Int J Adv Manuf Technol (2007) 33: 42–47 DOI 10.1007/s00170-006-0450-5
O R I G I N A L A RT I C L E
Kuo-Hsiung Wang . Jyh-Bin Ke . W. L. Pearn
Optimal management for a finite M/M/R queueing system
with two arrival modes
Received: 4 September 2005 / Accepted: 30 January 2006 / Published online: 30 March 2006 # Springer-Verlag London Limited 2006
Abstract We consider an M/M/R queueing system with finite capacityN, where customers have two arrival modes under steady-state conditions. It is assumed that each arrival mode is serviced by one or more servers, and that the two arrival modes have equal probabilities of receiving service. Arrival times of the customers and service times of the severs follow an exponential distribution. A cost model is developed to determine the optimal number of servers and the optimal system capacity. The minimum expected cost, the optimal number of servers, the optimal system capacity, and various system characteristics are obtained for some designated system parameters’ values. Sensitivity for the minimal cost is also investigated.
Keywords Cost . Optimization . Queue . Sensitivity analysis . Two arrival modes
1 Introduction
This paper considers an M/M/R queueing system with finite capacity N, where N is the maximum number of customers in the system. We assume that the customers have two arrival modes, and are serviced by one or more servers in the service facilities. We also assume that any one mode of arrival can be serviced by one or more servers, and that each arrival mode has an equal probability of receiving service.
Each customer has two independent arrival modes (mode 1 and mode 2). Both arrival modes 1 and 2 of customers follow a Poisson process with parametersλ1and
λ2, respectively. Suppose that both modes are equally
likely to be serviced next, when several customers are waiting for service. The service time of each server in the service modei has an exponential distribution with mean 1/μi, wherei=1, 2. The arriving customers join in a single waiting line based on the order of their arrivals; that is, in a first-come, first-served discipline. Each server services only one customer at a time. Customers who, upon entry into the service facility, find that the server is busy have to wait in the queue until the server is available.
Analytic steady-state solutions of a finite-capacity M/M/ R queueing system with two arrival modes have not been found. For cases with one single arrival mode, analytic steady- state solutions of an M/M/R queueing system have been provided by several authors, including Gross and Harris [1] and Kleinrock [2]. The past work for the two (or multiple) arrival modes situation may be divided into two parts, according to whether the queueing model is of infinite source or finite source. In the first category, we review a previous paper which deals with infinite source queues. Tijms [3] considered a finite-capacity queueing system with two arrival modes (which appeared as an exercise, p. 152, Exercise 2.23), but provided no cost analysis or solution methodologies. The second category of authors deals with papers treating the finite source model. Finite source models as applied to machine repair problems have been examined by several researchers. Without deriving analytic steady-state solutions, Benson and Cox [4] first considered the no-spare M/M/1 machine repair problem with two failure modes. Again, without providing analytic steady-state solutions, Elsayed [5] studied two repair policies for the no-spare M/M/1 machine repair problem with two failure modes. Analytic steady-state solutions of the cold-standby M/M/R machine repair problem with two failure modes were first derived by Wang [6]. Later on, Wang and Wu [7] investigated the M/ M/R machine repair problem with spares where machines have two failure modes. Spares are considered to be either cold-standby, warm-standby, or hot-standby. Wang and Lee [8] extended Wang’s model [6] to the M/M/R machine repair problem with multiple failure modes.
K.-H. Wang (*) . J.-B. Ke Department of Applied Mathematics, National Chung-Hsing University, Taichung, Taiwan, 402,
Republic of China
e-mail: khwang@amath.nchu.edu.tw W. L. Pearn
Department of Industrial Engineering and Management, National Chiao Tung University,
The birth-and-death process is used to derive analytic steady-state solutions to a finite capacity M/M/R queueing system with two arrival modes. This paper differs from past works in that: (a) the infinite-source queueing problem has distinct characteristics which are different from the machine repair problem; (b) it studies a finite-capacity M/M/R queueing system with two arrival modes; and (c) it performs a sensitivity analysis for the total expected cost with respect to specific values of the system parameters.
As an application of that problem, we consider a parking lot problem where there are two different classes of parkers, namely, long-period parkers and short-period parkers, arriving at a parking lot according to independent Poisson processes with rates λ1 and λ2 per hour,
respectively. The parking lot has space for N cars. There areR employees in the parking lot management office. The service times of those employees for each class of parkers follow an exponential distribution with mean 1/μi, where i=1, 2, respectively. The manager of the parking lot management office would like to know the system performance for the parking lot, such as the expected number of cars in the parking lot, the expected number of the busy employees, and the expected number of idle employees, for minimum cost.
We first develop analytic steady-state solutions for the finite M/M/R queueing system with two arrival modes by using the modified birth-and-death results. Next, a cost model is developed to determine the optimal values of the number of servers and the system capacity, simultaneously, in order to minimize the total expected cost per unit time. Finally, various system performance measures are evalu-ated under optimal operating conditions. We perform a sensitivity analysis for the minimal cost with respect to changes in specific values of some system parameters.
2 Steady-state results
The system can be analyzed as a continuous time parameter Markov chain with statesfði; jÞji þ j ¼ 0; 1; 2; . . . ; Ng; wherei denotes the number of customers of mode 1 and j is the number of customers of mode 2. For a steady-state condition, let P(i, j)≡probability that there are i and j customers of modes 1 and 2 in the system, respectively. The mean arrival rate,λi, jk, for arrival modek (k=1, 2) are as follows: λ1 i; j¼ λ1 if 0 i þ j < N 0 otherwise ( λ2 i; j¼ λ2 if 0 i þ j < N 0 otherwise (
Let n represent the number of customers of mode 1 (or mode 2) in the system. The mean service rate, μnk, for service modek (k=1, 2) are as follows:
μ1 n¼ nμ1 if 0 n R Rμ1 if R n N 0 otherwise 8 > > < > > : μ2 n¼ nμ2 if 0 n R Rμ2 if R n N 0 otherwise 8 > > < > > :
A special case of a finite-capacity M/M/R queueing system with arrival modes is shown in Fig. 1. The steady-state
Fig. 1 State-transition-rate dia-gram for an M/M/R/N queueing system with two arrival modes (R=3, N=5)
equations for P(i, j) for an M/M/R queueing system with finite capacityN and two arrival modes are given by:
λ1þ λ2 ð ÞP 0; 0ð Þ ¼ μ1P 1; 0ð Þ þ μ2P 0; 1ð Þ (1) λ1þ λ2þ minði; RÞμ1 ½ Pði; 0Þ ¼ λ1Pði 1; 0Þ þ μ2Pði; 1Þ þ minði þ 1; RÞμ1Pði þ 1; 0Þ 1 i N 1 (2) Rμ1P N; 0ð Þ ¼ λ1P N 1; 0ð Þ (3) λ1þ λ2þ minðj; RÞμ2 ½ Pð0; jÞ ¼ λ2Pð0; j 1Þ þ μ1Pð1; jÞ þ minðj þ 1; RÞμ2Pð0; j þ 1Þ 1 j N 1 (4) Rμ2P 0; Nð Þ ¼ λ2P 0; N 1ð Þ (5) λ1þ λ2þ minði; RÞμ1þ minðj; RÞμ2 ½ Pði; jÞ ¼ λ1Pði 1; jÞ þ λ2Pði; j 1Þ þ minði þ 1; RÞμ1Pði þ 1; jÞ þ minðj þ 1; RÞμ2Pði; j þ 1Þ 1 i; j N 1; 2 i þ j N 1 (6) minði; RÞμ1þ minðN i; RÞμ2 ½ Pði; N 1Þ ¼ λ1Pði 1; N 1Þ þ λ2Pði; N i 1Þ 1 i N 1 (7)
Solving Eqs.1,2,3,4,5,6,7recursively or by using the following known formula, which can be found in [6,7]:
P i; jð Þ ¼ Yi n¼1 λ1 n1; 0 μ1 n " # Yj n¼1 λ2 i; n1 μ2 n " # P 0; 0ð Þ (8)
wherea>b in the Q
b
l¼aðÞ notation indicates that the term is
1, we obtain respectively: P i; 0ð Þ ¼ ρi1 Qi n¼1min n; Rð Þ P 0; 0ð Þ 1 i N (9) P 0; jð Þ ¼ ρi2 Qj n¼1min n; Rð Þ P 0; 0ð Þ 1 j N (10) Pði; jÞ ¼ ρi1ρi2 Qi n¼1minðn; RÞ Qj n¼1 minðn; RÞ Pð0; 0Þ 1 i þ j N i; j ¼ 0; 1; 2; . . . ; N (11) whereρ1¼λ1 μ1; ρ2¼ λ2 μ2; and: Yn l¼1 minðl; RÞ ¼ n! if 1 n R R!RnR if R þ 1 n N (
It should be noted that Eq.9(or Eq.10) is identical to the results for a finite-capacity M/M/R queueing system with a single arrival mode (see Gross and Harris [1], p. 93). The steady-state solutions P(i, j) always exist because the number of states is finite. We use an efficient Matlab computer program to evaluateP(i, j) by using Eqs.9,10,11 and the following normalizing equation:
XN i¼0
XNi j¼0
P i; jð Þ ¼ 1 (12)
3 System performance measures
Our analysis is based on the following system performance measures of a finite M/M/R queueing system with two arrival modes. Let us note the following:
L1 Expected number of customers in the system for
arrival mode 1
L2 Expected number of customers in the system for
arrival mode 2
Ls Expected number of customers in the system
Lq Expected number of customers in the queue
E[I] Expected number of idle servers E[B] Expected number of busy servers
The expressions for L1, L2, Ls, Lq, E[I], and E[B] are given by: L1 ¼ XN i¼1 i XNi j¼0 P i; jð Þ " # (13) L2 ¼ XN j¼1 j XNj j¼0 P i; jð Þ " # (14) Ls¼ XN iþj¼0 i þ j ð ÞP i; jð Þ ¼ L1þ L2 (15) Lq¼ XN i¼Rþ1 ði RÞNR1X j¼0 Pði; jÞ þ XN j¼Rþ1 ðj RÞNR1X i¼0 Pði; jÞ N > R (16) E I½ ¼ XR1 iþj¼0 R max i; jð Þ ½ P i; jð Þ (17) E B½ ¼ R E I½ (18)
4 Cost sensitivity analysis
We develop a total expected cost function per unit time for an M/M/R queueing system with finite capacityN and two arrival modes, in whichR and N are two decision variables. Our objective is to determine the optimum number of serversR, say R*, and the optimal system capacity N, say N*, simultaneously, so as to minimize this function. Let C1
be the holding cost per unit time per customer present in the system,C2be the cost per unit time when one server is idle,
C3be the cost per unit time when one server is busy,C4be
the fixed cost for every customer’s space, and C4be the
fixed cost for every lost customer. Thus, the total expected cost function per unit time is given by:
F R; Nð Þ
¼ C1Lsþ C2E I½ þ C3E B½ þ C4N
þ λð 1þ λ2ÞC5PN (19)
wherePN ¼ P
iþjþNP i; jð Þ:
The cost parameters in Eq.19are assumed to be linear in the expected number of the indicated quantity. Substitution
of Eqs.11and15,16,17,18into Eq.19, the cost function F(R, N) is too detailed to be shown here. Hence, it would have been an arduous task or, at least, extremely difficult to develop the optimal solution (R*, N*) symbolically, due to the highly non-linear and complex nature of the optimiza-tion problem. To the best of the authors’ knowledge, no new and efficient methods to solve this optimization problem currently exist. This is due to the fact that there are two decision variables, R and N, involved in our model. Here, we should point out explicitly that the solution really gives the minimum value. Therefore, we will perform the numerical experiments to show that the cost function is really convex and that the solution gives a minimum. An efficient and direct procedure is used to obtain (R*, N*). Following Hilliard [9], we carry out the following steps for achieving the optimal value (R*, N*):
Step 1
Find the optimal system capacityN*, for R servers, i.e., min
N F R; Nð Þ ¼ F R; N
ð Þ
Step 2
Find the set of all minimum cost solutions forR=1, 2,…, N, i.e., Θ ¼ F R; Nf ð Þ : R ¼ 1; 2; . . . ; Ng Step 3
Find the optimal number of servers, R*, i.e., min
R Θ ¼ F R ; N
ð Þ.
The following numerical results are obtained for C1=
$8/h, C2=$10/h, C3=$30/h, C4=$5/h, and C5=$25/h. We
fix λ1=20 arrivals/h, μ0=10 services/h, λ2=10 arrivals/h,
μ2=10 services/h, vary the number of serversR from 1 to
6, and vary the system capacity N from 3 to 12. The expected cost F(R, N) is shown in Table 1 for various values ofR and N. We note that a minimum expected cost per hour of $160.54 is obtained withR*=4 and N*=9.
To find (R*, N*), we should show the existence of convexity or unimodality ofF(R, N). However, this task is difficult to implement. The functionF(R, N) is unimodal; that is, it has a single relative minimum. The numerical results shown in Table 1 can convince us that the cost function is convex. The minimum expected costF(R, N), the values of various system characteristicsLs,E[I], E[B],
and the sensitivities ofF(R, N) with respect to λ1,μ1,λ2,
andμ2, at the optimal value (R*, N*) are shown in Table2
for (λ2,μ2)=(20, 10) and different values of (λ1,μ1). From
Table2, as would be expected, we observe that: (1)F(R*, N*) increases as λ1increases orμ1decreases, which can be
easily predicted by the sign of its sensitivity; and (2) the optimum values of (R, N), (R*, N*) increases as λ1
increases. From the last three columns of Table2,R* does not change andN* rarely changes when μ1changes from
15 to 30. Intuitively, this seems to be too insensitive to changes inμ1. This phenomenon also can be seen from the
order of sensitivities. (i.e.,@F@μ1is the smallest in the last three columns of Table2).
The sensitivity analysis results are shown in Table3by increasing λ1from 0.01 to 10,000 and fixing μ1=20 and
(λ2,μ2)=(20, 10). We observe from Table3that: (1) asλ1
eventually reaches a stable level; (2) a similar trend also occurs for the impacts ofμ1andλ2onF(R, N), but not μ2;
(3) the system with smallerλ1has a light traffic at mode 1,
therefore,μ1 does not affectF(R, N); (4) the system with
largerλ1has a bottleneck at mode 1, therefore,μ2does not
affectF(R, N) and the values of λ1andλ2have the same
impact onF(R, N); and (5) the most dominant parameter is μ2for smallerλ1and shifts toμ1for largerλ1. (That is, we
can significantly reduce the system cost by increasing the values of these dominant parameters.)
Next, the sensitivity analysis results are shown in Table4 by increasingμ1from 0.01 to 10,000 and fixingλ1=15 and
(λ2,μ2)=(20, 10). We observe from Table4that: (1) asμ1
increases, the impact of μ1 on F(R, N) decreases and
reaches to zero eventually; (2) the similar trend also happens to the impacts ofλ1andλ2onF(R, N), but not μ2;
(3) the system with smallerμ1has a bottleneck at mode 1,
therefore,μ2 does not affectF(R, N); (4) the system with
larger μ1 has a bottleneck at mode 2, therefore, μ1 itself
does not affect F(R, N); and (5) the most dominant parameter isμ1for smallerμ1and shifts toμ2for largerμ1.
(That is, we can significantly reduce the system cost by increasing the value of these dominant parameters.)
5 Conclusions
In this paper, we modeled a finite M/M/R queueing system with two arrival modes, and obtained the steady-state analytic solutions. The considered model generalizes the existing M/M/R queueing system with infinite capacity and two arrival modes, the M/M/R queueing system with infinite capacity and a single arrival mode, and the finite M/ M/R queueing system with a single arrival mode. We have provided an efficient method to determine the optimal number of servers and the optimal system capacity simultaneously, in order to minimize the expected cost function, and evaluated various system performance mea-sures under the optimal operating conditions. We also performed a sensitivity analysis for the minimal expected cost with respect to specific values ofλ1,μ1, λ2, andμ2.
The results are useful for modeling banking service systems, computer jobs processing, performance
evalua-Table 4 Sensitivity analysis ofF(R, N) evaluated at (R, N)=(4, 8), λ1=15, (λ2,μ2)=(20, 10) and varyingμ1 μ1 F(4, 8) @F=@1 @F=@1 @F=@2 @F=@2 0.01 1,097 25.09 −235.5 24.93 0 0.1 1,075 25.90 −235.6 24.33 0 1 863.0 34.05 −235.6 18.37 −0.05 5 272.5 19.17 −49.65 7.70 −10.15 10 179.4 4.64 −6.00 5.91 −10.52 20 155.9 1.50 −0.90 5.06 −9.54 50 146.5 0.55 −0.11 4.68 −8.99 100 144.0 0.32 −0.02 4.57 −8.82 1,000 141.8 0.15 0 4.48 −8.68 10,000 141.6 0.14 0 4.47 −8.66
Table 3. Sensitivity analysis ofF(R, N) evaluated at (R, N)=(4, 8), μ1=10, (λ2,μ2)=(20, 10) and varyingλ1 λ1 F(4, 8) @F=@1 @F=@1 @F=@2 @F=@2 0.01 139.6 1.42 −0.001 3.87 −7.46 1 141.0 1.53 −0.14 3.92 −7.57 5 148.1 2.04 −0.91 4.29 −8.12 10 160.6 3.02 −2.64 4.95 −9.41 15 179.4 4.64 −6.00 5.91 −10.52 30 311.7 13.8 −33.55 9.47 −13.70 100 2,029 26.6 −115.3 15.75 −1.32 1,000 24,703 25 −102 24 0 10,000 249,721 25 −100 25 0
Table 1 The expected costF(R, N) for (λ1,μ1)=(20, 10), (λ2,μ2)= (10, 10) N R 1 2 3 4 5 6 3 496.00 367.33 351.84 – – – 4 483.65 311.26 268.80 270.68 – – 5 481.98 277.71 218.65 212.29 219.89 – 6 486.45 257.48 190.24 180.33 185.78 195.14 7 494.51 245.52 174.91 165.64 170.66 179.52 8 504.67 238.91 167.35 160.59 166.37 175.23 9 516.05 235.91 164.50 160.54 167.39 176.51 10 528.13 235.44 164.55 163.03 170.74 180.14 11 540.62 236.80 166.39 166.79 175.08 184.66 12 553.34 239.50 169.37 171.18 179.81 189.50
Table 2 System performance measures and sensitivities of a finite M/M/R queueing system with two arrival modes under optimal operating conditions for (λ2,μ2)=(20, 10) and various values of (λ1, μ1) (λ1,μ1) (30, 10) (20, 10) (15, 10) (15, 15) (15, 20) (15, 30) (R*, N*) (5, 12) (4, 11) (4, 10) (4, 9) (4, 8) (4, 8) F(R*, N*) 233.43 191.32 174.58 161.50 155.94 150.39 Ls 5.241 4.260 3.657 3.121 2.843 2.608 E[I] 1.608 1.291 1.543 1.757 1.852 1.920 E[B] 3.392 2.709 2.457 2.243 2.148 2.080 @F=@1 4.96 3.59 2.76 1.68 1.50 0.91 @F=@1 −14.06 −6.77 −3.87 −1.49 −0.90 −0.34 @F=@2 2.90 3.59 3.80 4.29 5.06 4.84 @F=@2 −5.25 −6.77 −7.25 −8.19 −9.43 −9.23
tions, parking lot services, automatic machine car wash services, and many related other applications.
References
1. Gross D, Harris CM (1998) Fundamentals of queueing theory, 3rd edn. Wiley, New York
2. Kleinrock L (1975) Queueing systems, vol. I: theory. Wiley, New York
3. Tijms HC (1986) Stochastic modeling and analysis: a computational approach. Wiley, New York
4. Benson F, Cox DR (1951) The productivity of machines requiring attention at random intervals. J Roy Stat Soc B 13:65–82
5. Elsayed EA (1981) An optimum repair policy for the machine interference problem. J Oper Res Soc 32:793–801
6. Wang K-H (1994) Profit analysis of the machine repair problem with cold standbys and two modes of failure. Microelectron Reliab 34:1635–1642
7. Wang K-H, Wu J-D (1995) Cost analysis of the M/M/R machine repair problem with spares and two modes of failure. J Oper Res Soc 46(6):783–790
8. Wang K-H, Lee H-C (1998) Cost analysis of the cold-standby M/M/R machine repair problem with multiple modes of failure. Microelectron Reliab 38(3):435–441
9. Hilliard JE (1976) An approach to cost analysis of maintenance float systems. IIE Trans 8:128–133
