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doi: 10.3934/jimo.2021163
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Synchronized abandonment in discrete-time renewal input queues with vacations

1. 

School of Computer Applications, Kalinga Institute of Industrial Technology, Bhubaneswar, India

2. 

Department of Electrical and Computer Engineering, University of Central Florida, USA

Received  June 2020 Revised  June 2021 Early access September 2021

We consider a discrete-time infinite buffer renewal input queue with multiple vacations and synchronized abandonment. Waiting customers get impatient during the server's vacation and decide whether to take service or abandon simultaneously at the vacation completion instants. Using the supplementary variable technique and difference operator method, we obtain an explicit expression to find the steady-state system-length distributions at pre-arrival, random, and outside observer's observation epochs. We provide the stochastic decomposition structure for the number of customers and discuss the various performance measures. With the help of numerical experiments, we show that the method formulated in this work is analytically elegant and computationally tractable. The results are appropriate for light-tailed inter-arrival distributions and can also be leveraged to find heavy-tailed inter-arrival distributions.

Citation: Veena Goswami, Gopinath Panda. Synchronized abandonment in discrete-time renewal input queues with vacations. Journal of Industrial and Management Optimization, doi: 10.3934/jimo.2021163
References:
[1]

I. AdanA. Economou and S. Kapodistria, Synchronized reneging in queueing systems with vacations, Queueing Syst., 62 (2009), 1-33.  doi: 10.1007/s11134-009-9112-2.

[2]

A. S. Alfa, Applied Discrete-Time Queues, 2$^{nd}$ edition, Springer, New York, USA, 2016. doi: 10.1007/978-1-4939-3420-1.

[3]

A. S. Alfa, Vacation models in discrete time, Queueing Syst, 44 (2003), 5-30.  doi: 10.1023/A:1024028722553.

[4]

E. Altman and U. Yechiali, Analysis of customers' impatience in queues with server vacations, Queueing Syst., 52 (2006), 261-279.  doi: 10.1007/s11134-006-6134-x.

[5]

F. Barbhuiya and U. Gupta, A discrete-time GI$^X$/Geo/1 queue with multiple working vacations under late and early arrival system, Methodol. Comput. Appl. Probab., 22 (2020), 599-624.  doi: 10.1007/s11009-019-09724-6.

[6]

H. Bruneel and B. G. Kim, Discrete-Time Models for Communication Systems Including ATM, Kluwer Academic, Boston, USA, 1993. doi: 10.1007/978-1-4615-3130-2.

[7]

K. C. ChaeD. E. Lim and W. S. Yang, The GI/M/1 queue and the GI/Geo/1 queue both with single working vacation, Performance Evaluation, 66 (2009), 356-367. 

[8]

M. L. ChaudhryU. C. Gupta and V. Goswami, On discrete-time multiserver queues with finite buffer: GI/Geom/m/N, Comput. Oper. Res., 31 (2004), 2137-2150.  doi: 10.1016/S0305-0548(03)00168-0.

[9]

A. Economou, The compound Poisson immigration process subject to binomial catastrophes, J. Appl. Probab., 41 (2004), 508-523.  doi: 10.1017/S0021900200014467.

[10]

A. Economou and S. Kapodistria, Synchronized abandonments in a single server unreliable queue, European Journal of Operational Research, 203 (2010), 143-155.  doi: 10.1016/j.ejor.2009.07.014.

[11]

S. W. Fuhrmann and R. B. Cooper, Stochastic decompositions in the M/G/1 queue with generalized vacations, Oper. Res., 33 (1985), 1117-1129.  doi: 10.1287/opre.33.5.1117.

[12]

V. Goswami and G. B. Mund, Analysis of a discrete-time GI/Geo/1/N queue with multiple working vacations, Journal of Systems Science and Systems Engineering, 19 (2010), 367-384. 

[13]

V. Goswami and G. B. Mund, Analysis of a discrete-time queues with batch renewal input and multiple vacations, J. Syst. Sci. Complex., 25 (2012), 486-503.  doi: 10.1007/s11424-012-0057-x.

[14]

C. M. HarrisP. H. Brill and M. J. Fischer, Internet-type queues with power-tailed inter-arrival times and computational methods for their analysis, INFORMS Journal on Computing, 12 (2000), 261-271.  doi: 10.1287/ijoc.12.4.261.11882.

[15] J. J. Hunter, Mathematical Techniques of Applied Probability. Vol. 2. Discrete Time Models - Techniques and Applications, Academic Press, New York, USA, 1983. 
[16]

S. Kapodistria, The M/M/1 queue with synchronized abandonments, Queueing Syst., 68 (2011), 79-109.  doi: 10.1007/s11134-011-9219-0.

[17]

S. KapodistriaT. Phung-Duc and J. Resing, Linear birth/immigration-death process with binomial catastrophes, Probab. Engrg. Inform. Sci., 30 (2016), 79-111.  doi: 10.1017/S0269964815000297.

[18]

J. Li and N. Tian, The discrete-time GI/Geo/1 queue with working vacations and vacation interruption, Appl. Math. Comput., 185 (2007), 1-10.  doi: 10.1016/j.amc.2006.07.008.

[19]

D.-E. LimD. H. LeeW. S. Yang and K.-C. Chae, Analysis of the GI/Geo/1 queue with N-policy, Appl. Math. Model., 37 (2013), 4643-4652.  doi: 10.1016/j.apm.2012.09.037.

[20]

W.-Y. LiuX.-L. Xu and N. Tian, Stochastic decompositions in the M/M/1 queue with working vacations, Oper. Res. Lett., 35 (2007), 595-600.  doi: 10.1016/j.orl.2006.12.007.

[21]

S. Nahmias, Perishable Inventory Systems, vol. 160 of International Series in Operations Research & Management Science, Springer, New York, USA, 2011.

[22]

S. Ndreca and B. Scoppola, Discrete time GI/Geom/1 queueing system with priority, European J. Oper. Res., 189 (2008), 1403-1408.  doi: 10.1016/j.ejor.2007.02.056.

[23]

T. Phung-Duc, Exact solutions for {M/M/c/setup} queues, Telecommunication Systems, 64 (2017), 309-324.  doi: 10.1007/s11235-016-0177-z.

[24]

H. Takagi, Queueing Analysis: A Foundation of Performance Evaluation, Volume 3: Discrete-Time Systems, North-Holland Publishing, Co., Amsterdam, 1993.

[25]

L. TaoL. ZhangX. Xu and S. Gao, The GI/Geo/1 queue with Bernoulli-schedule-controlled vacation and vacation interruption, Computers & Operations Research, 40 (2013), 1680-1692. 

[26]

N. Tian and Z. G. Zhang, The discrete-time GI/Geo/1 queue with multiple vacations, Queueing Syst., 40 (2002), 283-294.  doi: 10.1023/A:1014711529740.

[27]

N. Tian and Z. G. Zhang, Vacation Queueing Models: Theory and Applications, International Series in Operations Research & Management Science, 93. Springer, New York, 2006.

[28]

W. WeiQ. XuL. WangX. HeiP. ShenW. Shi and L. Shan, GI/Geom/1 queue based on communication model for mesh networks, International Journal of Communication Systems, 27 (2014), 3013-3029.  doi: 10.1002/dac.2522.

[29] M. E. Woodward, Communication and Computer Networks: Modelling with Discrete-Time Queues, Wiley-IEEE Computer Society Press, 1994. 
[30]

M. Yajima and T. Phung-Duc, A central limit theorem for a Markov-modulated infinite-server queue with batch Poisson arrivals and binomial catastrophes, Performance Evaluation, 129 (2019), 2-14. 

[31]

U. Yechiali, Queues with system disasters and impatient customers when system is down, Queueing Syst., 56 (2007), 195-202.  doi: 10.1007/s11134-007-9031-z.

show all references

References:
[1]

I. AdanA. Economou and S. Kapodistria, Synchronized reneging in queueing systems with vacations, Queueing Syst., 62 (2009), 1-33.  doi: 10.1007/s11134-009-9112-2.

[2]

A. S. Alfa, Applied Discrete-Time Queues, 2$^{nd}$ edition, Springer, New York, USA, 2016. doi: 10.1007/978-1-4939-3420-1.

[3]

A. S. Alfa, Vacation models in discrete time, Queueing Syst, 44 (2003), 5-30.  doi: 10.1023/A:1024028722553.

[4]

E. Altman and U. Yechiali, Analysis of customers' impatience in queues with server vacations, Queueing Syst., 52 (2006), 261-279.  doi: 10.1007/s11134-006-6134-x.

[5]

F. Barbhuiya and U. Gupta, A discrete-time GI$^X$/Geo/1 queue with multiple working vacations under late and early arrival system, Methodol. Comput. Appl. Probab., 22 (2020), 599-624.  doi: 10.1007/s11009-019-09724-6.

[6]

H. Bruneel and B. G. Kim, Discrete-Time Models for Communication Systems Including ATM, Kluwer Academic, Boston, USA, 1993. doi: 10.1007/978-1-4615-3130-2.

[7]

K. C. ChaeD. E. Lim and W. S. Yang, The GI/M/1 queue and the GI/Geo/1 queue both with single working vacation, Performance Evaluation, 66 (2009), 356-367. 

[8]

M. L. ChaudhryU. C. Gupta and V. Goswami, On discrete-time multiserver queues with finite buffer: GI/Geom/m/N, Comput. Oper. Res., 31 (2004), 2137-2150.  doi: 10.1016/S0305-0548(03)00168-0.

[9]

A. Economou, The compound Poisson immigration process subject to binomial catastrophes, J. Appl. Probab., 41 (2004), 508-523.  doi: 10.1017/S0021900200014467.

[10]

A. Economou and S. Kapodistria, Synchronized abandonments in a single server unreliable queue, European Journal of Operational Research, 203 (2010), 143-155.  doi: 10.1016/j.ejor.2009.07.014.

[11]

S. W. Fuhrmann and R. B. Cooper, Stochastic decompositions in the M/G/1 queue with generalized vacations, Oper. Res., 33 (1985), 1117-1129.  doi: 10.1287/opre.33.5.1117.

[12]

V. Goswami and G. B. Mund, Analysis of a discrete-time GI/Geo/1/N queue with multiple working vacations, Journal of Systems Science and Systems Engineering, 19 (2010), 367-384. 

[13]

V. Goswami and G. B. Mund, Analysis of a discrete-time queues with batch renewal input and multiple vacations, J. Syst. Sci. Complex., 25 (2012), 486-503.  doi: 10.1007/s11424-012-0057-x.

[14]

C. M. HarrisP. H. Brill and M. J. Fischer, Internet-type queues with power-tailed inter-arrival times and computational methods for their analysis, INFORMS Journal on Computing, 12 (2000), 261-271.  doi: 10.1287/ijoc.12.4.261.11882.

[15] J. J. Hunter, Mathematical Techniques of Applied Probability. Vol. 2. Discrete Time Models - Techniques and Applications, Academic Press, New York, USA, 1983. 
[16]

S. Kapodistria, The M/M/1 queue with synchronized abandonments, Queueing Syst., 68 (2011), 79-109.  doi: 10.1007/s11134-011-9219-0.

[17]

S. KapodistriaT. Phung-Duc and J. Resing, Linear birth/immigration-death process with binomial catastrophes, Probab. Engrg. Inform. Sci., 30 (2016), 79-111.  doi: 10.1017/S0269964815000297.

[18]

J. Li and N. Tian, The discrete-time GI/Geo/1 queue with working vacations and vacation interruption, Appl. Math. Comput., 185 (2007), 1-10.  doi: 10.1016/j.amc.2006.07.008.

[19]

D.-E. LimD. H. LeeW. S. Yang and K.-C. Chae, Analysis of the GI/Geo/1 queue with N-policy, Appl. Math. Model., 37 (2013), 4643-4652.  doi: 10.1016/j.apm.2012.09.037.

[20]

W.-Y. LiuX.-L. Xu and N. Tian, Stochastic decompositions in the M/M/1 queue with working vacations, Oper. Res. Lett., 35 (2007), 595-600.  doi: 10.1016/j.orl.2006.12.007.

[21]

S. Nahmias, Perishable Inventory Systems, vol. 160 of International Series in Operations Research & Management Science, Springer, New York, USA, 2011.

[22]

S. Ndreca and B. Scoppola, Discrete time GI/Geom/1 queueing system with priority, European J. Oper. Res., 189 (2008), 1403-1408.  doi: 10.1016/j.ejor.2007.02.056.

[23]

T. Phung-Duc, Exact solutions for {M/M/c/setup} queues, Telecommunication Systems, 64 (2017), 309-324.  doi: 10.1007/s11235-016-0177-z.

[24]

H. Takagi, Queueing Analysis: A Foundation of Performance Evaluation, Volume 3: Discrete-Time Systems, North-Holland Publishing, Co., Amsterdam, 1993.

[25]

L. TaoL. ZhangX. Xu and S. Gao, The GI/Geo/1 queue with Bernoulli-schedule-controlled vacation and vacation interruption, Computers & Operations Research, 40 (2013), 1680-1692. 

[26]

N. Tian and Z. G. Zhang, The discrete-time GI/Geo/1 queue with multiple vacations, Queueing Syst., 40 (2002), 283-294.  doi: 10.1023/A:1014711529740.

[27]

N. Tian and Z. G. Zhang, Vacation Queueing Models: Theory and Applications, International Series in Operations Research & Management Science, 93. Springer, New York, 2006.

[28]

W. WeiQ. XuL. WangX. HeiP. ShenW. Shi and L. Shan, GI/Geom/1 queue based on communication model for mesh networks, International Journal of Communication Systems, 27 (2014), 3013-3029.  doi: 10.1002/dac.2522.

[29] M. E. Woodward, Communication and Computer Networks: Modelling with Discrete-Time Queues, Wiley-IEEE Computer Society Press, 1994. 
[30]

M. Yajima and T. Phung-Duc, A central limit theorem for a Markov-modulated infinite-server queue with batch Poisson arrivals and binomial catastrophes, Performance Evaluation, 129 (2019), 2-14. 

[31]

U. Yechiali, Queues with system disasters and impatient customers when system is down, Queueing Syst., 56 (2007), 195-202.  doi: 10.1007/s11134-007-9031-z.

Figure 1.  Various time epochs in a late arrival system with delayed access (LAS-DA)
Figure 2.  Effect of $ \mu $ on customer abandonment in different queueing systems with $ \lambda = 0.2, \phi = 0.3, q = 0.6 $
Figure 3.  Effect of $ \mu $ on the mean sojourn time in different queueing systems with $ \lambda = 0.2, \phi = 0.3, q = 0.6 $
Figure 4.  Effect of $ \phi $ on customer abandonment in different queueing systems with $ \lambda = 0.2, \mu = 0.6, q = 0.6 $
Figure 5.  Effect of $ \phi $ on the mean sojourn time in different queueing systems with $ \lambda = 0.2, \mu = 0.6, q = 0.6 $
Figure 6.  Effect of $ q $ on customer abandonment in different queueing systems with $ \lambda = 0.2, \mu = 0.6, \phi = 0.3 $
Figure 7.  Effect of $ q $ on the mean sojourn time in different queueing systems with $ \lambda = 0.2, \mu = 0.6, \phi = 0.3 $
Figure 8.  $ \lambda $ vs $ L_s $ for different arrival streams with $ \mu = 0.6, \phi = 0.3, q = 0.5 $
Figure 9.  $ \lambda $ vs abandonment rate for systems with $ \mu = 0.6, \phi = 0.3, q = 0.5 $
Table 1.  System-length distributions at pre-arrival and arbitrary epochs for geometric and deterministic arrival distributions
Geo/Geo/1 D/Geo/1
n $ P_{n, 0}^- $ $ P_{n, 1}^- $ $ P_{n, 0} $ $ P_{n, 1} $ $ P_{n, 0}^- $ $ P_{n, 1}^- $ $ P_{n, 0} $ $ P_{n, 1} $
0 0.313492 0.313492 0.288979 0.138613
1 0.201531 0.064165 0.201531 0.064165 0.170639 0.152011 0.213012 0.143164
2 0.129555 0.030556 0.129555 0.030556 0.100761 0.080583 0.125781 0.108959
3 0.083286 0.014474 0.083286 0.014474 0.059498 0.035638 0.074273 0.051092
4 0.053541 0.006856 0.053541 0.006856 0.035133 0.015139 0.043857 0.022010
5 0.034419 0.003248 0.034419 0.003248 0.020746 0.006365 0.025897 0.009288
6 0.022127 0.001538 0.022127 0.001538 0.012250 0.002669 0.015292 0.003899
7 0.014224 0.000729 0.014224 0.000729 0.007234 0.001119 0.009030 0.001634
8 0.009144 0.000345 0.009144 0.000345 0.004271 0.000469 0.005332 0.000685
9 0.005878 0.000164 0.005878 0.000164 0.002522 0.000196 0.003149 0.000287
10 0.003779 0.000077 0.003779 0.000077 0.001489 0.000082 0.001859 0.000120
15 0.000415 0.000002 0.000415 0.000002 0.000107 0.000001 0.000133 0.000002
20 0.000046 0 0.000046 0 0.000008 0 0.000010 0
$\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$
$ \Sigma $ 0.877778 0.122222 0.877778 0.122222 0.705671 0.294329 0.658776 0.341224
$ L_s $=1.812532, $ W_s $= 9.062658 $ L_s $=1.964151, $ W_s $= 9.820755
Geo/Geo/1 D/Geo/1
n $ P_{n, 0}^- $ $ P_{n, 1}^- $ $ P_{n, 0} $ $ P_{n, 1} $ $ P_{n, 0}^- $ $ P_{n, 1}^- $ $ P_{n, 0} $ $ P_{n, 1} $
0 0.313492 0.313492 0.288979 0.138613
1 0.201531 0.064165 0.201531 0.064165 0.170639 0.152011 0.213012 0.143164
2 0.129555 0.030556 0.129555 0.030556 0.100761 0.080583 0.125781 0.108959
3 0.083286 0.014474 0.083286 0.014474 0.059498 0.035638 0.074273 0.051092
4 0.053541 0.006856 0.053541 0.006856 0.035133 0.015139 0.043857 0.022010
5 0.034419 0.003248 0.034419 0.003248 0.020746 0.006365 0.025897 0.009288
6 0.022127 0.001538 0.022127 0.001538 0.012250 0.002669 0.015292 0.003899
7 0.014224 0.000729 0.014224 0.000729 0.007234 0.001119 0.009030 0.001634
8 0.009144 0.000345 0.009144 0.000345 0.004271 0.000469 0.005332 0.000685
9 0.005878 0.000164 0.005878 0.000164 0.002522 0.000196 0.003149 0.000287
10 0.003779 0.000077 0.003779 0.000077 0.001489 0.000082 0.001859 0.000120
15 0.000415 0.000002 0.000415 0.000002 0.000107 0.000001 0.000133 0.000002
20 0.000046 0 0.000046 0 0.000008 0 0.000010 0
$\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$
$ \Sigma $ 0.877778 0.122222 0.877778 0.122222 0.705671 0.294329 0.658776 0.341224
$ L_s $=1.812532, $ W_s $= 9.062658 $ L_s $=1.964151, $ W_s $= 9.820755
Table 2.  System-length distributions at pre-arrival and arbitrary epochs for negative binomial and discrete phase-type arrival distributions
NB/Geo/1 PH/Geo/1
$ n $ $ P_{n, 0}^- $ $ P_{n, 1}^- $ $ P_{n, 0} $ $ P_{n, 1} $ $ P_{n, 0}^- $ $ P_{n, 1}^- $ $ P_{n, 0} $ $ P_{n, 1} $
0 0.288979 0.138613 0.231585 0.228347
1 0.170639 0.152011 0.213012 0.143164 0.148479 0.119171 0.149592 0.117569
2 0.100761 0.080583 0.125781 0.108959 0.095196 0.099585 0.095909 0.099895
3 0.059498 0.035638 0.074273 0.051092 0.061034 0.062761 0.061491 0.063295
4 0.035133 0.015139 0.043857 0.022010 0.039131 0.035351 0.039425 0.035744
5 0.020746 0.006365 0.025897 0.009288 0.025089 0.018767 0.025277 0.019004
6 0.012250 0.002669 0.015292 0.003899 0.016085 0.009615 0.016206 0.009745
7 0.007234 0.001119 0.009030 0.001634 0.010313 0.004813 0.010390 0.004881
8 0.004271 0.000469 0.005332 0.000685 0.006612 0.002371 0.006662 0.002406
9 0.002522 0.000196 0.003149 0.000287 0.004239 0.001155 0.004271 0.001173
10 0.001489 0.000082 0.001859 0.000120 0.002718 0.000558 0.002738 0.000567
15 0.000107 0.000001 0.000133 0.000002 0.000294 0.000014 0.000297 0.000014
20 0.000008 0 0.000010 0 0.000032 0 0.000032 0
$\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$
$ \Sigma $ 0.705671 0.294329 0.658776 0.341224 0.645338 0.354662 0.645201 0.354799
$ L_s $=1.964151, $ W_s $=9.820755 $ L_s $=2.041164, $ W_s $=10.205818
NB/Geo/1 PH/Geo/1
$ n $ $ P_{n, 0}^- $ $ P_{n, 1}^- $ $ P_{n, 0} $ $ P_{n, 1} $ $ P_{n, 0}^- $ $ P_{n, 1}^- $ $ P_{n, 0} $ $ P_{n, 1} $
0 0.288979 0.138613 0.231585 0.228347
1 0.170639 0.152011 0.213012 0.143164 0.148479 0.119171 0.149592 0.117569
2 0.100761 0.080583 0.125781 0.108959 0.095196 0.099585 0.095909 0.099895
3 0.059498 0.035638 0.074273 0.051092 0.061034 0.062761 0.061491 0.063295
4 0.035133 0.015139 0.043857 0.022010 0.039131 0.035351 0.039425 0.035744
5 0.020746 0.006365 0.025897 0.009288 0.025089 0.018767 0.025277 0.019004
6 0.012250 0.002669 0.015292 0.003899 0.016085 0.009615 0.016206 0.009745
7 0.007234 0.001119 0.009030 0.001634 0.010313 0.004813 0.010390 0.004881
8 0.004271 0.000469 0.005332 0.000685 0.006612 0.002371 0.006662 0.002406
9 0.002522 0.000196 0.003149 0.000287 0.004239 0.001155 0.004271 0.001173
10 0.001489 0.000082 0.001859 0.000120 0.002718 0.000558 0.002738 0.000567
15 0.000107 0.000001 0.000133 0.000002 0.000294 0.000014 0.000297 0.000014
20 0.000008 0 0.000010 0 0.000032 0 0.000032 0
$\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$
$ \Sigma $ 0.705671 0.294329 0.658776 0.341224 0.645338 0.354662 0.645201 0.354799
$ L_s $=1.964151, $ W_s $=9.820755 $ L_s $=2.041164, $ W_s $=10.205818
Table 3.  System-length distributions at pre-arrival and arbitrary epochs for discrete standard log normal (SLN) and discrete Weibull (Wb) arrival distributions
SLN/Geo/1 Wb/Geo/1
$ n $ $ P_{n, 0}^- $ $ P_{n, 1}^- $ $ P_{n, 0} $ $ P_{n, 1} $ $ P_{n, 0}^- $ $ P_{n, 1}^- $ $ P_{n, 0} $ $ P_{n, 1} $
0 0.181276 0.248765 0.103221 0.132529
1 0.123225 0.108283 0.112313 0.104447 0.085123 0.113309 0.082833 0.109539
2 0.083763 0.108336 0.076346 0.099997 0.070198 0.088184 0.068309 0.084544
3 0.056939 0.081316 0.051897 0.073931 0.057890 0.062539 0.056332 0.059909
4 0.038705 0.054268 0.035277 0.048965 0.047740 0.043931 0.046455 0.042080
5 0.026310 0.033964 0.023980 0.030504 0.039369 0.030828 0.038310 0.029529
6 0.017885 0.020412 0.016301 0.018276 0.032467 0.021630 0.031593 0.020719
7 0.012157 0.011930 0.011081 0.010658 0.026774 0.015177 0.026054 0.014537
8 0.008264 0.006832 0.007532 0.006094 0.022080 0.010649 0.021486 0.010200
9 0.005618 0.003853 0.005120 0.003432 0.018208 0.007472 0.017718 0.007157
10 0.003819 0.002146 0.003480 0.001910 0.015016 0.005242 0.014612 0.005021
15 0.000554 0.000102 0.000505 0.000091 0.005727 0.000891 0.005573 0.000854
20 0.000080 0.000004 0.000073 0.000004 0.002184 0.000152 0.002126 0.000145
30 0.000002 0 0.000002 0 0.000318 0.000004 0.000309 0.000004
40 0 0 0 0 0.000046 0 0.000045 0
60 0 0 0 0 0.000001 0 0.000001 0
$\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$
$ \Sigma $ 0.566067 0.433933 0.599480 0.400520 0.588712 0.411288 0.604956 0.395044
$ L_s $=2.280859, $ W_s $=10.610305 $ L_s $=4.048721, $ W_s $=7.961503
SLN/Geo/1 Wb/Geo/1
$ n $ $ P_{n, 0}^- $ $ P_{n, 1}^- $ $ P_{n, 0} $ $ P_{n, 1} $ $ P_{n, 0}^- $ $ P_{n, 1}^- $ $ P_{n, 0} $ $ P_{n, 1} $
0 0.181276 0.248765 0.103221 0.132529
1 0.123225 0.108283 0.112313 0.104447 0.085123 0.113309 0.082833 0.109539
2 0.083763 0.108336 0.076346 0.099997 0.070198 0.088184 0.068309 0.084544
3 0.056939 0.081316 0.051897 0.073931 0.057890 0.062539 0.056332 0.059909
4 0.038705 0.054268 0.035277 0.048965 0.047740 0.043931 0.046455 0.042080
5 0.026310 0.033964 0.023980 0.030504 0.039369 0.030828 0.038310 0.029529
6 0.017885 0.020412 0.016301 0.018276 0.032467 0.021630 0.031593 0.020719
7 0.012157 0.011930 0.011081 0.010658 0.026774 0.015177 0.026054 0.014537
8 0.008264 0.006832 0.007532 0.006094 0.022080 0.010649 0.021486 0.010200
9 0.005618 0.003853 0.005120 0.003432 0.018208 0.007472 0.017718 0.007157
10 0.003819 0.002146 0.003480 0.001910 0.015016 0.005242 0.014612 0.005021
15 0.000554 0.000102 0.000505 0.000091 0.005727 0.000891 0.005573 0.000854
20 0.000080 0.000004 0.000073 0.000004 0.002184 0.000152 0.002126 0.000145
30 0.000002 0 0.000002 0 0.000318 0.000004 0.000309 0.000004
40 0 0 0 0 0.000046 0 0.000045 0
60 0 0 0 0 0.000001 0 0.000001 0
$\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$
$ \Sigma $ 0.566067 0.433933 0.599480 0.400520 0.588712 0.411288 0.604956 0.395044
$ L_s $=2.280859, $ W_s $=10.610305 $ L_s $=4.048721, $ W_s $=7.961503
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