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Approximations for the waiting time distribution in polling models with and without state-dependent setups
Operations Research Letters 28 (2001) 113–123
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Approximations for the waiting time distribution in polling models with and without state-dependent setups Tava Lennon Olsen ∗ John M. Olin School of Business, Washington University in St. Louis, Campus Box 1133, St. Louis, MO 63130-4899, USA Received 1 December 1997; received in revised form 1 April 1999; accepted 1 January 2001
Abstract A polling model is a queueing model where many job classes share a single server and a setup time is incurred whenever the server changes class. Polling models are applicable to many computing, telecommunications, and manufacturing environments. The scheduling method considered in this paper is a common policy known as cyclic, serve to exhaustion (CSE). Recently, Co4man, Puhalskii and Reiman (CPR) have developed a heavy-tra5c approximation for the waiting time distribution in a CSE polling model. This paper presents three new approximations. Firstly, the CPR approximation and the traditional (non-heavy-tra5c) polling model literature are combined to obtain a re7nement of the CPR approximation. This re7nement is much more accurate under conditions of moderate loading. Next, an approximation is made for the distribution of the number of jobs present in a queue upon it being polled. Lastly, the previous two approximations are combined to form an approximation for the waiting time distribution when setups are not performed for queues containing no jobs. A simulation study is undertaken c 2001 Elsevier Science B.V. All rights reserved. to evaluate these three approximations.
A polling model is a queueing model where n job classes share a single server and a setup time is incurred whenever the server changes class. Polling models are applicable to many computing, telecommunications, and manufacturing environments. In particular, local area networks and token rings are often modeled using polling models. An excellent survey of polling models may be found in both [14,15] which serve as a supplements to [13]. The scheduling method considered in this paper is a common policy known as cyclic, serve-to-exhaustion. Under this policy, arrivals of jobs of each of the n ∗
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job classes are held in a queue dedicated to that class. These queues are arranged in a cycle. When the server becomes free it advances to the next class in the cycle and begins a setup. All jobs of the server’s current class are served (including those that arrive during the current service period) before the server moves on to the next class. Jobs within a job class are served on a 7rst-in-7rst-out (FIFO) basis. This paper presents waiting time distribution approximations for two cyclic serve-to-exhaustion submodels. The 7rst submodel is the one traditionally studied where setups are incurred regardless of whether or not jobs are waiting in the current class. We will refer to this model as having empty-queue setups. The second submodel is the one where the
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server can ascertain whether there are jobs present in a queue and will not set up for that queue if it is empty. We will refer to this model as having positive-queue setups. In this model the server idles at its most recently served queue if the entire system is empty. Clearly, which submodel is applicable depends on the speci7c application being studied. Recently, Co4man, Puhalskii, and Reiman (CPR) have developed a heavy-tra5c approximation for the waiting time distribution in cyclic, serve-to-exhaustion polling models with empty-queue setups [4]. The CPR approximation is the 7rst explicit (closed-form) approximation for the waiting time distribution. Others have provided numerical algorithms for approximating the waiting time distribution (see [7] or [3]), but there is much value in having an explicit approximation, even if it is less exact. Explicit approximations lead to insights into how the system depends on di4erent parameters that are not possible with numerical algorithms. They are also easily implementable and useful for “back of the envelope” calculations. Furthermore, the heavy-tra5c approximation depends only on the 7rst two moments of the input distributions and not on their speci7c distributions. Unfortunately, the CPR approximation is extremely inaccurate at light to moderate loads. This paper presents a re7nement of the CPR approximation that is more accurate (and is still explicit). A search of the literature has revealed three papers which derive approximations for the mean waiting time in polling models with positive-queue setups, namely Ferguson [9], Bradlow and Byrd [2], and Gupta and Srinivasan [10]. In addition, Eisenberg (see, for example, previous work [5]) has derived an exact algorithm for 7nding the mean waiting time in polling systems but to our knowledge this paper has not yet been completed. The paper most related to our work is [2] which will be discussed further in Section 4. We have found no previous results for approximating the waiting time distribution (as opposed to the mean only) in systems with positive-queue setups. This paper is organized as follows. The notation used is introduced in Section 1. Section 2 presents the CPR heavy-tra5c approximation and re7nes it using non-heavy-tra5c theory. An approximation for the queue-length distribution for systems with empty-queue setups is presented in Section 3. This distribution is used in Section 4 to mod-
ify the empty-queue approximation to systems with positive-queue setups. Finally, Section 5 presents a simulation study of both waiting time approximations.
1. Notation Polling models with empty-queue and positive-queue setups have been described in the previous section. This section outlines the notation used and describes the assumptions made. Jobs are assumed to arrive at each queue i; 1 6 i 6 n, according to a Poisson process of rate i . Arrival processes are assumed to be mutually indepenn dent. Total arrival rate for the system is = i=1 i . Service times at queue i have mean bi and variance i2 . Utilization at queue i; i , equals n i bi and system utilization (excluding setups) = i=1 i . For stability we assume that ¡ 1 (see [13]). The generic setup time at queue i; 1 6 i 6 n, is represented by Si and may have a positive probability of being zero in systems with positive-queue setups. The mean and second moment of the setup time at queue i, given that a setup occurs, are si and si(2) , respectively. The only assumptions made regarding the distribution of the setup and service times are that the means and variances are 7nite and that all distributions are independent. We de7ne a cycle at queue i as the time between two successive arrivals of the server to queue i. Total steady-state nsetup time around a cycle is represented by S (= i=1 Si ). A cycle is therefore made up of setup time S, the time to serve all queues, and possibly some idling time when the whole system is empty. For empty-queue setups we let s = E[S] = n the case of (2) s and s = E[S 2 ]. i i=1 We de7ne intervisit time as the time that elapses from the server’s leaving queue i to its next commencement of service at queue i. Thus, the intervisit time for queue i equals the cycle time for queue i minus the time spent serving queue i. Steady-state intervisit time is represented by Ii . It is well known (see [13]) that, for systems with empty-queue setups, E[Ii ]=(1− i )s=(1−). The steady-state time-averaged intervisit time Ii∗ for queue i (see, for example [1]) is de7ned such that if Ii has p.d.f. fIi (x) then the time-averaged random variable Ii∗ has p.d.f. fIi∗ (x) = xfIi (x)=E[Ii ]:
(1.1)
T.L. Olsen / Operations Research Letters 28 (2001) 113–123
Steady-state waiting time at queue i; Wi , is de7ned as the time from a job’s arrival at queue i until it commences service n in steady state. De7ne 2 = i=1 i (b2i + i2 ). In our approximation, 2 will be the only place that the higher order moments of the service time distribution appear and may be thought of as a variance term. The last system parameters which need to be introduced are and ∗ which are essentially measures of the imbalance in the system. They are largest when the loads (i ’s) are identical at all queues. De7ne =
n−1 n i=1 j=i+1
i j =
n 1 i ( − i ) 2 i=1
and ∗ =
n
1 1 i (1 − i ) = (1 − ) + : 2 2
(1.2)
i=1
The 100pth percentile for the waiting time distribution at queue i is de7ned as the number Pi (p) such that Prob(Wi 6 Pi (p)) = p. Lastly, we de7ne a symmetric system as one where all queues have identical arrival rates, setup time distributions, and service time distributions. Conversely, an asymmetric system allows for di4erences among the queues. 2. Refined heavy-traffic approximation Using heavy-tra5c theory, CPR [4] have developed an approximation for the waiting time distribution for jobs in cyclic serve-to-exhaustion polling models with empty-queue setups. Some of the results cited here are not made explicitly in [4] (although they follow naturally) and further details may be found in Lennon [11]. Recall that Ii∗ is the steady-state time-averaged intervisit time at queue i; 1 6 i 6 n. The work of CPR shows that, for all i; Ii∗ may be approximated by a gamma random variable with scale parameter i = 2(1 − )=(2 (1 − i )) and shape parameter = 2s=2 + 1. Let U be a uniform[0,1] random variable which is independent of Ii∗ . Then the CPR steady-state waiting time approximation at queue i, Wi , is UIi∗ . The CPR heavy-tra5c approximation for the steady-state waiting time distribution can be used as an approximation for the waiting time distribution
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in non-heavy-tra5c systems. However, it tends to perform poorly in moderate tra5c. We will therefore adjust the CPR approximation so that its mean corresponds to known explicit results. Probably the most accurate explicit approximation for the mean waiting time in asymmetric systems has been derived by Everitt [6]. Everitt’s approximation is exact for symmetric systems and states that E[S 2 ] E[S] 2 1 − i + + : E[Wi ] ≈ 2∗ 2(1 − ) 2E[S] (1 − ) (2.1) It is straightforward to show that the di4erence between Everitt’s mean approximation and the CPR mean waiting time approximation converges to zero as increases to 1. For the re7ned approximation we will assume that the waiting time distribution is still of the form of a gamma random variable multiplied by a uniform[0,1] random variable. This implies that, for queue i, d
Wi ≈ UXia where Xia is a gamma random variable with scale parameter ia and shape parameter ia . The parameters ia and ia will be chosen so that the mean corresponds to Everitt’s approximation. Namely, such that, s(2) s ia 2 1 − i + + : = 2ia 2∗ 2(1 − ) 2s (1 − ) Clearly, there are in7nitely many di4erent ways the parameters ia and ia could be chosen to satisfy the above condition. We will only consider parameters that are correct under heavy tra5c. In other words, parameters such that ia =i → 1 and ia = → 1 as → 1. Note that, because the CPR mean is asymptotically identical to Everitt’s as ↑ 1, these are not independent requirements. Such an adjustment may be achieved by comparing the formulas for Everitt’s and CPR’s approximations for mean waiting time. CPR’s approximation for the mean waiting time is 2 1 − i s wi = : + 2 2(1 − ) (1 − ) Comparing this with (2.1) suggests that a natural adjusted approximation for ia is 2s=(2 ) + 1. Thus, the shape parameter remains identical except that the extra factor of that appears in Everitt’s mean approximation gets carried over to the denominator of
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the shape parameter. If we let pi = i = then n pi (1 − pi )s ia = ni=1 + 1: ( i=1 pi (bi + i2 =bi ) Note that does not appear in this equation, but if is scaled so that the pi and service times remain constant then ia is also constant in . The given ia implies that 2 + 2s 2∗ (1 − ) ia = 2 : (1 − i ) 2 + 2s + (1 − )s(2) =s Notice that ∗ ia =i =
2 + 2s 2 + 2s + (1 − )s(2) =s
→1
as ↑ 1 as desired. We tried a number of other adjustments (which are omitted due to space considerations) but this one was the most robust. 3. Queue-length distribution This section develops an approximation for the distribution of the number of jobs in a queue when the server commences service for that queue in a polling model with empty-queue setups. Later, in order to obtain an approximation for the waiting time distribution for polling models with positive-queue setups, the probability that queue i is empty when the server arrives must be calculated. We will approximate this probability in systems with positive-queue setups by the result for systems with empty-queue setups. The queue-length distribution may also be interesting in its own right. In particular, it may be used heuristically to design bu4er sizes in local area networks. In production applications it may be used to calculate optimal base-stock levels in make-to-stock systems as described in [8]. Let Ni be the steady-state number of jobs at queue i when the server commences service (or moves on if Ni = 0) after having setup for queue i. Then Ni is the number of jobs that have arrived during the last intervisit time for queue i. The memoryless property for Poisson arrivals combined with the fact that queue i is empty when the server leaves it, implies that the steady-state intervisit time Ii is independent of the
arrival stream to queue i. Thus, −i Ii (i Ii )x e P(Ni = x) = E x! ∞ −i y e (i y)x = fIi (y) dy: (3.1) x! 0 Hence, an approximate distribution for Ni may be obtained from an approximate distribution of Ii . Eq. (1.1) shows that the distribution of Ii may be obtained from the distribution of the time-averaged intervisit time Ii∗ . Also, recall from Section 2 that the heavy-tra5c approximation for the distribution of Ii∗ is a gamma random variable with scale parameter i and shape parameter . Thus, from (3.1), E[Ii ]i P(Ni = x) ≈ ∞ −(i +i )y e (i y)x i (i y)−1 × dy: x! () 0 (3.2) The terms may be factored so that the quantity within the integral is a gamma density and hence integrates to 1. Furthermore, E[Ii ] = (1 − i )s=(1 − ) = =i . Thus, straight-forward algebra yields an approximation for the distribution of Ni to be as follows: x i (x + ) i P(Ni = x) ≈ : (3.3) i + i i + i x! () The adjusted parameters ia and ia may, of course, be used in this approximation. If is an integer, then this is the negative binomial probability distribution. It is interesting to note that a mixture of negative binomials appears in the numerical procedure for approximating queue length as described by Federgruen and Katalan [7]. 4. Polling models with positive-queue setups This section uses the theory of the previous two sections to derive an approximation for the waiting time (both mean and distribution) in polling models with positive-queue setups. The approximation requires the numerical solution of a set of equations. Thus far all approximations in this paper have been explicit. If one requires an explicit formula in the positive-queue setups case, then a crude approximation is also presented here.
T.L. Olsen / Operations Research Letters 28 (2001) 113–123
To obtain an approximation for the mean waiting time in systems with positive-queue setups we will use Everitt’s approximation (2.1) but calculate S as the total amount of setup in a cycle with positive-queue setups. Thus, we need only obtain estimates for E[S] and E[S 2 ] to form an approximation for mean waiting time. We de7ne se = E[S] for systems with positive-queue setups. If ei is the probability that queue i is not empty when the server arrives at it, then se =
n
ei s i :
(4.1)
i=1
We will discuss approximating ei later in this section. Observe that
2 n E[S 2 ] = E Si i=1
=E
n
Si2
n +E Si Sj :
i=1
i=1
j=i
Now, in general, Si and Sj are not independent. However, for systems with empty queue setups they are independent. We shall form our approximation by assuming negligible dependence of Si and Sj . It should be noted that this assumption only a4ects the term E[S 2 ]=(2E[S]) and thus becomes negligible as ↑ 1. This assumption implies that 2
E[S ] ≈
n
ei si(2)
+
i=1
=
n i=1
n i=1
ei si(2)
−
n
e i si
e j sj
j=i
(ei si )2 + se2 :
(4.2)
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adjusted empty-queue approximation we have that ia =
2∗ (1 − ) 2 (1 − i ) 2 + 2E[S] × 2 + 2E[S] + (1 − )E[S 2 ]=E[S] (4.4)
and ia =
2E[S] + 1: 2
(4.5)
Thus, Eq. (4.1)–(4.5) may be solved numerically to form approximations for E[S], E[S 2 ]; ia ; ia , and ei for systems with positive queue setups. In theory we have presented a system of 2n + 3 equations in 2n + 3 unknowns which may be somewhat di5cult to solve. However, in practice, we found that a simple 7xed point iterative algorithm always converged. This algorithm 7rst calculates the empty-queue ia and ia . It uses these to calculate the ei from Eq. (4.3) then substitutes the ei back into Eqs. (4.4) and (4.5). These substitutions are continued until convergence is obtained. Notice that, with approximations for ia and ia , we have an approximation for the whole waiting time distribution not just the mean. It should be noted that this method of approximating polling models with positive-queue setups is somewhat close to that used by Bradlow and Byrd [2]. They describe the probability of queue i being empty as ei ≈ E[e−i Ci ], where cycle time is assumed to be exponentially distributed. It is unclear why they use cycle time as opposed to intervisit time in the above equation. Section 5 shows that our approximation compares very favorably to the Bradlow and Byrd approximation.
i=1
Thus it remains to approximate ei . We use the results of the previous section to approximate the queue-length distribution in systems with positive-queue setups. Thus, from (3.3) with x = 0, i −i Ii 1 − ei = P(Ni = 0) = Ee ≈ : (4.3) i + i This completes a crude but explicit approximation for the mean waiting time. A more re7ned approximation may be formed by noting the interdependence of i ; , and ei . Using our
5. Testing the accuracy of the approximations This section examines the accuracy of our empty-queue and positive-queue waiting time approximations for di4erent polling models. The approximations are compared to the results of a simulation program written in C. The simulation was run for at least 10,000,000 time units and sometimes longer for higher utilizations. Also, if mean service time was greater than one time unit, then the run length was scaled up similarly.
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Table 1 Percentage error in estimating the 95th percentile of waiting time in CPR approximation si = 0:2
= 0:95 = 0:9 = 0:8 = 0:7 = 0:6 = 0:5
si = 1
si = 5
n=5
n = 10
n = 20
n=5
n = 10
n = 20
n=5
n = 10
n = 20
6.61 16.11 37.92 65.62 104.10 157.54
4.48 12.39 28.38 48.83 76.60 114.56
4.42 9.36 20.36 35.28 53.95 79.79
2.89 8.98 19.76 33.06 49.19 72.21
1.67 4.83 12.40 20.00 29.83 42.75
0.12 3.04 7.29 11.79 17.93 25.35
0.26 −0:64 0.86 1.80 4.20 7.29
0.37 −0:73 −0:33 0.63 1.45 3.30
−0:03 −0:90 −0:35 0.04 0.33 1.50
Table 2 Percentage error in estimating the 95th percentile of waiting time in adjusted approximation si = 0:2
= 0:95 = 0:9 = 0:8 = 0:7 = 0:6 = 0:5
si = 1
si = 5
n=5
n = 10
n = 20
n=5
n = 10
n = 20
n=5
n = 10
n = 20
−1:78 −1:77 −2:75 −4:15 −5:74 −8:10
−1:58 −0:55 −0:74 −1:03 −1:06 −1:54
0.38 0.55 0.74 1.64 2.15 3.37
−0:68 1.14 2.27 3.35 4.15 5.31
−0:52 0.22 2.08 2.67 3.44 4.11
−0:88 0.37 1.22 1.76 2.40 3.07
−0:20 −1:62 −1:59 −2:54 −3:40 −4:43
0.07 −1:38 −1:90 −2:23 −3:22 −3:97
−0:03 −1:33 −1:33 −1:70 −2:40 −2:92
5.1. Empty-queue approximation Tables 1 and 2 show the percentage error in the 95th percentile of the waiting time for the CPR and the adjusted approximation, respectively. These tables use a symmetric 10 queue polling model when the number of queues changes from 5, to 10, to 20. Percentage error is de7ned as 100(Estimate − Simulated value)=Simulated value. Setup and service times are taken to be exponentially distributed. Mean service time is 1 and mean setup times are 0.2, 1, and 5 as given in the table. The adjusted approximation clearly outperforms the CPR approximation except in the cases with the most setup. As would be expected (because this is a heavy-tra5c approximation), both approximations generally improve as increases. It appears that the accuracy of the adjusted approximation improves but subsequently deteriorates as the setup time is increased. The approximations appear to improve as n increases when other parameter values are kept con-
stant (and arrival rates are scaled down to keep load constant). This is probably because an increase in n results in an increase in e4ective load in the system. In general (although not always), we found that the tendency to overestimate waiting time increased as larger and larger percentiles were estimated. In other words the waiting time approximation tends to have a heavier tail than the actual distribution. This same tendency was found in the CPR approximation in an even more pronounced form. Table 3 illustrates this for adjusted approximation for the symmetric 10 queue model with a mean setup time of 1 h per queue. Thus far only symmetric polling models have been considered. The approximation can be expected to perform better for symmetric models than for asymmetric models because the mean waiting time is guaranteed to be exact in the symmetric case, see (2.1). Table 4 shows the percentage error in the 10 queue polling model when the 7rst queue has a fraction p of all arrivals and the remaining fraction (1−p) is spread equally over the other queues. The errors shown are
T.L. Olsen / Operations Research Letters 28 (2001) 113–123
119
Table 3 Percentage error in estimating di4erent percentiles of waiting time
= 0:95 = 0:9 = 0:8 = 0:7 = 0:6 = 0:5
50th perc.
60th perc.
70th perc.
80th perc.
90th perc.
95th perc.
−0:51 −0:76 −1:15 −1:99 −2:76 −3:63
−0:41 −0:54 −0:58 −1:18 −1:71 −2:50
−0:38 −0:34 0:04 −0:30 −0:60 −1:12
−0:36 −0:09 0:71 0:65 0:68 0.50
−0:41 0.09 1.50 1.87 2.32 2.57
−0:52 0.22 2:08 2:67 3.44 4.11
Table 4 Percentage error in estimating the 95th percentile of waiting time si = 0:2
= 0:95 = 0:9 = 0:8 = 0:7 = 0:6 = 0:5
si = 1
si = 5
p = 0:55
p = 0:325
p = 0:1
p = 0:55
p = 0:325
p = 0:1
p = 0:55
p = 0:325
p = 0:1
3.37 5.24 8.92 10.52 10.50 9.22
14.03 14.71 14.61 12.82 10.73 9.36
−1:58 −0:55 −0:74 −1:03 −1:06 −1:54
2.60 4.90 8.24 10.46 11.65 12.18
0.17 1.83 3.77 5.23 11.21 11.32
−0:52 0.22 2.08 2.67 3.44 4.11
1.03 −0:17 0.40 −0:28 −1:22 −2:31
−0:52 −1:74 −1:99 −2:74 −3:59 −4:34
0.07 −1:38 −1:90 −2:23 −3:22 −3:97
the percentage error in estimating the 95th percentile of the waiting time for the 7rst queue. In general, the queues with lighter loads had smaller percentage errors. There are no clear trends that emerge with these errors. Table 5 shows the percentage error in the 10 queue polling model when the mean setup time of the 7rst queue is a fraction q of s, the mean total setup time per cycle, and the other queues have identical setup times. The errors shown are the percentage error in estimating the 95th percentile of the waiting time for the 7rst queue. Again, there are no clear trends that emerge with these errors. 5.2. Positive-queue approximation This section 7rst evaluates our mean waiting time approximation by using the test cases of Bradlow and Byrd [2]. Second, we evaluate our approximation for estimating the percentiles of the waiting time distribution. Bradlow and Byrd tested their approximation with an asymmetric system of 55 queues. In this model, queues 11, 22, 33, 44, and 55 equally contribute to half
the utilization and the remaining 50 queues contribute the other half. Service times are exponential with mean 512 s and setup times are deterministic with mean 300 s. Table 6 shows the mean waiting time and percentage error for the Bradlow and Byrd approximation, our approximation and lastly for Everitt’s approximation for systems with empty-queue setups. We only show results for the average error in the estimated mean waiting time for queues 11, 22, 33, 44, and 55 which tended to be larger than the errors for the lightly loaded queues. Notice that for = 0:95 there is no need to adjust for positive-queue setups as Everitt’s approximation performs very well. However, for systems with less utilization the adjustment can make a huge di4erence (over 1000% for = 0:3). Also notice how our approximation clearly outperforms that of Bradlow and Byrd for moderate to high utilizations. In order to test our approximation for the percentiles of the waiting time we use the same test cases as in Section 5.1. Table 7 uses the symmetric polling model of Table 2. It shows the percentage error when the number of queues changes from 5, to 10, to 20. It is no longer true (as with empty-queue setups) that the
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T.L. Olsen / Operations Research Letters 28 (2001) 113–123
Table 5 Percentage error in estimating the 95th percentile of waiting time s=2
= 0:95 = 0:9 = 0:8 = 0:7 = 0:6 = 0:5
s = 10
s = 50
q = 0:55
q = 0:325
q = 0:1
q = 0:55
q = 0:325
q = 0:1
q = 0:55
q = 0:325
q = 0:1
−0:83 −0:35 0.07 0.77 1.32 1.03
−1:98 −0:14 −0:57 −0:34 0.00 −0:28
−1:58 −0:55 −0:74 −1:03 −1:06 −1:54
−1:48 −0:14 1.19 1.26 1.74 1.58
−0:97 0.42 2.00 2.35 3.05 3.97
−0:52 0.22 2.08 2.67 3.44 4.11
−0:89 −3:27 −5:41 −6:02 −7:24 −8:94
−0:27 −2:05 −2:53 −2:98 −3:84 −4:53
0.07 −1:38 −1:90 −2:23 −3:22 −3:97
Table 6 Estimating the mean waiting time in systems with 55 queues
= 0:3 = 0:5 = 0:65 = 0:8 = 0:85 = 0:9 = 0:95
Bradlow and Byrd
Our approximation
Everitt
8.44 14.25 31.53 168.15 302.20 572.29 1371.67
4.74 13.03 45.88 242.12 422.82 744.98 1580.04
116.49 161.75 229.61 399.21 531.10 794.87 1586.14
(15.47) (−19:06) (−35:99) (−28:81) (−27:86) (−22:95) (−13:03)
(−35:13) (−25:97) (−6:87) (2.50) (0.93) (0.30) (0.19)
(1493.96) (818.98) (366.11) (69.01) (26.78) (7.01) (0.57)
Table 7 Percentage error in estimating the 95th percentile of waiting time si = 0:2 = 0:95 = 0:9 = 0:8 = 0:7 = 0:6 = 0:5
si = 1
si = 5
n=5
n = 10
n = 20
n=5
n = 10
n = 20
n=5
n = 10
n = 20
−2:25 −4:21 −8:69 −14:28 −19:64 −24:87
−2:25 −4:67 −10:92 −17:69 −23:18 −28:02
−2:12 −5:13 −14:11 −21:31 −26:04 −30:04
−0:52 0.07 −0:23 −3:52 −9:60 −11:21
−0:12 0.56 0.43 −2:99 −10:72 −22:87
0.35 0.48 0.47 −2:41 −10:17 −26:91
0.55 0.91 2.87 4.42 5.93 6.78
0.47 0.50 1.52 2.49 3.40 3.89
0.38 0.34 0.78 1.29 1.86 2.21
Table 8 Percentage error in estimating di4erent percentiles of waiting time
= 0:95 = 0:9 = 0:8 = 0:7 = 0:6 = 0:5
Mean
50th perc.
60th perc.
70th perc.
80th perc.
90th perc.
95th perc.
−0:36 −0:47 −0:55 −1:26 −5:18 −17:37
−0:63 −1:37 −1:39 0.57 1.97 −7:06
−0:53 −0:99 −1:32 −1:01 −2:03 −11:11
−0:41 −0:60 −1:06 −2:19 −5:40 −15:04
−0:27 −0:25 −0:70 −2:97 −8:12 −18:67
−0:13 0.19 −0:06 −3:20 −10:22 −21:72
−0:12 0.56 0.43 −2:99 −10:72 −22:87
T.L. Olsen / Operations Research Letters 28 (2001) 113–123
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Table 9 Percentage error in estimating the 95th percentile of waiting time si = 0:2 = 0:95 = 0:9 = 0:8 = 0:7 = 0:6 = 0:5
si = 1
si = 5
p = 0:55
p = 0:325
p = 0:1
p = 0:55
p = 0:325
p = 0:1
p = 0:55
p = 0:325
p = 0:1
2.13 0.40 −1:69 −5:68 −11:58 −18:58
−0:90 −2:78 −8:05 −13:13 −18:22 −23:58
−2:25 −4:67 −10:92 −17:69 −23:18 −28:02
2.86 4.17 3.52 −1:52 −7:79 −12:72
0.15 1.79 1.62 −2:35 −10:11 −19:71
−0:12 0.56 0.43 −2:99 −10:72 −22:87
1.51 2.51 5.07 6.47 6.56 4.85
0.05 0.95 2.68 3.90 4.74 4.38
0.47 0.50 1.52 2.49 3.40 3.89
Table 10 Percentage error in estimating the 95th percentile of waiting time s=2 = 0:95 = 0:9 = 0:8 = 0:7 = 0:6 = 0:5
s = 10
s = 50
q = 0:55
q = 0:325
q = 0:1
q = 0:55
q = 0:325
q = 0:1
q = 0:55
q = 0:325
q = 0:1
−1:67 −3:96 −9:55 −14:41 −18:29 −21:84
−1:82 −4:20 −10:59 −16:82 −22:24 −27:59
−2:25 −4:67 −10:92 −17:69 −23:18 −28:02
−0:45 0.81 0.98 −1:14 −5:87 −11:45
−0:60 0.70 0.80 −2:32 −9:59 −20:47
−0:12 0.56 0.43 −2:99 −10:72 −22:87
−0:28 0.32 1.75 2.67 3.66 4.33
−0:29 0.37 1.83 2.59 3.52 4.13
0.47 0.50 1.52 2.49 3.40 3.89
approximation improves as n increases when other parameter values are kept constant. Although an increase in n still results in an increase in e4ective load in the system it also increases the individual probability that any given queue is empty. The tendency to overestimate large waiting time percentiles still applies when the system is heavily loaded (and hence behaves similarly to a system with empty-queue setups). However, this is not the case for lightly loaded systems as illustrated in Table 8. This table shows the percentage errors for the symmetric 10 queue model with a mean setup time of 1 h per queue. The 7rst column is the percentage error in estimating the mean waiting time. As found previously, the mean waiting time approximation performs very well for moderate to heavy loads. As would be expected, when the mean approximation performs poorly, so does the estimate for the various percentiles. Notice, however, that errors remain of the same order of magnitude as the mean waiting time error as far as the 95th percentile. Tables 9 and 10 evaluate the set of instances of Tables 3 and 4, respectively. As could be expected, when both load and mean setup time are light there is
signi7cant error. Increasing mean setup time tends to improve the estimate but this is probably mostly because this decreases the e4ect of skipping over empty queues. Eventually we would expect to see a degradation in performance with increased setup because this is what we saw for systems with empty-queue setups. When the load is light making the system more asymmetric, improves the estimate but when the load is heavy the reverse is true. 5.3. Tests with industry data set This section tests our approximations using industry data from a fence making operation. The environment that this data arises from is given in Olsen [12]. The data set is highly asymmetric. It consists of 19 di4erent products with an average mean service time of 0:1 h and a range from 0.02 to 0.27. The average mean setup time over all products is 2:09 h with a range from 0.17 to 11.3. The arrival rates at the di4erent queues are very diverse and range from 0.17% of arrivals to 29.9%. The data set contains the relative arrival rates along with the mean service and setup times but not the system utilization or the actual setup and service time
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Table 11 Percentage error in industry data set CPR Approx.
Adjusted approx
Mean abs. s = 39:7 s = 39:7 s = 19:9 s = 19:9 s = 19:9 s = 19:9
exp exp exp exp det. det.
eq pq eq pq eq pq
0.61 4.28 1.25 18.92 0.58 18.18
95th perc. avg. 0.61 −4:28 1.25 18.92 0.58 18.18
Mean
95th perc.
abs.
avg.
abs.
avg.
abs.
avg.
2.57 17.53 4.90 16.49 2.97 14.06
2.57 −17:53 4.90 16.49 2.97 14.06
0.15 1.37 0.18 2.75 0.17 2.76
0.01 1.34 0.04 2.08 −0:02 2.07
1.03 13.46 1.67 1.49 1.27 2.37
1.03 −13:46 1.67 −1:49 1.27 −2:37
distributions. Table 11 shows the percentage errors for this data set for the light utilization of = 0:5. As would be expected, higher utilizations tended to have smaller errors and are not shown. Table 11 gives percentage errors for six di4erent scenarios. The columns labeled “avg.” correspond to an averaging of the errors over all queues, while the columns labeled “abs.” correspond to an averaging of the absolute errors over all queues. The 7rst four rows use exponential service and setup times, while the last two rows use deterministic service and setup time. The odd rows are for systems with empty-queue setups, while the even rows correspond systems with positive queue setups. The 7rst two rows have setup times as given above, while the last four rows test the e4ect of scaling down setup times by a factor of 2. The approximations can be seen to perform somewhat comparably to the more symmetric examples from the previous sections. In conclusion, it can be seen that both the empty queue and positive queue approximations perform very well. While maintaining the closed form characteristics of the CPR approximation, the empty-queue approximation provides a signi7cantly better estimate for the waiting time distribution. The mean waiting time approximation for positive-queue setups is slightly less complicated than that of Bradlow and Byrd [2] and yet signi7cantly outperforms their approximation in moderate to heavy load. The estimate for the waiting time distribution provides estimates that are certainly of the correct order of magnitude and are sometimes very good. Because the approximation arises from heavy tra5c it performs best in moderate to heavy loads.
Acknowledgements I would like to thank Marty Reiman for sharing his results prior to their publication and for the insights he gave on this model. I am grateful to Mike Harrison and Peter Glynn for their advice on this topic. I would like to thank Stevan Vlaovic for his help with some of the computational results. Finally, thanks go to the anonymous referee whose comments greatly improved this paper. References [1] F. Baccelli, P. BrSemaud, Elements of Queueing Theory, Springer, Berlin, 1991. [2] H.S. Bradlow, H.F. Byrd, Mean waiting time evaluation of packet switches for centrally controlled pbx’s, Performance Evaluation 7 (4) (1987) 309–327. [3] G.L. Choudhury, W. Whitt, Computing transient and steady-state distributions in polling models by numerical transform inversion, Working paper, AT&T Bell Laboratories, Holmdel, NJ, 1994. [4] E. Co4man Jr., A. Puhalskii, M. Reiman, Polling systems in heavy-tra5c: a Bessel process limit, Math. Oper. Res. 23 (1998) 257–304. [5] M. Eisenberg, The polling system with a stopping server, Queueing Systems Theory Appl. 18(3– 4) (1994) 387–431. [6] D. Everitt, Simple approximations for token rings, IEEE Trans. Commun. COM-34(7) (1986) 719–721. [7] A. Federgruen, Z. Katalan, Approximating queue size and waiting time distributions in general polling systems, Queueing Systems Theory Appl. 18 (1994) 353–386. [8] A. Federgruen, Z. Katalan, The stochastic economic lot scheduling problem: cyclical base stock policies with idle times, Manage. Sci. 42 (6) (1996) 783–796. [9] M.J. Ferguson, Mean waiting time for a token ring with nodal dependent overheads, in: M. Akiyama (Ed.), Teletra5c Issues in an Advanced Information Society, ITC-11, Elsevier
T.L. Olsen / Operations Research Letters 28 (2001) 113–123 Science Publishers B.V., North-Holland, Amsterdam, 1985, pp. 634–640. [10] D. Gupta, M.M. Srinivasan, Polling systems with state-dependent setup times, Queueing Systems Theory Appl. 22 (1996) 403–423. [11] T.M. Lennon, Response-time approximations for multi-server polling models, with manufacturing applications, Ph.D. Thesis, Stanford University, 1994. [12] T.L. Olsen, A practical scheduling method for multiclass production systems with setups, Manage. Sci. 45(1) (1999) 116–130.
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[13] H. Takagi, Analysis of Polling Systems, MIT Press, Cambridge, MA, 1986. [14] H. Takagi (Ed.), Stochastic Analysis of Computer and Communication Systems, Elsevier Science Publishers B.V., North-Holland, Amsterdam, 1990 (Chapter 1). [15] J.H. Dshalalow (Ed.), Frontiers in Queueing, CRC Press, Inc, Boca Raton, FL, 1997 (Chapter 5).
www.elsevier.com/locate/dsw
Approximations for the waiting time distribution in polling models with and without state-dependent setups Tava Lennon Olsen ∗ John M. Olin School of Business, Washington University in St. Louis, Campus Box 1133, St. Louis, MO 63130-4899, USA Received 1 December 1997; received in revised form 1 April 1999; accepted 1 January 2001
Abstract A polling model is a queueing model where many job classes share a single server and a setup time is incurred whenever the server changes class. Polling models are applicable to many computing, telecommunications, and manufacturing environments. The scheduling method considered in this paper is a common policy known as cyclic, serve to exhaustion (CSE). Recently, Co4man, Puhalskii and Reiman (CPR) have developed a heavy-tra5c approximation for the waiting time distribution in a CSE polling model. This paper presents three new approximations. Firstly, the CPR approximation and the traditional (non-heavy-tra5c) polling model literature are combined to obtain a re7nement of the CPR approximation. This re7nement is much more accurate under conditions of moderate loading. Next, an approximation is made for the distribution of the number of jobs present in a queue upon it being polled. Lastly, the previous two approximations are combined to form an approximation for the waiting time distribution when setups are not performed for queues containing no jobs. A simulation study is undertaken c 2001 Elsevier Science B.V. All rights reserved. to evaluate these three approximations.
A polling model is a queueing model where n job classes share a single server and a setup time is incurred whenever the server changes class. Polling models are applicable to many computing, telecommunications, and manufacturing environments. In particular, local area networks and token rings are often modeled using polling models. An excellent survey of polling models may be found in both [14,15] which serve as a supplements to [13]. The scheduling method considered in this paper is a common policy known as cyclic, serve-to-exhaustion. Under this policy, arrivals of jobs of each of the n ∗
Tel.: +1-314-935-4732; fax: +1-314-935-6359. E-mail address: [email protected] (T.L. Olsen).
job classes are held in a queue dedicated to that class. These queues are arranged in a cycle. When the server becomes free it advances to the next class in the cycle and begins a setup. All jobs of the server’s current class are served (including those that arrive during the current service period) before the server moves on to the next class. Jobs within a job class are served on a 7rst-in-7rst-out (FIFO) basis. This paper presents waiting time distribution approximations for two cyclic serve-to-exhaustion submodels. The 7rst submodel is the one traditionally studied where setups are incurred regardless of whether or not jobs are waiting in the current class. We will refer to this model as having empty-queue setups. The second submodel is the one where the
c 2001 Elsevier Science B.V. All rights reserved. 0167-6377/01/$ - see front matter PII: S 0 1 6 7 - 6 3 7 7 ( 0 1 ) 0 0 0 5 8 - X
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server can ascertain whether there are jobs present in a queue and will not set up for that queue if it is empty. We will refer to this model as having positive-queue setups. In this model the server idles at its most recently served queue if the entire system is empty. Clearly, which submodel is applicable depends on the speci7c application being studied. Recently, Co4man, Puhalskii, and Reiman (CPR) have developed a heavy-tra5c approximation for the waiting time distribution in cyclic, serve-to-exhaustion polling models with empty-queue setups [4]. The CPR approximation is the 7rst explicit (closed-form) approximation for the waiting time distribution. Others have provided numerical algorithms for approximating the waiting time distribution (see [7] or [3]), but there is much value in having an explicit approximation, even if it is less exact. Explicit approximations lead to insights into how the system depends on di4erent parameters that are not possible with numerical algorithms. They are also easily implementable and useful for “back of the envelope” calculations. Furthermore, the heavy-tra5c approximation depends only on the 7rst two moments of the input distributions and not on their speci7c distributions. Unfortunately, the CPR approximation is extremely inaccurate at light to moderate loads. This paper presents a re7nement of the CPR approximation that is more accurate (and is still explicit). A search of the literature has revealed three papers which derive approximations for the mean waiting time in polling models with positive-queue setups, namely Ferguson [9], Bradlow and Byrd [2], and Gupta and Srinivasan [10]. In addition, Eisenberg (see, for example, previous work [5]) has derived an exact algorithm for 7nding the mean waiting time in polling systems but to our knowledge this paper has not yet been completed. The paper most related to our work is [2] which will be discussed further in Section 4. We have found no previous results for approximating the waiting time distribution (as opposed to the mean only) in systems with positive-queue setups. This paper is organized as follows. The notation used is introduced in Section 1. Section 2 presents the CPR heavy-tra5c approximation and re7nes it using non-heavy-tra5c theory. An approximation for the queue-length distribution for systems with empty-queue setups is presented in Section 3. This distribution is used in Section 4 to mod-
ify the empty-queue approximation to systems with positive-queue setups. Finally, Section 5 presents a simulation study of both waiting time approximations.
1. Notation Polling models with empty-queue and positive-queue setups have been described in the previous section. This section outlines the notation used and describes the assumptions made. Jobs are assumed to arrive at each queue i; 1 6 i 6 n, according to a Poisson process of rate i . Arrival processes are assumed to be mutually indepenn dent. Total arrival rate for the system is = i=1 i . Service times at queue i have mean bi and variance i2 . Utilization at queue i; i , equals n i bi and system utilization (excluding setups) = i=1 i . For stability we assume that ¡ 1 (see [13]). The generic setup time at queue i; 1 6 i 6 n, is represented by Si and may have a positive probability of being zero in systems with positive-queue setups. The mean and second moment of the setup time at queue i, given that a setup occurs, are si and si(2) , respectively. The only assumptions made regarding the distribution of the setup and service times are that the means and variances are 7nite and that all distributions are independent. We de7ne a cycle at queue i as the time between two successive arrivals of the server to queue i. Total steady-state nsetup time around a cycle is represented by S (= i=1 Si ). A cycle is therefore made up of setup time S, the time to serve all queues, and possibly some idling time when the whole system is empty. For empty-queue setups we let s = E[S] = n the case of (2) s and s = E[S 2 ]. i i=1 We de7ne intervisit time as the time that elapses from the server’s leaving queue i to its next commencement of service at queue i. Thus, the intervisit time for queue i equals the cycle time for queue i minus the time spent serving queue i. Steady-state intervisit time is represented by Ii . It is well known (see [13]) that, for systems with empty-queue setups, E[Ii ]=(1− i )s=(1−). The steady-state time-averaged intervisit time Ii∗ for queue i (see, for example [1]) is de7ned such that if Ii has p.d.f. fIi (x) then the time-averaged random variable Ii∗ has p.d.f. fIi∗ (x) = xfIi (x)=E[Ii ]:
(1.1)
T.L. Olsen / Operations Research Letters 28 (2001) 113–123
Steady-state waiting time at queue i; Wi , is de7ned as the time from a job’s arrival at queue i until it commences service n in steady state. De7ne 2 = i=1 i (b2i + i2 ). In our approximation, 2 will be the only place that the higher order moments of the service time distribution appear and may be thought of as a variance term. The last system parameters which need to be introduced are and ∗ which are essentially measures of the imbalance in the system. They are largest when the loads (i ’s) are identical at all queues. De7ne =
n−1 n i=1 j=i+1
i j =
n 1 i ( − i ) 2 i=1
and ∗ =
n
1 1 i (1 − i ) = (1 − ) + : 2 2
(1.2)
i=1
The 100pth percentile for the waiting time distribution at queue i is de7ned as the number Pi (p) such that Prob(Wi 6 Pi (p)) = p. Lastly, we de7ne a symmetric system as one where all queues have identical arrival rates, setup time distributions, and service time distributions. Conversely, an asymmetric system allows for di4erences among the queues. 2. Refined heavy-traffic approximation Using heavy-tra5c theory, CPR [4] have developed an approximation for the waiting time distribution for jobs in cyclic serve-to-exhaustion polling models with empty-queue setups. Some of the results cited here are not made explicitly in [4] (although they follow naturally) and further details may be found in Lennon [11]. Recall that Ii∗ is the steady-state time-averaged intervisit time at queue i; 1 6 i 6 n. The work of CPR shows that, for all i; Ii∗ may be approximated by a gamma random variable with scale parameter i = 2(1 − )=(2 (1 − i )) and shape parameter = 2s=2 + 1. Let U be a uniform[0,1] random variable which is independent of Ii∗ . Then the CPR steady-state waiting time approximation at queue i, Wi , is UIi∗ . The CPR heavy-tra5c approximation for the steady-state waiting time distribution can be used as an approximation for the waiting time distribution
115
in non-heavy-tra5c systems. However, it tends to perform poorly in moderate tra5c. We will therefore adjust the CPR approximation so that its mean corresponds to known explicit results. Probably the most accurate explicit approximation for the mean waiting time in asymmetric systems has been derived by Everitt [6]. Everitt’s approximation is exact for symmetric systems and states that E[S 2 ] E[S] 2 1 − i + + : E[Wi ] ≈ 2∗ 2(1 − ) 2E[S] (1 − ) (2.1) It is straightforward to show that the di4erence between Everitt’s mean approximation and the CPR mean waiting time approximation converges to zero as increases to 1. For the re7ned approximation we will assume that the waiting time distribution is still of the form of a gamma random variable multiplied by a uniform[0,1] random variable. This implies that, for queue i, d
Wi ≈ UXia where Xia is a gamma random variable with scale parameter ia and shape parameter ia . The parameters ia and ia will be chosen so that the mean corresponds to Everitt’s approximation. Namely, such that, s(2) s ia 2 1 − i + + : = 2ia 2∗ 2(1 − ) 2s (1 − ) Clearly, there are in7nitely many di4erent ways the parameters ia and ia could be chosen to satisfy the above condition. We will only consider parameters that are correct under heavy tra5c. In other words, parameters such that ia =i → 1 and ia = → 1 as → 1. Note that, because the CPR mean is asymptotically identical to Everitt’s as ↑ 1, these are not independent requirements. Such an adjustment may be achieved by comparing the formulas for Everitt’s and CPR’s approximations for mean waiting time. CPR’s approximation for the mean waiting time is 2 1 − i s wi = : + 2 2(1 − ) (1 − ) Comparing this with (2.1) suggests that a natural adjusted approximation for ia is 2s=(2 ) + 1. Thus, the shape parameter remains identical except that the extra factor of that appears in Everitt’s mean approximation gets carried over to the denominator of
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the shape parameter. If we let pi = i = then n pi (1 − pi )s ia = ni=1 + 1: ( i=1 pi (bi + i2 =bi ) Note that does not appear in this equation, but if is scaled so that the pi and service times remain constant then ia is also constant in . The given ia implies that 2 + 2s 2∗ (1 − ) ia = 2 : (1 − i ) 2 + 2s + (1 − )s(2) =s Notice that ∗ ia =i =
2 + 2s 2 + 2s + (1 − )s(2) =s
→1
as ↑ 1 as desired. We tried a number of other adjustments (which are omitted due to space considerations) but this one was the most robust. 3. Queue-length distribution This section develops an approximation for the distribution of the number of jobs in a queue when the server commences service for that queue in a polling model with empty-queue setups. Later, in order to obtain an approximation for the waiting time distribution for polling models with positive-queue setups, the probability that queue i is empty when the server arrives must be calculated. We will approximate this probability in systems with positive-queue setups by the result for systems with empty-queue setups. The queue-length distribution may also be interesting in its own right. In particular, it may be used heuristically to design bu4er sizes in local area networks. In production applications it may be used to calculate optimal base-stock levels in make-to-stock systems as described in [8]. Let Ni be the steady-state number of jobs at queue i when the server commences service (or moves on if Ni = 0) after having setup for queue i. Then Ni is the number of jobs that have arrived during the last intervisit time for queue i. The memoryless property for Poisson arrivals combined with the fact that queue i is empty when the server leaves it, implies that the steady-state intervisit time Ii is independent of the
arrival stream to queue i. Thus, −i Ii (i Ii )x e P(Ni = x) = E x! ∞ −i y e (i y)x = fIi (y) dy: (3.1) x! 0 Hence, an approximate distribution for Ni may be obtained from an approximate distribution of Ii . Eq. (1.1) shows that the distribution of Ii may be obtained from the distribution of the time-averaged intervisit time Ii∗ . Also, recall from Section 2 that the heavy-tra5c approximation for the distribution of Ii∗ is a gamma random variable with scale parameter i and shape parameter . Thus, from (3.1), E[Ii ]i P(Ni = x) ≈ ∞ −(i +i )y e (i y)x i (i y)−1 × dy: x! () 0 (3.2) The terms may be factored so that the quantity within the integral is a gamma density and hence integrates to 1. Furthermore, E[Ii ] = (1 − i )s=(1 − ) = =i . Thus, straight-forward algebra yields an approximation for the distribution of Ni to be as follows: x i (x + ) i P(Ni = x) ≈ : (3.3) i + i i + i x! () The adjusted parameters ia and ia may, of course, be used in this approximation. If is an integer, then this is the negative binomial probability distribution. It is interesting to note that a mixture of negative binomials appears in the numerical procedure for approximating queue length as described by Federgruen and Katalan [7]. 4. Polling models with positive-queue setups This section uses the theory of the previous two sections to derive an approximation for the waiting time (both mean and distribution) in polling models with positive-queue setups. The approximation requires the numerical solution of a set of equations. Thus far all approximations in this paper have been explicit. If one requires an explicit formula in the positive-queue setups case, then a crude approximation is also presented here.
T.L. Olsen / Operations Research Letters 28 (2001) 113–123
To obtain an approximation for the mean waiting time in systems with positive-queue setups we will use Everitt’s approximation (2.1) but calculate S as the total amount of setup in a cycle with positive-queue setups. Thus, we need only obtain estimates for E[S] and E[S 2 ] to form an approximation for mean waiting time. We de7ne se = E[S] for systems with positive-queue setups. If ei is the probability that queue i is not empty when the server arrives at it, then se =
n
ei s i :
(4.1)
i=1
We will discuss approximating ei later in this section. Observe that
2 n E[S 2 ] = E Si i=1
=E
n
Si2
n +E Si Sj :
i=1
i=1
j=i
Now, in general, Si and Sj are not independent. However, for systems with empty queue setups they are independent. We shall form our approximation by assuming negligible dependence of Si and Sj . It should be noted that this assumption only a4ects the term E[S 2 ]=(2E[S]) and thus becomes negligible as ↑ 1. This assumption implies that 2
E[S ] ≈
n
ei si(2)
+
i=1
=
n i=1
n i=1
ei si(2)
−
n
e i si
e j sj
j=i
(ei si )2 + se2 :
(4.2)
117
adjusted empty-queue approximation we have that ia =
2∗ (1 − ) 2 (1 − i ) 2 + 2E[S] × 2 + 2E[S] + (1 − )E[S 2 ]=E[S] (4.4)
and ia =
2E[S] + 1: 2
(4.5)
Thus, Eq. (4.1)–(4.5) may be solved numerically to form approximations for E[S], E[S 2 ]; ia ; ia , and ei for systems with positive queue setups. In theory we have presented a system of 2n + 3 equations in 2n + 3 unknowns which may be somewhat di5cult to solve. However, in practice, we found that a simple 7xed point iterative algorithm always converged. This algorithm 7rst calculates the empty-queue ia and ia . It uses these to calculate the ei from Eq. (4.3) then substitutes the ei back into Eqs. (4.4) and (4.5). These substitutions are continued until convergence is obtained. Notice that, with approximations for ia and ia , we have an approximation for the whole waiting time distribution not just the mean. It should be noted that this method of approximating polling models with positive-queue setups is somewhat close to that used by Bradlow and Byrd [2]. They describe the probability of queue i being empty as ei ≈ E[e−i Ci ], where cycle time is assumed to be exponentially distributed. It is unclear why they use cycle time as opposed to intervisit time in the above equation. Section 5 shows that our approximation compares very favorably to the Bradlow and Byrd approximation.
i=1
Thus it remains to approximate ei . We use the results of the previous section to approximate the queue-length distribution in systems with positive-queue setups. Thus, from (3.3) with x = 0, i −i Ii 1 − ei = P(Ni = 0) = Ee ≈ : (4.3) i + i This completes a crude but explicit approximation for the mean waiting time. A more re7ned approximation may be formed by noting the interdependence of i ; , and ei . Using our
5. Testing the accuracy of the approximations This section examines the accuracy of our empty-queue and positive-queue waiting time approximations for di4erent polling models. The approximations are compared to the results of a simulation program written in C. The simulation was run for at least 10,000,000 time units and sometimes longer for higher utilizations. Also, if mean service time was greater than one time unit, then the run length was scaled up similarly.
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T.L. Olsen / Operations Research Letters 28 (2001) 113–123
Table 1 Percentage error in estimating the 95th percentile of waiting time in CPR approximation si = 0:2
= 0:95 = 0:9 = 0:8 = 0:7 = 0:6 = 0:5
si = 1
si = 5
n=5
n = 10
n = 20
n=5
n = 10
n = 20
n=5
n = 10
n = 20
6.61 16.11 37.92 65.62 104.10 157.54
4.48 12.39 28.38 48.83 76.60 114.56
4.42 9.36 20.36 35.28 53.95 79.79
2.89 8.98 19.76 33.06 49.19 72.21
1.67 4.83 12.40 20.00 29.83 42.75
0.12 3.04 7.29 11.79 17.93 25.35
0.26 −0:64 0.86 1.80 4.20 7.29
0.37 −0:73 −0:33 0.63 1.45 3.30
−0:03 −0:90 −0:35 0.04 0.33 1.50
Table 2 Percentage error in estimating the 95th percentile of waiting time in adjusted approximation si = 0:2
= 0:95 = 0:9 = 0:8 = 0:7 = 0:6 = 0:5
si = 1
si = 5
n=5
n = 10
n = 20
n=5
n = 10
n = 20
n=5
n = 10
n = 20
−1:78 −1:77 −2:75 −4:15 −5:74 −8:10
−1:58 −0:55 −0:74 −1:03 −1:06 −1:54
0.38 0.55 0.74 1.64 2.15 3.37
−0:68 1.14 2.27 3.35 4.15 5.31
−0:52 0.22 2.08 2.67 3.44 4.11
−0:88 0.37 1.22 1.76 2.40 3.07
−0:20 −1:62 −1:59 −2:54 −3:40 −4:43
0.07 −1:38 −1:90 −2:23 −3:22 −3:97
−0:03 −1:33 −1:33 −1:70 −2:40 −2:92
5.1. Empty-queue approximation Tables 1 and 2 show the percentage error in the 95th percentile of the waiting time for the CPR and the adjusted approximation, respectively. These tables use a symmetric 10 queue polling model when the number of queues changes from 5, to 10, to 20. Percentage error is de7ned as 100(Estimate − Simulated value)=Simulated value. Setup and service times are taken to be exponentially distributed. Mean service time is 1 and mean setup times are 0.2, 1, and 5 as given in the table. The adjusted approximation clearly outperforms the CPR approximation except in the cases with the most setup. As would be expected (because this is a heavy-tra5c approximation), both approximations generally improve as increases. It appears that the accuracy of the adjusted approximation improves but subsequently deteriorates as the setup time is increased. The approximations appear to improve as n increases when other parameter values are kept con-
stant (and arrival rates are scaled down to keep load constant). This is probably because an increase in n results in an increase in e4ective load in the system. In general (although not always), we found that the tendency to overestimate waiting time increased as larger and larger percentiles were estimated. In other words the waiting time approximation tends to have a heavier tail than the actual distribution. This same tendency was found in the CPR approximation in an even more pronounced form. Table 3 illustrates this for adjusted approximation for the symmetric 10 queue model with a mean setup time of 1 h per queue. Thus far only symmetric polling models have been considered. The approximation can be expected to perform better for symmetric models than for asymmetric models because the mean waiting time is guaranteed to be exact in the symmetric case, see (2.1). Table 4 shows the percentage error in the 10 queue polling model when the 7rst queue has a fraction p of all arrivals and the remaining fraction (1−p) is spread equally over the other queues. The errors shown are
T.L. Olsen / Operations Research Letters 28 (2001) 113–123
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Table 3 Percentage error in estimating di4erent percentiles of waiting time
= 0:95 = 0:9 = 0:8 = 0:7 = 0:6 = 0:5
50th perc.
60th perc.
70th perc.
80th perc.
90th perc.
95th perc.
−0:51 −0:76 −1:15 −1:99 −2:76 −3:63
−0:41 −0:54 −0:58 −1:18 −1:71 −2:50
−0:38 −0:34 0:04 −0:30 −0:60 −1:12
−0:36 −0:09 0:71 0:65 0:68 0.50
−0:41 0.09 1.50 1.87 2.32 2.57
−0:52 0.22 2:08 2:67 3.44 4.11
Table 4 Percentage error in estimating the 95th percentile of waiting time si = 0:2
= 0:95 = 0:9 = 0:8 = 0:7 = 0:6 = 0:5
si = 1
si = 5
p = 0:55
p = 0:325
p = 0:1
p = 0:55
p = 0:325
p = 0:1
p = 0:55
p = 0:325
p = 0:1
3.37 5.24 8.92 10.52 10.50 9.22
14.03 14.71 14.61 12.82 10.73 9.36
−1:58 −0:55 −0:74 −1:03 −1:06 −1:54
2.60 4.90 8.24 10.46 11.65 12.18
0.17 1.83 3.77 5.23 11.21 11.32
−0:52 0.22 2.08 2.67 3.44 4.11
1.03 −0:17 0.40 −0:28 −1:22 −2:31
−0:52 −1:74 −1:99 −2:74 −3:59 −4:34
0.07 −1:38 −1:90 −2:23 −3:22 −3:97
the percentage error in estimating the 95th percentile of the waiting time for the 7rst queue. In general, the queues with lighter loads had smaller percentage errors. There are no clear trends that emerge with these errors. Table 5 shows the percentage error in the 10 queue polling model when the mean setup time of the 7rst queue is a fraction q of s, the mean total setup time per cycle, and the other queues have identical setup times. The errors shown are the percentage error in estimating the 95th percentile of the waiting time for the 7rst queue. Again, there are no clear trends that emerge with these errors. 5.2. Positive-queue approximation This section 7rst evaluates our mean waiting time approximation by using the test cases of Bradlow and Byrd [2]. Second, we evaluate our approximation for estimating the percentiles of the waiting time distribution. Bradlow and Byrd tested their approximation with an asymmetric system of 55 queues. In this model, queues 11, 22, 33, 44, and 55 equally contribute to half
the utilization and the remaining 50 queues contribute the other half. Service times are exponential with mean 512 s and setup times are deterministic with mean 300 s. Table 6 shows the mean waiting time and percentage error for the Bradlow and Byrd approximation, our approximation and lastly for Everitt’s approximation for systems with empty-queue setups. We only show results for the average error in the estimated mean waiting time for queues 11, 22, 33, 44, and 55 which tended to be larger than the errors for the lightly loaded queues. Notice that for = 0:95 there is no need to adjust for positive-queue setups as Everitt’s approximation performs very well. However, for systems with less utilization the adjustment can make a huge di4erence (over 1000% for = 0:3). Also notice how our approximation clearly outperforms that of Bradlow and Byrd for moderate to high utilizations. In order to test our approximation for the percentiles of the waiting time we use the same test cases as in Section 5.1. Table 7 uses the symmetric polling model of Table 2. It shows the percentage error when the number of queues changes from 5, to 10, to 20. It is no longer true (as with empty-queue setups) that the
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T.L. Olsen / Operations Research Letters 28 (2001) 113–123
Table 5 Percentage error in estimating the 95th percentile of waiting time s=2
= 0:95 = 0:9 = 0:8 = 0:7 = 0:6 = 0:5
s = 10
s = 50
q = 0:55
q = 0:325
q = 0:1
q = 0:55
q = 0:325
q = 0:1
q = 0:55
q = 0:325
q = 0:1
−0:83 −0:35 0.07 0.77 1.32 1.03
−1:98 −0:14 −0:57 −0:34 0.00 −0:28
−1:58 −0:55 −0:74 −1:03 −1:06 −1:54
−1:48 −0:14 1.19 1.26 1.74 1.58
−0:97 0.42 2.00 2.35 3.05 3.97
−0:52 0.22 2.08 2.67 3.44 4.11
−0:89 −3:27 −5:41 −6:02 −7:24 −8:94
−0:27 −2:05 −2:53 −2:98 −3:84 −4:53
0.07 −1:38 −1:90 −2:23 −3:22 −3:97
Table 6 Estimating the mean waiting time in systems with 55 queues
= 0:3 = 0:5 = 0:65 = 0:8 = 0:85 = 0:9 = 0:95
Bradlow and Byrd
Our approximation
Everitt
8.44 14.25 31.53 168.15 302.20 572.29 1371.67
4.74 13.03 45.88 242.12 422.82 744.98 1580.04
116.49 161.75 229.61 399.21 531.10 794.87 1586.14
(15.47) (−19:06) (−35:99) (−28:81) (−27:86) (−22:95) (−13:03)
(−35:13) (−25:97) (−6:87) (2.50) (0.93) (0.30) (0.19)
(1493.96) (818.98) (366.11) (69.01) (26.78) (7.01) (0.57)
Table 7 Percentage error in estimating the 95th percentile of waiting time si = 0:2 = 0:95 = 0:9 = 0:8 = 0:7 = 0:6 = 0:5
si = 1
si = 5
n=5
n = 10
n = 20
n=5
n = 10
n = 20
n=5
n = 10
n = 20
−2:25 −4:21 −8:69 −14:28 −19:64 −24:87
−2:25 −4:67 −10:92 −17:69 −23:18 −28:02
−2:12 −5:13 −14:11 −21:31 −26:04 −30:04
−0:52 0.07 −0:23 −3:52 −9:60 −11:21
−0:12 0.56 0.43 −2:99 −10:72 −22:87
0.35 0.48 0.47 −2:41 −10:17 −26:91
0.55 0.91 2.87 4.42 5.93 6.78
0.47 0.50 1.52 2.49 3.40 3.89
0.38 0.34 0.78 1.29 1.86 2.21
Table 8 Percentage error in estimating di4erent percentiles of waiting time
= 0:95 = 0:9 = 0:8 = 0:7 = 0:6 = 0:5
Mean
50th perc.
60th perc.
70th perc.
80th perc.
90th perc.
95th perc.
−0:36 −0:47 −0:55 −1:26 −5:18 −17:37
−0:63 −1:37 −1:39 0.57 1.97 −7:06
−0:53 −0:99 −1:32 −1:01 −2:03 −11:11
−0:41 −0:60 −1:06 −2:19 −5:40 −15:04
−0:27 −0:25 −0:70 −2:97 −8:12 −18:67
−0:13 0.19 −0:06 −3:20 −10:22 −21:72
−0:12 0.56 0.43 −2:99 −10:72 −22:87
T.L. Olsen / Operations Research Letters 28 (2001) 113–123
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Table 9 Percentage error in estimating the 95th percentile of waiting time si = 0:2 = 0:95 = 0:9 = 0:8 = 0:7 = 0:6 = 0:5
si = 1
si = 5
p = 0:55
p = 0:325
p = 0:1
p = 0:55
p = 0:325
p = 0:1
p = 0:55
p = 0:325
p = 0:1
2.13 0.40 −1:69 −5:68 −11:58 −18:58
−0:90 −2:78 −8:05 −13:13 −18:22 −23:58
−2:25 −4:67 −10:92 −17:69 −23:18 −28:02
2.86 4.17 3.52 −1:52 −7:79 −12:72
0.15 1.79 1.62 −2:35 −10:11 −19:71
−0:12 0.56 0.43 −2:99 −10:72 −22:87
1.51 2.51 5.07 6.47 6.56 4.85
0.05 0.95 2.68 3.90 4.74 4.38
0.47 0.50 1.52 2.49 3.40 3.89
Table 10 Percentage error in estimating the 95th percentile of waiting time s=2 = 0:95 = 0:9 = 0:8 = 0:7 = 0:6 = 0:5
s = 10
s = 50
q = 0:55
q = 0:325
q = 0:1
q = 0:55
q = 0:325
q = 0:1
q = 0:55
q = 0:325
q = 0:1
−1:67 −3:96 −9:55 −14:41 −18:29 −21:84
−1:82 −4:20 −10:59 −16:82 −22:24 −27:59
−2:25 −4:67 −10:92 −17:69 −23:18 −28:02
−0:45 0.81 0.98 −1:14 −5:87 −11:45
−0:60 0.70 0.80 −2:32 −9:59 −20:47
−0:12 0.56 0.43 −2:99 −10:72 −22:87
−0:28 0.32 1.75 2.67 3.66 4.33
−0:29 0.37 1.83 2.59 3.52 4.13
0.47 0.50 1.52 2.49 3.40 3.89
approximation improves as n increases when other parameter values are kept constant. Although an increase in n still results in an increase in e4ective load in the system it also increases the individual probability that any given queue is empty. The tendency to overestimate large waiting time percentiles still applies when the system is heavily loaded (and hence behaves similarly to a system with empty-queue setups). However, this is not the case for lightly loaded systems as illustrated in Table 8. This table shows the percentage errors for the symmetric 10 queue model with a mean setup time of 1 h per queue. The 7rst column is the percentage error in estimating the mean waiting time. As found previously, the mean waiting time approximation performs very well for moderate to heavy loads. As would be expected, when the mean approximation performs poorly, so does the estimate for the various percentiles. Notice, however, that errors remain of the same order of magnitude as the mean waiting time error as far as the 95th percentile. Tables 9 and 10 evaluate the set of instances of Tables 3 and 4, respectively. As could be expected, when both load and mean setup time are light there is
signi7cant error. Increasing mean setup time tends to improve the estimate but this is probably mostly because this decreases the e4ect of skipping over empty queues. Eventually we would expect to see a degradation in performance with increased setup because this is what we saw for systems with empty-queue setups. When the load is light making the system more asymmetric, improves the estimate but when the load is heavy the reverse is true. 5.3. Tests with industry data set This section tests our approximations using industry data from a fence making operation. The environment that this data arises from is given in Olsen [12]. The data set is highly asymmetric. It consists of 19 di4erent products with an average mean service time of 0:1 h and a range from 0.02 to 0.27. The average mean setup time over all products is 2:09 h with a range from 0.17 to 11.3. The arrival rates at the di4erent queues are very diverse and range from 0.17% of arrivals to 29.9%. The data set contains the relative arrival rates along with the mean service and setup times but not the system utilization or the actual setup and service time
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T.L. Olsen / Operations Research Letters 28 (2001) 113–123
Table 11 Percentage error in industry data set CPR Approx.
Adjusted approx
Mean abs. s = 39:7 s = 39:7 s = 19:9 s = 19:9 s = 19:9 s = 19:9
exp exp exp exp det. det.
eq pq eq pq eq pq
0.61 4.28 1.25 18.92 0.58 18.18
95th perc. avg. 0.61 −4:28 1.25 18.92 0.58 18.18
Mean
95th perc.
abs.
avg.
abs.
avg.
abs.
avg.
2.57 17.53 4.90 16.49 2.97 14.06
2.57 −17:53 4.90 16.49 2.97 14.06
0.15 1.37 0.18 2.75 0.17 2.76
0.01 1.34 0.04 2.08 −0:02 2.07
1.03 13.46 1.67 1.49 1.27 2.37
1.03 −13:46 1.67 −1:49 1.27 −2:37
distributions. Table 11 shows the percentage errors for this data set for the light utilization of = 0:5. As would be expected, higher utilizations tended to have smaller errors and are not shown. Table 11 gives percentage errors for six di4erent scenarios. The columns labeled “avg.” correspond to an averaging of the errors over all queues, while the columns labeled “abs.” correspond to an averaging of the absolute errors over all queues. The 7rst four rows use exponential service and setup times, while the last two rows use deterministic service and setup time. The odd rows are for systems with empty-queue setups, while the even rows correspond systems with positive queue setups. The 7rst two rows have setup times as given above, while the last four rows test the e4ect of scaling down setup times by a factor of 2. The approximations can be seen to perform somewhat comparably to the more symmetric examples from the previous sections. In conclusion, it can be seen that both the empty queue and positive queue approximations perform very well. While maintaining the closed form characteristics of the CPR approximation, the empty-queue approximation provides a signi7cantly better estimate for the waiting time distribution. The mean waiting time approximation for positive-queue setups is slightly less complicated than that of Bradlow and Byrd [2] and yet signi7cantly outperforms their approximation in moderate to heavy load. The estimate for the waiting time distribution provides estimates that are certainly of the correct order of magnitude and are sometimes very good. Because the approximation arises from heavy tra5c it performs best in moderate to heavy loads.
Acknowledgements I would like to thank Marty Reiman for sharing his results prior to their publication and for the insights he gave on this model. I am grateful to Mike Harrison and Peter Glynn for their advice on this topic. I would like to thank Stevan Vlaovic for his help with some of the computational results. Finally, thanks go to the anonymous referee whose comments greatly improved this paper. References [1] F. Baccelli, P. BrSemaud, Elements of Queueing Theory, Springer, Berlin, 1991. [2] H.S. Bradlow, H.F. Byrd, Mean waiting time evaluation of packet switches for centrally controlled pbx’s, Performance Evaluation 7 (4) (1987) 309–327. [3] G.L. Choudhury, W. Whitt, Computing transient and steady-state distributions in polling models by numerical transform inversion, Working paper, AT&T Bell Laboratories, Holmdel, NJ, 1994. [4] E. Co4man Jr., A. Puhalskii, M. Reiman, Polling systems in heavy-tra5c: a Bessel process limit, Math. Oper. Res. 23 (1998) 257–304. [5] M. Eisenberg, The polling system with a stopping server, Queueing Systems Theory Appl. 18(3– 4) (1994) 387–431. [6] D. Everitt, Simple approximations for token rings, IEEE Trans. Commun. COM-34(7) (1986) 719–721. [7] A. Federgruen, Z. Katalan, Approximating queue size and waiting time distributions in general polling systems, Queueing Systems Theory Appl. 18 (1994) 353–386. [8] A. Federgruen, Z. Katalan, The stochastic economic lot scheduling problem: cyclical base stock policies with idle times, Manage. Sci. 42 (6) (1996) 783–796. [9] M.J. Ferguson, Mean waiting time for a token ring with nodal dependent overheads, in: M. Akiyama (Ed.), Teletra5c Issues in an Advanced Information Society, ITC-11, Elsevier
T.L. Olsen / Operations Research Letters 28 (2001) 113–123 Science Publishers B.V., North-Holland, Amsterdam, 1985, pp. 634–640. [10] D. Gupta, M.M. Srinivasan, Polling systems with state-dependent setup times, Queueing Systems Theory Appl. 22 (1996) 403–423. [11] T.M. Lennon, Response-time approximations for multi-server polling models, with manufacturing applications, Ph.D. Thesis, Stanford University, 1994. [12] T.L. Olsen, A practical scheduling method for multiclass production systems with setups, Manage. Sci. 45(1) (1999) 116–130.
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