2#oN)/(1!o2!o3!2!oN)

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and so ¹H"(1!o2!o3!2!oN)¹1/(r#¹1). The expected length of a vacation with respect to sub-system 1 is E[Z]"r#E[¹@2]#E[¹@3]#2#E[¹@N] "r(o2#o3#2#oN)/(1!o)#r.

(27)

As we mentioned at the beginning of this section, we are unable to develop a simple expression for E[Z2], so we assume that Z is constant and thus E[Z2]"E[Z]2, with E[Z] given by Eq. (27). If the tra$c loads in the other systems are either very light or very heavy, then the random variable Z will tend to be constant. Thus we feel that this assumption would be su$cient to develop an accurate approximation. Our simulation comparisons have shown this to be the case. Let us now summarize our approximation. For the sub-system of interest, we use Eq. (23) to approximate the expected waiting time for a customer from that subsystem, where the constants b1 and b2 are given by Eqs. (19) and (20). In those equations, b and z are ratios of the second moments of the underlying random variables to their means. For b and the sub-system of interest, this calculation is straightforward. Since we have assumed that the vacation times are constant, then z"E[Z]. Next, we use Eqs. (26) and(27), suitably modi"ed to re#ect the sub-system of interest, to compute E[Z] and ¹H. If the service distribution of the sub-system of interest is exponential, then E[S]!E[S3]"0 and a need not be found. If, however, the service distribution is constant and ¹ is an integer multiple of the mean service time, then we use Eq. (24) for E[=]. For other cases, we suggest following our comments just after Eq. (24). Before proceeding with some comparisons with the simulation results, a discussion about existence conditions is in order. For a sub-system to have a well de"ned steady-state distribution, its tra$c load has to be less than ¹H. We can have some sub-systems blow up and still have a well de"ned steady-state distribution for the other sub-systems, since the respective arrival streams are independent. For the sub-systems that do not have a well de"ned steady state, we set E[¹@j] to ¹j. To illustrate our approximation, let us consider an example with two sub-systems (N"2). We assume that the service distribution for each sub-system is exponential with mean one time unit (b"1), so that b(2)"2 and b"2. The walk time between the sub-systems is set at 2.5 time units and hence 5 time units are needed for the cycle (r"5). The maximum on-time for each sub-system will be 5 time units (¹i"5, i"1, 2). If sub-system 1 is the one of interest, specializing the results of this section leads to ¹H"(1!o2)¹1/(r#¹1)

(28)

E[Z]"ro2/(1!o1!o2)#r"r(1!o1)/(1!o1!o2).

(29)

and

We assume further that the vacation time is constant, so that E[Z2]"E[Z]2 or z"E[Z]. In order to determine the expected waiting time using Eq. (23), we have to calculate bi, i"1, 2, via Eqs. (19) and (20), and then use Eqs. (28) and (29) to determine ¹H and E[Z].

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In Table 4, we present comparisons of our approximation with results obtained from a simulation, where we have used the actual loads generated in the simulation when we computed the approximation. For the last two cases, the load of sub-system 1 during a cycle is greater than 5 and so its waiting time is in"nite. But sub-system 2's average waiting time does exist. In order to use the approximation for all of these cases, we set the vacation time to 10 time units; the walktime to 5; and the maximum on-time for system 1 also to 5. We see that the approximation does extremely well. The worst error is in the (0.3, 0.2) case for sub-system 2, where the approximation and the simulation di!er by (8.457!7.952)/7.952"6.4%.

5. Concluding remarks In this paper we have developed a computationally attractive, interpolation approximation for the average waiting time at a particular sub-system in a multi-queue con"guration operating under a time-limited polling discipline. We have compared the approximation with results obtained from a simulator and other published results. These comparisons have lead us to believe that the approximation is quite accurate. When we couple the accuracy with the simplicity of the approximation, we can appreciate the power of the approximation for this class of problems.

References [1] Fuhrmann SW. A note on the M/G/1 queue with server vacations. Operations Research 1984;32:1368}73. [2] Boxma OJ, Groenendijk WP. Pseudo-conservation law in cyclic service systems. Journal Applied Probability 1987;24:949}64. [3] Fischer MJ. Analysis and design of loop service systems via a di!usion approximation. Operations Research 1977;25:269}78. [4] Fischer MJ. An Approximation to queueing systems with interruptions. Management Science 1977;25:338}44. [5] Burman DY, Smith DR. Asymptotic analysis of a queueing model with bursty tra$c. Bell Systems Technology Journal 1983;62:1433}53. [6] Fleming PJ, Simon B. Interpolation approximations of sojourn time distributions. Operations Research 1991;39:251}60. [7] Chen H, Whitt W. Di!usion analysis of a queueing model with bursty tra$c. Bell System Technical Journal 1993;62:1433}53. [8] Co!man Jr. EG, Fayolle G, Mitrani I. Two queues with alternating service periods. Performance 1987;87:227}37. [9] Cooper RB. Introduction to queueing theory, 2nd Edition, New York: Elsevier, 1984. [10] Kella O, Whitt W. Di!usion approximations for queues with server vacations. Advances Applied Probability 1990;22:706}29. [11] Leung KK, Eisenberg M. A single-server queue with vacations and non-gated time-limited service. Performance Evaluation 1991;12:115}25. [12] Leung KK. Cyclic-service systems with nonpreemptive, time-limited service. IEEE Trans. Commun. 1994;42:2521}4. [13] Reiman MI, Simon B. An interpolation approximation for queueing systems with Poisson input. Operations Research 1987;36:454}69. [14] Tagagi H. Mean message waiting times in symmetric multiqueue systems with cyclic service. Performance Evaluation 1985;5:271}7. [15] Whitt W. An interpolation approximation for the mean workload in a GI/G/1 queue. Operations Research 1989;37:936}52.

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[16] Xie J. Cyclic service systems with limited services disciplines, Ph.D. dissertation, Dept. of Operations Research and Engineering, George Mason University, Fairfax, VA, 1992. [17] Xie J, Fischer MJ, Harris CM. Workload and waiting time in a "xed time-limited loop system. Computers and Operations Research 1997;24:789}803.

Martin J. Fischer is a Fellow at Mitretek Systems in McLean, Virginia. He received the B.A. and M.S. degrees in Mathematics from the University of New Hampshire and Florida State University, respectively; and the M.S. and Ph.D. degrees in Operations Research from Southern Methodist University. He has published approximately 35 journal articles, most of them dealing with the development of network design and analysis tools and their application. His work includes a 1998 paper in this journal on queueing approximations by interpolation, coauthored with Prof. Harris. Currently, he is also an adjunct professor in the Department of Systems Engineering and Operations Research at George Mason University. Carl M. Harris is the BDM International Professor of Operations Research and Information Technology in the Department of Systems Engineering and Operations Research, George Mason University. He received the B.S. degree from Queens College (New York), and the M.S. and Ph.D. degrees from the Polytechnic Institute of Brooklyn, all in Mathematics. He is a coauthor of three editions of the textbook Fundamentals of Queueing Theory, and a coeditor of The Encyclopedia of Operations Research and Management Science. He was President of the Operations Research Society of America in 1990}1991. His current research interests include applied probability and statistics, queueing theory, simulation, and public systems analysis. Jennifer Xie is a senior program analyst at the United States Postal Service in Washington, DC. Her previous a$liations include Research Management Consultants (McLean, VA), Washington Gas Light Company (Washington, DC), and Project Design Systems (Arlington, VA). She received her B.S. degree in Electrical Engineering from Jiangsu Institute of Technology (China), M.S. degree in Systems Engineering from Hohai University (China), and Ph.D. in Information Technology from George Mason University.