Production Systems with Parallel Heterogeneous Servers of Limited Capacity: Accurate Modeling and Performance Analysis

Abstract

Heterogeneous systems of limited capacity have general applications in manufacturing, but also in logistic or service systems due to the differences in server or workstation performance or work assignment; this is in close relationship with system flexibility, where saturation and blocking are ordinary situations of systems with high demand and limited capacity, and thus, accurate loss quantification is essential for performance evaluation. Multi-class systems of limited capacity have been studied much less than parallel homogeneous systems (Erlang models). In this context, accurate models for parallel heterogeneous ordered-entry systems were developed: without any prior queue, i.e., M/Mi/c/c, and with a k-capacity queue, i.e., M/Mi/c/c + k. These new matrix models gave an exact state formulation, and their accuracy was verified using discrete event simulation and comparison with literature results. Also, the effect of the queue capacity was studied in relationship to the pattern of service rates. Next, the heterogeneous recirculating system model was also developed with good approximation results. Finally, the proposed models were applied to evaluate systems with non-exponential service times using a new hybrid methodology by combining the Markovian model and the Monte Carlo method (MCM) for normal or lognormal service times, which also yielded useful good approximations to the simulated system.

1. Introduction

Parallel server systems with limited capacity are present in many operational situations of interest in manufacturing [1], but also in transportation or traffic [2], service [3], or computer systems [4]. Their study using Markovian network models requires accurate loss quantification to properly evaluate their performance, and the ordinary model of these systems consists of enumerating all the states for the size of the system in order to calculate the state probabilities that sum to 1. Among them, there is interest in the probabilities of the full and empty states that allow for estimating the maximum expected capacity or idleness. In particular, the analysis of the parallel systems with homogeneous or identical servers facilitates analytical explicit expressions of the state probabilities. Therefore, based on the PASTA property [5], the probability of the full-size system is established in manuals as the estimation of the arrival rate percentage that becomes lost due to blocking when the system is busy, and thus, the effective throughput is the difference from the offered load rate. For parallel systems with identical servers, the Erlang systems of Poisson arrivals and service rates are well known and of great interest for service systems, like call centers [6]. In addition, the important property of performance equivalence of M/G/c/c and M/M/c/c [3] gives extra opportunities for analysis and decision making with service time distributions other than the exponential one.

Even though parallel homogeneous systems have been studied in depth (Erlang models), many real systems have heterogeneous servers, where non-uniform human behavior [7] or task variety is present. The practice of ordered entry involves assigning work upon arrival to the first idle server in the order of arrangement. This is a frequent natural assignation, for instance, in conveyor systems or call centers. In this context, some former studies are in more direct connection with the present work. Initial research of parallel heterogeneous systems can be found in [8], who analyzed ordered entry with the heterogeneous systems with the random allocation of service. His work assumed no queue in front of the servers, i.e., M/Mi/n. His error analysis of the heterogeneous vs. homogeneous system concluded that a heterogeneous system cannot be replaced by one of the equal servers, whose mean service rate is the average of the mean service rate of each server. The authors of [9,10] studied a parallel system with homogeneous servers and ordered entry with application to conveyor systems. Singh (1970, 1971) [11,12] compared homogeneous and heterogeneous Markovian queuing systems and obtained the steady-state probabilities of a heterogeneous three-server system, i.e., M/Mi/3. Later, some other researchers dealt with the analysis of the server arrangement for performance optimization. Elsayed (1983) [13] and Yao (1987) [14] studied the optimal allocation of buffers in front of the servers but without the quantification of blocking. Yao (1986) [15] and Saglam and Shahbazov (2007) [16] paid attention to minimizing the overflow of the system. Further extensions in parallel heterogeneous system studies can be found in Boxma et al. (1994) [17], who gave a comprehensive view of solution methods for the epoch. Isguder and Uzunoglu (2014) [18] dealt with the semi-Markov process of GI/M/n/0 and loss quantification; more recently, Melikov et al. (2020) [19] presented an exact formulation but with an approximate resolution for ordered entry. Specifically, for systems using the ordered entry process, one can also refer to the work of Cooper (1976) [20], who concluded the need for a multi-dimensional birth-and-death process to represent the system. Matsuy and Fukuta (1977) [21] studied the state probabilities of a parallel system of heterogeneous multiservers (multichannel) with ordered entry discipline and no waiting line, i.e., M/Mi/n. With direct application to conveyor systems, they found that the optimal order for maximum throughput while minimizing the overflow is the faster-to-slower server arrangement. Nath and Enns (1981) [22] showed that the overflow (loss) is minimal for M/Mi/c under the faster service rule. Pourbabai and Sonderman (1986) [23] studied approximate expressions of the heterogeneous server G/G/n.

With a focus on heterogeneous systems with retrial, surveys by Muth and White (1979) [24] and Nazzal and El-Nashar (2007) [25] are noted for simulation works. Nawijn (1983, 1984) [26,27] studied the analytical model of two heterogeneous servers with recirculation, while Pourbabai (1987) [28] studied the asymptotic performances of random access and an ordered entry G/M/K/O with approximate expressions. Recently, Boysen et al. (2019) [29] presented a revision focused on the operation and not on modeling. Even though closed-loop or recirculating systems have been studied much less than ordinary loss systems, the recirculating flow represents additional difficulties in system modeling. Using stochastic models, Muth and White (1979) [24] analyzed a conveyor with one loading station. Soderman (1982) [30] modeled the recirculating conveyor systems with a single loading station and a single unloading station as GI/M/1/1 using the superposition of the recirculating rate with the new arrivals. He approximated the joint interarrival distribution as a hyperexponential with an iterative method. Schmidt and Jackman (2000) [31] modeled a recirculating system as a queue network based on an approximate method of solving tandem finite queues with blocking by Brandwajan and Jow (1988) [32]. Hsieh and Bozer (2005) [33] used conditional probabilities to approach the recirculating overflow and considered multiple unloading stations. Haghighi and Mishev (2006) [34] revised the formulation of Disney’s model in a two-station multiserver model with balking and reneging. Retrial queues or recirculating systems are also of particular interest in modeling service systems, like call centers (Gans et al. (2003) [6]) or in merging systems with recirculation (Van der Gaast et al. (2018) [35]).

Together with the analytical models, time-consuming simulation allows for the evaluation of the system performance of complex systems, alleviating the difficulties of heterogeneous systems. Proper simulation can also provide experimental verification of analytical models. Today’s growing computing capabilities are paving the way for the increasing use of simulation. Nevertheless, the possibilities of analytical models can be better valued in optimization analysis, transient behavior, or system control, in particular when quick solutions are better than long simulation runs, with a particular interest in the decision making of evolving flexible systems.

Heterogeneous systems have no accurate analytical models available, like homogenous ones. We contributed to reducing this gap in this study with a set of new scalable models that gives accurate quantification of the loss of the parallel system with heterogeneous servers of limited capacity, and even including recirculation. It allows for the calculation of performance metrics based on the state probabilities solved from the transition matrix. We developed a detailed justification with a novel approach to evaluate the loss of the system applied to parallel heterogeneous servers and we also verified its agreement with simulation results with differences under 0.5%, followed by the analysis of system performance. The model contributes to overcoming the low accuracy of other loss estimations that are only based on the probability of a busy system from conventional transition state diagrams, whose states only enumerate the size of the system. Its application to recirculating systems also gives a good approximation, within 1% across most of the arrival rate range. Finally, these good results suggest approaching some more general heterogeneous systems with service rate distributions other than Poisson. This work contributes to stochastic system modeling through a new approach to the state transition diagram with the incorporation of a blocking state that is different from the full state (maximum size), allowing for an accurate evaluation of heterogeneous systems loss. A novelty of this method is that it facilitates the proper application of the PASTA (Poisson arrivals see time averages) property with outstanding accurate results, as verified with exponential rates, but also allows for the extension of the models with good approximation to recirculating systems and non-exponential service time systems using a hybrid methodology of the model together with the Monte Carlo method.

This paper is organized as follows: In Section 2, we present parallel systems of identical servers without a waiting line and the classical transition diagram of a loss system, namely, model A, where the PASTA property was applied for heterogeneous servers and only gives a rough approximation to the simulated system. Next, a new approach to evaluate loss was introduced in model B. It was verified against former analytical calculations from the literature and through experimental simulation, with very accurate results in both cases. It follows the detailed analysis of the loss and full-size probabilities of the system across the arrival rate range. Then, Section 3 introduces model E, which includes a waiting queue of limited capacity in front of the servers, and the influence of the maximum queue size was analyzed for optimal configuration. In Section 4, the recirculating system was modeled by adapting model E, and its performance was checked against simulation results. Next, in Section 5, the combined application of model B with the Monte Carlo method evaluated the use of the transition states matrix of Markovian chains to evaluate systems with non-exponential service time distributions. In the Conclusions section, a synthesis of contributions, results, and potential future works is outlined.

2. Methodology of Parallel System Modeling

2.1. Loss in Parallel Systems without Queue

We considered a workstation with a mean service rate μ with a Poisson distribution (service interdeparture time of exponential distribution), with a total capacity of k units, in a queue of finite capacity k − 1, and with an arrival flow rate λ. It was represented by the distribution of state probabilities M/M/1/K shown in (1), with ρ = λ/μ and inventory L [1]:

$$i=0,\dots k; p_i=\{\begin{array}{l}{\rho }^i\frac{\left(1-\rho \right)}{1-{\rho }^{k+1}}\hspace{1em}if\hspace{1em}\rho \ne 1\\ \frac{1}{k+1}\hspace{1em}if\hspace{1em}\rho =1\end{array}\}\hspace{1em}L={\displaystyle \sum _{i=0}^{k}i\cdot }{p}_i$$

(1)

If the server has infinite capacity, the departure rate will be equal to the arrival rate with the same distribution (Burke, 1956) [36]. Because of the finite capacity of the buffer queue, the effective input rate will be lower than λ; therefore, when the system is full with k units, a new incoming unit cannot enter the system and is lost or discarded from the arrival flow. Due to the memoryless property of the arrival rate of exponential time distribution, the loss does not modify the exponential interarrival time distribution, though it has an effective rate λ_eff lower than λ due to loss. The effective rate is λ_eff = λ(1 − p_k) based on the PASTA property ([2] p. 128; [1] p. 14 and 102; [3] p. 73).

Next, we considered the model of homogeneous servers M/M/m/m, with the rate of arrivals λ and k identical parallel servers with the service rate μ, which all had Poisson distributions (exponential interarrival time) with a k maximum size or capacity of the system. This is frequently named the Erlang first model and the probability of the full system is given by the well-known Erlang-B formula B(m,r) (2), where the traffic flow ratio is r = λ/μ and the utilization ρ = λ/(mμ). We ran the experimental simulation of the M/M/1/m and the Erlang systems through the discrete event simulation software Arena by Rockwell, with 30 regenerative replications of 1000 h and a previous 1000 h warm-up period. This confirmed the full agreement of the mean results from the simulation and the analytical model, which were always within 1% difference and inside the half-width interval of simulation variability that included the 95% simulation shots in all the regime ranges. The use of Little’s law based on this effective input rate allowed for the calculation of the waiting times or cycle time inside the system CT (queue and server) from L = λ_eff · CT.

$$p_i=\frac{\frac{{(\lambda /\mu )}^n}{n!}}{{\displaystyle {\sum }_{i=0}^m\frac{{(\lambda /\mu )}^i}{i!}}};\hspace{1em}i=0,\dots m;\hspace{1em}and\hspace{1em}B(m,r)\equiv p_m=\frac{\frac{r^m}{m!}}{{\displaystyle {\sum }_{i=0}^m\frac{r^i}{i!}}}; r=\lambda /\mu$$

(2)

2.2. System of Heterogeneous Parallel Servers

Next, we considered a multiclass or heterogeneous server system with an arrival rate λ with a Poisson distribution. We considered a three-parallel-server system with service rates μ₁, μ₂, and μ₃, also with Poisson distributions. Parts or customers were serviced in entry-ordered discipline. The state diagram of the system, namely, model A, is given in Figure 1.

Each server can only allocate one unit and the three-tuple indicates which server is in service (1) or empty (0), read from right to left, like a binary number. Thus, (110) denotes server 1 is empty (0) and servers 2 and 3 are in service. The state (1, 1, 1) is the state of a full system.

$$\begin{array}{l}A\cdot p=c\\ \\ \left[\begin{array}{cccccccc}-\lambda & \mu 1& \mu 2& 0& \mu 3& 0& 0& 0\\ \lambda & -(\lambda +\mu 1)& 0& \mu 2& 0& \mu 3& 0& 0\\ 0& 0& -(\lambda +\mu 2)& \mu 1& 0& 0& \mu 3& 0\\ 0& \lambda & \lambda & -(\lambda +\mu 1+\mu 2)& 0& 0& 0& 0\\ 0& 0& 0& 0& -(\lambda +\mu 3)& \mu 1& 0& 0\\ 0& 0& 0& 0& \lambda & -(\lambda +\mu 1+\mu 3)& 0& \mu 2\\ 0& 0& 0& 0& 0& 0& -(\lambda +\mu 2+\mu 3)& \mu 1\\ 1& 1& 1& 1& 1& 1& 1& 1\end{array}\right]\cdot \left[\begin{array}{c}{p}_{000}\\ p_{001}\\ p_{010}\\ p_{011}\\ p_{100}\\ p_{101}\\ p_{110}\\ p_{111}\end{array}\right]=\left[\begin{array}{c}0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 1\end{array}\right]\\ \end{array}$$

(3)

The resolution in the stationary regime of the Markovian chains system can be expressed by their state equations in the form of the transition matrix (3) and by solving a homogeneous linear system of equations. An ordinary alternative involves replacing the last equation with the product form condition of probabilities that sum to 1 (3).

The model results and system simulations are included in

Table 1. Dimensionless throughput R* = R/Σμ of three servers from simulation, M/M/3/3 (Erlang-B), model A, and model B.

Arrival Rate	Service Rates	Simul		Erlang-B	Model A	Model B	Simul vs. Model A	Simul vs. Model B
λ	(μ1, μ2, μ3)	Mean	Half Width				(%)	(%)
0.1	(1, 1, 1)	0.0332	0.0002	0.0333	0.0333	0.0333	−0.40	−0.39
0.5		0.1644	0.0003	0.1646	0.1665	0.1646	−1.25	−0.08
1		0.3125	0.0004	0.3125	0.3254	0.3125	−4.14	−0.01
3		0.6540	0.0009	0.6538	0.6838	0.6538	−4.55	0.03
5		0.7838	0.0014	0.7839	0.8035	0.7839	−2.51	−0.01
10		0.8936	0.0014	0.8931	0.9004	0.8931	−0.76	0.05
30		0.9656	0.0032	0.9657	0.9567	0.9657	0.93	0.00
0.1	(0.25, 0.75, 2)	0.0332	0.0002		0.0333	0.0333	−0.40	−0.35
0.5		0.1606	0.0002		0.1621	0.1607	−0.91	−0.04
1		0.2944	0.0005		0.2973	0.2944	−0.99	0.00
3		0.5966	0.0008		0.5757	0.5964	3.50	0.03
5		0.7263	0.0013		0.6789	0.7261	6.52	0.02
10		0.8518	0.0016		0.7711	0.8518	9.47	−0.01
30		0.9487	0.0031		0.8392	0.9489	11.55	−0.02
0.1	(2, 0.75, 0.25)	0.0332			0.0333	0.0333	−0.50	−0.50
0.5		0.1652			0.1638	0.1652	0.84	−0.03
1		0.3138			0.2970	0.3138	5.34	−0.02
3		0.6319			0.5396	0.6318	−0.57	0.02
5		0.7508			0.6356	0.7513	2.12	−0.07
10		0.8630			0.7349	0.8626	14.84	0.05
30		0.9507			0.8223	0.9507	13.51	0.00
0.1	(1.3, 0.4, 1.3)	0.0332			0.0333	0.0333	−0.30	−0.30
0.5		0.1642			0.1637	0.1642	0.31	0.01
1		0.3090			0.3028	0.3090	1.99	−0.01
3		0.6367			0.5845	0.6368	8.20	−0.01
5		0.7655			0.6922	0.7658	9.58	−0.03
10		0.8803			0.7965	0.8798	9.52	0.05
30		0.9604			0.8830	0.9602	8.06	0.03
0.1	(0.5, 2, 0.5)	0.0332			0.0333	0.0333	−0.30	−0.30
0.5		0.1638			0.1636	0.1639	0.10	−0.04
1		0.3070			0.3000	0.3071	2.28	−0.03
3		0.6212			0.5279	0.6212	15.02	0.00
5		0.7449			0.5862	0.7451	21.31	−0.02
10		0.8614			0.6279	0.8611	27.10	0.04
30		0.9506			0.6541	0.9512	31.20	−0.06

, where R* = R/Σμ.

In the case where μ₁ = μ₂ = μ₃, it can be directly compared with the Erlang first model for k = 3 using (2). The throughput was calculated for model A by applying the PASTA property, and thus, λ_eff = λ(1 − p₁₁₁) = λ(1 − p_f), where p_f is the probability of the maximum size or full system.

There were significant differences between the experimentally simulated throughput and that calculated by model A, which mostly overestimated the throughput when the arrival rates were high (saturation). Note that in this conventional transition diagram that enumerates the states of the servers, the probability of the full system with all busy servers estimated the probability of loss. The transition to the full system (111) happened at the rate λ (birth) from the system of size 2—with states (110), (011), or (101)—and the system leaves (death) the full state from size 3 to 2 at a rate Σμ.

We also considered the homogeneous system M/M/m/m with m = 3, and thus, we could apply the Erlang-B Formula (2) for the calculation of the effective throughput λ_eff = λ(1 − p₁₁₁) = λ(1 − B(3,r)); see Table 1. For the Erlang model, the results were completely in agreement with those from the simulation, and thus, the identical set of servers of M/M/m/m behaved with no difference from an ordered entry system, but since the servers were indistinguishable, the order became irrelevant and there would not be any difference when serving according to ordered entry, random entry, or another service discipline [31].

Next, we considered a new formulation of the transition diagram, namely, model B. Different from model A, when the system is full, the lost load is redirected to a virtual server with a queue of infinite capacity, herein called the loss server, where arrivals enter when the real servers of the system are occupied (Figure 2). When arrivals are lost, the busy system enters the state >111 = b.

The transition matrix is given by (4), and the mean number of customers L in the system or the utilization U was calculated using (5). Note that when the system falls into the loss server, state >(111) = b, the three-server subsystem is occupied and the calculation of L in the servers by (5) is taken into account. In the transition diagram, the transition (birth) from the busy system (111) into the loss state happened at the rate λ; therefore, the arrival goes into the loss server when servers are busy and the system leaves this loss state from size 3 to 2 at the rate Σμ as soon any server is available to serve, as represented.

The results of the simulation, as shown in Table 1, were obtained with Arena software by Rockwell Inc. using λ = μ_i = 1, a simulation run of 30 replications of 1000 h, 1000 h of warm-up for the stationary regime, and where the throughput is expressed as dimensionless by dividing by Σμ_i. The effective throughput rate was calculated using λ_eff = λ(1 − p_busy), where p_busy is the percentage of time the three-server system is busy. In the Erlang model of identical servers, p_busy = B(3,r) using (2); for model A with three servers, p_busy = p₁₁₁ = p_f from (3); and in the case of model B, with three servers and the loss server, p_busy = p₁₁₁ + p_>111 = p_f + p_b using (4). Model A is a representation of a loss system that evaluates the throughput from the offered load λ. The direct calculation based on the PASTA property with the effective offered load λ_eff = λ(1 − p_f) was only a rough approximation for heterogeneous parallel servers, as the experimental results in Table 1 show for model A.

$$\begin{array}{l}A\cdot p=c\\ \\ \left[\begin{array}{ccccccccc}-\lambda & \mu 1& \mu 2& 0& \mu 3& 0& 0& 0& 0\\ \lambda & -(\lambda +\mu 1)& 0& \mu 2& 0& \mu 3& 0& 0& 0\\ 0& 0& -(\lambda +\mu 2)& \mu 1& 0& 0& \mu 3& 0& 0\\ 0& \lambda & \lambda & -(\lambda +\mu 1+\mu 2)& 0& 0& 0& 0& \mu 3\\ 0& 0& 0& 0& -(\lambda +\mu 3)& \mu 1& 0& 0& 0\\ 0& 0& 0& 0& \lambda & -(\lambda +\mu 1+\mu 3)& 0& \mu 2& \mu 2\\ 0& 0& 0& 0& 0& 0& -(\lambda +\mu 2+\mu 3)& \mu 1& \mu 1\\ 0& 0& 0& \lambda & 0& \lambda & \lambda & -(\lambda +\mu 1+\mu 2+\mu 3)& 0\\ 1& 1& 1& 1& 1& 1& 1& 1& 1\end{array}\right]\cdot \left[\begin{array}{c}{p}_{000}\\ p_{001}\\ p_{010}\\ p_{011}\\ p_{100}\\ p_{101}\\ p_{110}\\ p_{111}\\ p_{111b}\end{array}\right]=\left[\begin{array}{c}0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 1\end{array}\right]\end{array}$$

(4)

$$\begin{array}{c}L={\displaystyle \sum _{i=0}^{k+1}i\cdot }{p}_i=0\cdot p_{000}+1\cdot (p_{001}+p_{010}+p_{100})+2\cdot (p_{110}+p_{101}+p_{011})+3\cdot p_f+3\cdot p_b\\ U=L/3\end{array}$$

(5)

The results indicate that the description of the system states and their probabilities should take into account not only the states of the servers but also the associated allocation of all the arrivals and the different independent states in order to be considered a product form the set of state probabilities that sum to 1. In addition to the server’s busy state (111), the lost customers enter the loss server. In the transitions diagram of model B, >(111) = b is a state of full size for the servers. The system transits into the state >(111) with the rate λ when the arrivals overflow the servers, and the net input into the servers is λ_eff, while the rate through the loss server is λ − λ_eff. Based on the PASTA property, the split of the Poisson rate for all the servers, including the loss server, remains a Poisson distribution [1]. Based on [36], the net output rate through the loss server, which includes a queue of infinite capacity, is also λ − λ_eff, providing that the service rate of the loss server is greater than λ − λ_eff. This loss server has infinite capacity and we associates an arbitrarily sufficient high service rate >(λ − λ_eff) to comply with it. Finally, the total output of the whole system is λ, with the sum of the servers and loss server together.

The system transitions into the loss server with probability p_b and rate λ after the system is full (111), and the system leaves the state at the rate of the sum of the service rates of the three servers; therefore, as soon any server releases a customer after a service, the number of customers in the servers becomes m-1, as represented in the transition states diagram. Note that by including the loss server in model B, the system is not a loss system anymore, and thus, the calculation of the states associated with the offered rate is accurate. The probability of state >(111) = b is the percentage of time that the arrivals fall into the loss server, but this always happens after the servers are occupied. In model B, the PASTA property was applied to the three real servers defined in the transition diagram: the offered rate to the servers is λ(1 − p_>111) = λ(1 − p_b) before the application of the PASTA property. After applying the PASTA property to the three-server input, the effective rate through the servers becomes λ_eff = λ(1 − p_>111 − p₁₁₁) = λ(1 − p_b − p_f) by discounting the percentage of time the servers occupy p₁₁₁. The arrival net rate into the loss server is finally λ(p_b + p_f), which is an accurate account of the time percentage that the total arrival rate λ (offered load) does not enter the servers, which was verified in the experiment through the accurate results of Table 1. There is a need to consider every single arrival in terms of the probabilities that arise from the multi-dimensional queue nature of the ordered entry for heterogeneous systems. Meanwhile, Erlang systems of indistinguishable homogeneous servers do not require it.

In the timeline of the arrival rates, there are intervals of time when the servers are occupied and no arrivals are trying to enter the system with probability p_f, and others where the arrivals try to enter and are rejected to the loss server with probability p_b. In both intervals, no entrance of customers into the server system is possible, and thus, they are discounted to get the effective arrival rate into the servers. In model B, every arrival is included in the set of states for accurate accounting, and thus, the system evaluates every customer destination. In general, an ordinary transition matrix that only enumerates the states of the servers, namely, model A, overestimates the throughput (underestimates the probability of occupied servers) for a high offered load by applying the PASTA property (see Table 1). In summary, including the time percentage of the loss state (loss server) provides an accurate throughput calculation of systems of finite capacity, such as model B.

The evolution of the probabilities or percentage of time for model B of full servers p_f, loss from blocking when servers are busy (blocking) p_b, and the system being occupied p_occ = pb + pf are represented in Figure 3. In addition, the probability of an empty system p0 is represented in the results. While λ/Σμ < 1, the probability of the system being busy p_b was lower than the probability of having full servers p_f. When λ/Σμ ≈ 1, both were equally probable. It is remarkable that the probability of full servers pf continued to grow with increasing levels of arrival rate up to the point it reached a maximum at about λ/Σμ ≈ 1.75. After that maximum, when some extra load was offered, it went directly to the loss server, and p_f decreased while p_b continuously grew. The percentage of time the servers were occupied with no customers trying to enter the servers, i.e., p_f, decreased monotonically after that maximum; therefore, for an infinite arrival rate, it was inferred that there was no chance of a full system without loss, and thus, p_f → 0, while the probability of loss from blocking, i.e., p_b, continued to grow. Thus, when an extremely high load was offered, i.e., λ → ∞, the loss from the blocking probability p_b became the only probable situation.

Also, from the results in Figure 3, at low λ, the probability of an empty system p₀ was higher than the probability of an occupied system p_occ. At about λ/Σμ ≈ 0.6, both were equally probable, and when λ/Σμ > 0.7 (approx.), the probability of an empty system p₀ became lower than the probability of occupied servers p_f. In an attempt to maximize the utilization of servers, increasing the arrival rate λ over that point increased the utilization. Nevertheless, if the increase in cost from losing a customer is higher than the reduction in cost by increasing the utilization, operating the system over that point might not be convenient. Increasing the installed capacity to avoid customer loss instead of increasing server utilization might be a better option. Note that at λ/Σμ ≈ 0.6, the perceived saturation of servers was low, i.e., only about 18% of the time the servers were observed to be full (p_f) or empty (p₀). Nevertheless, judging the service performance of a loss system based on the observed utilization of the servers seemed to be clearly misleading: for instance, at a high offered load of λ/Σμ = 4, less than 20% of the time, the servers were fully occupied (p_f), with the appearance of extra idle capacity, while, in fact, the system was saturated and the offered load loss pb was more than 60%.

Model B considers heterogeneous servers, whereas the Erlang model M/M/k/k just with k identical servers is a particular case. In this case, the full system state can be described properly by λ_eff = λ(1 − p_busy) using (1) and (2); the results are included in Table 1. The results of model B for (k = 3, μ₁ = μ₂ = μ₃ = μ = 1) of identical servers were also those of the Erlang-B formula. It resulted in p_busy = B(3,λ/μ) in Figure 3, but it was also verified using the numerical values, which were identical to 15 decimal places.

The calculations based on the simulation confirmed the accuracy of model B, i.e., M/Mi/k/k, to describe the heterogeneous parallel server systems. From the former results, it is valuable to compare the overflow results of M/Mi/k calculated from the analytical results by Matsui and Fukuta (1977) [21] and those obtained for the full system p_busy = p_b + p_f from model B (

Table 2. Loss probability comparison of analytical literature models with model B.

Dataset Source	λ	(μ1, μ2, μ3)	Loss from Simulation	poverflow M/Mi/k (k = 3)	pocc = pf + pb M/Mi/k/k Model B (k = 3)
[21]		(1.2, 1, 0.8)		0.0109	0.01086
		(1, 1.2, 0.8)		0.0118	0.01179
	0.5	(1.2, 0.8, 1)		0.0119	0.01189
		(0.8, 1.2, 1)		0.0140	0.01402
		(1, 0.8, 1.2)		0.0140	0.01397
		(0.8, 1, 1.2)		0.0152	0.01520
		(1.2, 1, 0.8)		0.0575	0.05749
		(1, 1.2, 0.8)		0.0600	0.06005
	1.0	(1.2, 0.8, 1)		0.0609	0.06092
		(0.8, 1.2, 1)		0.0661	0.06611
		(1, 0.8, 1.2)		0.0669	0.06688
		(0.8, 1, 1.2)		0.0696	0.06956
		(1.2, 1, 0.8)		0.2049	0.20490
		(1, 1.2, 0.8)		0.2082	0.20816
	2.0	(1.2, 0.8, 1)		0.2094	0.20940
		(0.8, 1.2, 1)		0.2153	0.21533
		(1, 0.8, 1.2)		0.2170	0.21700
		(0.8, 1, 1.2)		0.2198	0.21978
[18]	90	(60, 45, 10)	0.27166	0.2999	0.27189
		(60, 10, 45)	0.28577	0.1986	0.28593

). The probabilities were identical. Based on the independence of events of the exponential interarrival times, the results from the arrangement of arrivals in an infinite waiting queue or providing new arrivals after the loss were equivalent in terms of the overflow or loss. It is noteworthy that the M/Mi/k system has an infinite capacity queue in front of the servers. It is known that this system model is unstable when λ ≥ Σμ, as the waiting line grows indefinitely. Meanwhile, the M/Mi/k/k in model B redirects the overflow to the loss server without this limitation

3. Parallel Heterogeneous Servers with a Common Waiting Queue

Model B includes servers without any prior queue. Next, we considered model E, Figure 4, where the servers have a common queue of limited capacity in front of them, and thus, the incoming arrivals are lost only when the queue is full. The method of assignation is ordered entry. The whole system has a total capacity of m + k, where k is the capacity of the queue and m is the number of servers. For the case of m = 3, the transition rate diagram of Figure 4 represents the states of the system, where the state >(3) + k is the state where the loss probability is accounted for (p_b). In the birth and death transitions, the death of state b falls into the (3) + k − 1 state because as soon as any customer is serviced in any server, a customer enters the server and a new allocation in the queue of capacity k becomes available. Its transition rate matrix k = 3 is A in (6).

$$\begin{array}{l}A\cdot p=c\\ \\ \left[\begin{array}{cccccccccccc}-\lambda & \mu 1& \mu 2& 0& \mu 3& 0& 0& 0& 0& 0& 0& 0\\ \lambda & -(\lambda +\mu 1)& 0& \mu 2& 0& \mu 3& 0& 0& 0& 0& 0& 0\\ 0& 0& -(\lambda +\mu 2)& \mu 1& 0& 0& \mu 3& 0& 0& 0& 0& 0\\ 0& \lambda & \lambda & -(\lambda +\mu 1+\mu 2)& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& \lambda & -(\lambda +\mu 3)& \mu 1& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& \lambda & -(\lambda +\mu 1+\mu 3)& 0& \mu 2& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& -(\lambda +\mu 2+\mu 3)& \mu 1& \mu 1& 0& 0& 0\\ 0& 0& 0& 0& 0& \lambda & \lambda & -(\lambda +{\displaystyle \sum \mu })& {\displaystyle \sum \mu }& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& \lambda & -(\lambda +{\displaystyle \sum \mu })& {\displaystyle \sum \mu }& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& \lambda & -(\lambda +{\displaystyle \sum \mu })& {\displaystyle \sum \mu }& {\displaystyle \sum \mu }\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& \lambda & -(\lambda +{\displaystyle \sum \mu })& 0\\ 1& 1& 1& 1& 1& 1& 1& 1& 1& 1& 1& 1\end{array}\right]\cdot \left[\begin{array}{c}{p}_{000}\\ p_{100}\\ p_{110}\\ p_{010}\\ p_{011}\\ p_{001}\\ p_{101}\\ p_{111}\\ p_{111+1}\\ p_{111+2}\\ p_f\\ p_b\end{array}\right]=\left[\begin{array}{c}0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 1\end{array}\right]\end{array}$$

(6)

In order to evaluate the performance of the model,

Table 3. Dimensionless throughput R*= R/Σμ of model E (m = 3, k = 3) vs. simulation.

Arrival Rate	Service Rates	Simul		Mod E	Model E vs. Simul
λ	(μ1, μ2, μ3)	Mean	Half Width	R*	[%]
1	(1, 1, 1)	0.3323	0.0005	0.3326	−0.09
3		0.8304	0.0009	0.8302	0.03
5		0.9629	0.0014	0.9628	0.01
1	(0.25, 0.75, 2)	0.3320	0.0005	0.3320	0.00
3		0.8174	0.0009	0.8174	0.00
5		0.9557	0.0010	0.9553	0.05
1	(2, 0.75, 0.25)	0.3331	0.0005	0.3326	0.13
3		0.8251	0.0009	0.8250	0.00
5		0.9580	0.0013	0.9584	−0.05
1	(1.3, 0.4, 1.3)	0.3322	0.0005	0.3325	−0.09
3		0.8261	0.0007	0.8262	−0.01
5		0.9601	0.0011	0.9603	−0.02
1	(0.5, 2, 0.5)	0.3324	0.0005	0.3324	0.01
3		0.8227	0.0009	0.8227	0.00
5		0.9576	0.0012	0.9576	0.00

shows the comparison of the throughput of model E vs. the simulation results, where the dimensionless throughput is R* = R/Σμ = λ(1 − p_f − p_b)/Σμ. There was complete agreement in the results and the proper representation of the parallel loss system with a buffer using model E. Model B and model E of multiclass or heterogeneous servers are presented for m = 3 and they were accurate.

The accuracy of this methodology can be used for large systems, such as M/Mi/m/m + k, at the cost of the description of intermediate states in the state transition matrix and computational effort because the size of the transition matrix is of the order 2^m. Nevertheless, the construction for a higher queue capacity k is a much simpler birth and death sequence of extra states in the queue. We developed the algorithm to automate the generation of elements of the transition matrix for an arbitrary number of servers m. Even though the coding of the algorithm was out of the scope of this study, it is worth mentioning some basic insights. The matrix is sparse and can be formulated as the sum of three rate matrices: rates out (negative) from the state (allocated in the diagonal of the matrix), rate into the state (positive) from the death of other states, and rate to the state (positive) from other state births. The states are also automatically enumerated from the binary number representation (m digits) of every server state (0-1, idle or busy). Even when programmed in a matrix-efficient code, like Matlab, this exact transition matrix construction of size order 2^m is heavy for a moderate system size m. Therefore, these transition matrix models are of particular interest for systems with a moderate number of heterogeneous servers, where the accurate assessment of individual servers’ performance differences is an objective.

For the sake of presenting the model generalization for any size of the system, Figure 5 includes the operating curves R* = R/Σμ vs. offered arrival rate λ, as calculated for parallel systems of m = 6, 8, and 10 servers, in each case with queue capacity k = m/2 and Σμ = 8. The service rates are in descending arithmetic progression from the first server μ₁ = 1.5(Σμ/m) for faster to slower (f-t-s) and ascending to the last server with μ_m = 1.5(Σμ/m) in the slower-to-faster (s-t-f) arrangement.

Then, we examined how the pattern of the servers influenced the performance of the system (Figure 6).

With a slight imbalance, some extra throughput was obtained when the order from faster to slower was applied, but when the imbalance was high—see, for instance, (2, 0.75, 0.5)—the benefit could only be obtained under medium to low utilization (low arrival rate). The system underperformed for the balanced service rates (1, 1, 1) with the same average for high utilization. When the utilization tended to saturation (see Figure 6 over 90% utilization), the throughput was quite similar, attenuating the effect of the server’s arrangement. However, for a given average arrival rate of the mean service time, the maximum throughput was reached by homogenous (1, 1, 1) or slightly heterogeneous servers. A high imbalance in service rates reduced the throughput. As a rough estimation, reductions up to the order of 5% in R* = R/Σμ can be observed in Figure 6, depending on the pattern of multiclass servers that caused significant heterogeneity for the performance and not only the average of service rates.

Next, the performance improvement by adding a queue was evaluated using model E for different values of the maximum queue capacity (Figure 7). For k = 0, model E was equivalent to model B. The operative curves of Figure 7 show that the size of the buffer increased the throughput, but more relevant, for heterogeneous service rates, the performance could reach higher results with imbalanced service rates arranged from slower to faster, i.e., (0.25, 0.75, 2), than the balanced distribution (1, 1, 1) since this provided a buffer of enough capacity. In the case of heterogeneous servers without any queue, ordering the servers from faster to slower gave the best results for the throughput.

When heterogeneous servers were combined with a prior queue, the performance was slightly better with assignation from slower to faster service rates when there was no queue (k = 0), but in combination with a minimum queue, the results became similar, and with sufficient capacity (k = 4), the performances of heterogeneous and homogeneous systems were indistinguishable (see Figure 8). A properly dimensioned queue before the multiclass parallel servers can provide improvements in performance, overwhelming the heterogeneity of service times. This applies to traffic problems, like call centers, where the use of an interactive voice response to the arriving customers can handle a waiting queue to be serviced. Former research established priority for the faster agents in [37], as was also mentioned in [14,16], or in the problem of fair agents [7]. Our results show that a proper queue or buffer can compensate for heterogeneity.

Conversely to manufacturing systems of tangible products, where the cycle time of customers in the system can be secondary after throughput, in service manufacturing or pure service systems, the sojourn time in the system can be a priority and fundamental metric. For the system under study with m = 3, multiclass servers with a buffer of maximum size k before them, the time in the system, also called the sojourn time or cycle time CT, is calculated using Little’s law (7) and is represented in Figure 8. At low λ, the better results (minimum CT) were obtained with faster-to-slower server priority, while in the rest of the range, the better configuration was the homogeneous arrangement of servers. While the queue capacity improved throughput by mitigating loss with a high offered load, it also increased CT.

$$\begin{array}{l}L(m=3,k)={\displaystyle \sum _{i=0}^{m+k+1}i\cdot }{p}_i=0\cdot p_{000}+1\cdot (p_{100}+p_{010}+p_{010})+2\cdot (p_{110}+p_{101}+p_{011})+3\cdot p_{111}+4\cdot p_{111+1}+\dots (3+k)\cdot p_f+(3+k)p_f\\ CT={\lambda }_{ef}/L=\lambda (1-p_{111+k}-p_{>111+k})/L=\lambda (1-p_f-p_f)/L\end{array}$$

(7)

4. Systems of Parallel Heterogeneous Servers with Recirculation

Next, we considered the case when the rejected arrival rate recirculated and merged with the offered input flow into the system instead of blocking and direct loss. The system can represent a conveyor with recirculation or another system with retrial. The recirculating rate had priority over the external input and the system remained a loss system. In general, it is known that the merging input rate will not have a Poisson distribution [36]. Models B and E demonstrated their accuracy when evaluating the overflow of a parallel system of limited capacity.

We modified model E to include a recirculation flow, as represented in Figure 9. When the conveyor recirculates the non-served arrivals, the offered load increases to include the flow recirculated in the time percentage the system is full (p_f), and thus, the expected recirculated rate is λp_f, and the net offered load to the system becomes λ_r = λ(1 + p_f) instead of λ. Thus, considering the findings from model B, the loss of this net offered load will be λ_r (p_f + p_b) = λ(1 + p_f) (p_f + p_b), but in fact, the loss of the recirculating fraction λ p_f (p_f + p_b) does not exit the system, but it remains in recirculation; therefore, the net loss out of the system will be λ(1 + p_f)(p_f + p_b) − λ p_f (p_f + p_b) = λ (p_f + p_b), which is consistent with the net balance with the external offered load λ, which was already verified in models B and E. In consequence, the output rate is expected to be λ_r(1 − p_f − p_b) = λ(1 + p_f) (1 − p_f − p_b). The recirculation does not generate any extra state in the transition state diagram of servers in model E (see Figure 4), but increases the offered load to the servers. In general, the probabilities p_f and p_b with recirculation will reach different values than those without recirculation.

We formulated the recirculation model through the transition state matrix as a product form problem with the same states as model E; therefore, the size of the system provided a complete description of states with the probability summing to 1, but with the correction of the offered load at the entrance of the conveyor. Thus, the resulting matrix for the conveyor of maximum capacity 3, with three servers and ordered entry service was also (6), but the offered load was λ_r = λ(1 + p_f) instead of λ. The state probability was the solution of A·p = c, where A = A(λ, μ_i, p_f). The solution was obtained from the iteration on p_f as a fixed-point problem that converged very quickly. In general, the expression of the effective rate offered and the recirculating rate through the conveyor involved conditional probabilities, but the proposed transition state diagram was completed by incorporating the loss server state, and the iteration as a fixed-point problem provided the state probabilities solution.

In

Table 4. Throughput R* = R/Σμ model E (m = 3, k = 3) vs. simulation, with and without recirculation.

μ	R* Simul w/o Recirc.	R* Model E	Simul w/o vs. Model E%	R* Simul w/Recirc.		R* Model E Recirc.	Simul w/recirc. vs. Model E Recirc. %
				Mean	Half Width
(1, 1, 1)	0.3325	0.3326	−0.03	0.3342	0.0092	0.3331	0.32
(1, 1, 1)	0.6349	0.6346	0.06	0.6531	0.0018	0.6508	0.34
(1, 1, 1)	0.8302	0.8302	0.01	0.8592	0.0245	0.8668	−0.89
(1, 1, 1)	0.9221	0.9229	−0.08	0.9478	0.0319	0.9507	−0.31
(1, 1, 1)	0.9634	0.9628	0.06	0.9869	0.0283	0.9786	0.84
(1, 1, 1)	0.9895	0.9890	0.05	0.9900	0.0425	0.9940	−0.40
(1.2, 1, 0.8)	0.3328	0.3326	0.05	0.3378	0.0089	0.3332	1.37
(1, 1.2, 0.8)	0.6357	0.6349	0.13	0.6522	0.0146	0.6510	0.19
(1.2, 0.8, 1)	0.8303	0.8303	0.00	0.8636	0.0198	0.8669	−0.38
(0.8, 1.2, 1)	0.9214	0.9221	−0.07	0.9419	0.0400	0.9501	−0.86
(1, 0.8, 1.2)	0.9621	0.9623	−0.02	0.9867	0.0150	0.9781	0.86
(0.8, 1, 1.2)	0.9886	0.9888	−0.02	0.9989	0.0400	0.9938	0.51

, the results for the system of three servers with a queue (conveyor) of maximum capacity 3 are included, with and without recirculation. The output rate was evaluated using the average result of simulation over 600,000 recirculating cycles with a warm-up of 60,000 cycles, with 10 independent replications. Model E with recirculation produced simulation results within 1% in most of the ranges of operation, with a maximum below 1.4%.

5. Hybrid Modeling of Heterogeneous Servers with General Service Time Distributions

5.1. Uncertainty of the Offered Load

The stationary solution of the M/M/c/c + k system of identical servers has been studied analytically and is included in manuals of Markovian queues. The analytical multiclass, multichannel, or heterogeneous servers in model E can be used to study the performance of M/Mi/3/3 + k, but when using the Monte Carlo method (MCM), it can approach M/Gi/3/3 + k. The combination of MCM and model E can offer the resulting distribution of the stationary solution for each system setup of mean arrival rate and service rates Gi coming from general (non-exponential) distributions. Regardless of the distribution from which the service rates come, model E offers one solution (stationary), and thus, the variability effect of service rates under general distribution on performance can be assessed by generating the resulting distribution of performance results.

We proposed the use of the precise stationary solution from the transition matrix in combination with MCM to evaluate the effect of the uncertainty in the offered load or the service rates, which approached the solution of the system for other service distributions. The transition matrix was developed with the strong assumption of events being independent or memoryless, and thus, the independent state probabilities were evaluated in a Markovian process. The MCM offers the distribution of solutions from independent trials of shots to the non-exponential distributions of arrival and/or service rates. The MCM is a standard method for uncertainty evaluation in other fields, like metrology, and its use together with the analytical Markovian model can allow for evaluating the effect of service time variability or arrival rate uncertainty without the time-consuming discrete-event simulation to obtain the stationary solution under general distributions. For instance, it might be of interest in facilitating system control or decision making in logistic or manufacturing activities in cyberphysical systems. While the transition matrix gives the stationary solution of the temporal series, the output of the MCM is a distribution around the average stationary solutions that show the expected spread around the mean value under the ergodic assumption, which is due to the variation in the service times under different distributions. The hybrid method is considered very appropriate because the variability effect in the response to the uncertainty of the variables is also based on its variance, like the arrival or service rates variability. In addition, it is easy to implement for the proposed analytical model to produce a quick resolution, with no need for solution initialization or other previous algorithm adjustments, and thus, the method directly propagates the spread of stochastic variables through the analytical model into the solution.

Hybrid modeling of accurate Markovian chain models in combination with the MCM can facilitate approximate studies by alleviating the limitation of the distribution with memory that non-exponential distributions impose because each shot in the MCM simulation will be calculated independently with the accurate stationary result (expected asymptotic mean in time of an ergodic process), and the resulting distribution from the MCM will be the convolution of stationary results generated from shots to the distribution of the arrival rate or the service rate. The alternative of direct discrete-event simulation will require looking for the average solution when a simultaneous change occurs in the mean values and their variance; therefore, the effect of the mean and variance are combined in every solution of the simulation run. In addition, real systems are better represented for decision making by the mean value of service time or demand in short-term periods with some uncertainty, but with a drift in longer temporal periods. The accurate transition rate matrix of Markovian chains combined with MCM simulation represents a convenient and controlled way of separating the variability in different time scales to deal with the offered load uncertainty or analyze the mean effects on system performance for non-exponential service time distributions.

Figure 10 presents the results for model B with a standard deviation of 10% of the offered load and mean service rate configurations (1, 1, 1) and (2, 0.75, 0.25) of the same average service rate 1. Each point in Figure 10 is the result of 10,000 shots using the MCM simulation. While the overall mean performance seemed to be slightly higher for (1, 1, 1) across the range, depending on the particular level of offered load, it could be reversed or simply indistinguishable due to the uncertainty of the arrival rate.

5.2. Application to Non-Exponential Service Time Distributions

In

Table 5. Simulation and model B throughputs for 3 parallel servers with exponential arrival rate and service rate distributions: exponential, normal N(⋅), and lognormal LogN(⋅).

		Exp (μ)	N (μ, 0.3μ)			LogN (μ, 0.3μ)
Arrival Rate	Service Rates	Simul Exp (μ)	Simul N (μ, 0.3μ)	Mod B	Simul N(·) vs. Mod B	Simul LogN (μ, 0.3μ)	Mod B	Simul LogN(·) vs. Mod B
λ	(μ1, μ2, μ3)				(%)			(%)
0.1	(1, 1, 1)	0.0996	0.0997	0.1000	−0.33	0.0996	0.1000	−0.42
0.5		0.4933	0.4930	0.4921	0.17	0.4932	0.4925	0.14
1		0.9374	0.9376	0.9289	0.93	0.9375	0.9304	0.76
3		1.9621	1.9619	1.9298	1.63	1.9615	1.9333	1.44
5		2.3515	2.3518	2.3157	1.53	2.3522	2.3184	1.44
10		2.6807	2.6793	2.6601	0.72	2.6796	2.6604	0.72
30		2.8969	2.8972	2.8851	0.42	2.8964	2.8856	0.37
0.1	(0.25, 0.75, 2)	0.0996	0.0995	0.0999	−0.38	0.0995	0.0999	−0.40
0.5		0.4819	0.4803	0.4799	0.08	0.4806	0.4802	0.09
1		0.8833	0.8761	0.8753	0.08	0.8758	0.8758	−0.01
3		1.7897	1.7740	1.7590	0.85	1.7737	1.7632	0.59
5		2.1788	2.1678	2.1448	1.06	2.1677	2.1488	0.87
10		2.5553	2.5526	2.5300	0.88	2.5524	2.5316	0.81
30		2.8461	2.8467	2.8341	0.44	2.8470	2.8296	0.61
0.1	(2, 0.75, 0.25)	0.0995	0.0997	0.1000	−0.34	0.0995	0.1000	−0.46
0.5		0.4955	0.4972	0.4937	0.71	0.4973	0.4943	0.60
1		0.9413	0.9486	0.9288	2.09	0.9493	0.9313	1.89
3		1.8958	1.90585	1.8605	2.38	1.9059	1.8643	2.18
5		2.2524	2.2588	2.2120	2.07	2.2586	2.2151	1.92
10		2.5890	2.5894	2.5653	0.93	2.5894	2.5662	0.89
30		2.8521	2.8519	2.8381	0.48	2.8520	2.8390	0.45
0.1	(1.3, 0.4, 1.3)	0.0997	0.0998	0.1000	−0.23	0.0998	0.1000	−0.15
0.5		0.4925	0.4933	0.4904	0.58	0.4931	0.4910	0.43
1		0.9269	0.9277	0.9172	1.13	0.9278	0.9189	0.96
3		1.9101	1.9092	1.8781	1.63	1.9096	1.8819	1.45
5		2.2966	2.2960	2.2607	1.53	2.2961	2.2638	1.41
10		2.6408	2.6393	2.6197	0.74	2.6395	2.6198	0.75
30		2.8812	2.8804	2.8655	0.52	2.8806	2.8667	0.48
0.1	(0.5, 2, 0.5)	0.0997	0.0999	0.1000	−0.09	0.0998	0.1000	−0.19
0.5		0.4913	0.4928	0.4895	0.67	0.4927	0.4901	0.53
1		0.9211	0.9234	0.9111	1.34	0.9236	0.9130	1.15
3		1.8635	1.8626	1.8325	1.61	1.8625	1.8361	1.42
5		2.2348	2.2327	2.1983	1.54	2.2327	2.2013	1.40
10		2.5843	2.5827	2.5583	0.94	2.5829	2.5595	0.91
30		2.8519	2.8532	2.8418	0.40	2.8531	2.8413	0.42

, the results of the simulation and model B are compared for exponential service time and also normal or lognormal distributions. The normal distribution can be appropriated for logistic or manufacturing systems, where the coefficient of variation (CV) in the range from 0.2 to 0.5 is realistic [38]. A 0.3 value was set for this study. The lognormal distribution was found to properly fit some service times, like call centers. Again, CV = 0.3 was used, based on [39]. For the application of the MCM, each point was obtained from 10,000 shots to the distribution.

We note that sampling the service times from a normal distribution of a given mean provides a normal distribution of service times with the same mean. In the case of the lognormal distribution, in order to provide a mean service time m and coefficient of variation CV, the sampling distribution must sample from the mean μ and standard deviation σ given by (8).

$$\mu =\ln \left[\frac{m}{\sqrt{C{V}^2+1}}\right]; \sigma =\sqrt{\ln (C{V}^2+1)}$$

(8)

As a consequence of non-exponential service times, the throughput did not vary in a significant way with respect to the exponential times of the same average; therefore, the independence of exponential arrivals seemed to dominate over the non-exponential service times. The average difference between the discrete events simulation of the system and model B with MCM was 0.8% (ranging from −0.4% to 2.4%) for the normal distribution and 0.7% (ranging from −0.5% to 2.2%) for the lognormal distribution. The pattern of homogeneous server rates (1, 1, 1) offered better throughput in all the cases under analysis. The difference with the solution of exponential service times for every server’s arrangement was very low; therefore, these results initially support the use of the transition matrix as a first approach to study non-homogeneous server systems M/Gi/k/k. Further testing would consolidate this possibility. While the equivalence of homogenous systems M/M/k/k and M/G/k/k is well established, the performance of heterogeneous systems seems to also be mainly influenced by the average values and with little dependence on the service time distribution.

6. Conclusions

We present the accurate analytical modeling of heterogeneous or multi-class parallel server systems of limited capacity through their transition rate matrix. The loss mechanisms were modeled inside the system, and thus, the transition rate matrix precisely captured the loss percentage, with accurate results checked with former analytical results and experimentally through simulation. Model B revises the usual application of the PASTA property to the offered load by including the loss evaluation as a state in the transition state diagram. This state diagram construction demonstrated its accuracy for heterogeneous server parallel systems, where the Erlang system is a particular case with homogeneous servers (B-Erlang formula). The addition of a queue of finite capacity in front of the servers can also be accurately modeled with the same methodology, i.e., model E. The precise evaluation using this model was outstanding in all regimes, from low demand to leveled load/capacity or system oversaturation, which allowed for qualifying it as an exact model for the state probabilities, in agreement with the simulation also undertaken inside the intervals of confidence. It was verified that a small queue very quickly approached the heterogeneous to homogenous performances of the same average service rate, with practical significance in logistic, manufacturing, or service operations. The assessment of recirculating systems presented higher difficulty, even though the model was developed successfully through the transition matrix as an open system with a modified offered load that included recirculation. Even when the flow was strictly non-exponential, the iterative process of resolution as a product form problem included all the state probabilities of every offered arrival, even the loss state that established continuity, as it provided a fair good approximation of the output rate from a simple and scalable model.

These models allowed for the easy analysis of performance in a full range of arrival rates, with insights into system behavior for ordered entry assignments. In real systems, the homogeneity of servers can be a rough assumption. Many problems of interest in logistics, manufacturing systems, call centers, or computer servers, among others, include heterogeneous service times across servers, and thus, this analytical model of heterogeneous servers can evaluate their behavior better.

Based on this methodology, the hybrid modeling of the transition matrix along the application of the Monte Carlo method (MCM) can be an intermediate practical technique between analytical models and pure discrete-event simulation. The transition matrix gives a basic behavior structure under the assumptions of events independence in a Markovian chain, and the MCM allows for the introduction of the variability around the stationary solution for uncertainty estimation or for non-exponential service times. The initial test of normal and lognormal distributions showed useful approximations. Future works of hybrid modeling that combine the transition matrix of limited complexity with the MCM can be a useful starting point for further research, integrating available growing computation capabilities with simple structured models for the study of complex systems. In particular, in the dynamic situation of short-term changes in a manufacturing system, where the stationary regime is less representative and the opportunities for system control in the transient regime can be better facilitated by the state transition matrix instead of the stationary long-term solution of simulation. This is foreseen to be of particular interest for cyberphysical systems or digital twins of growing importance, where not only the massive amounts of data but appropriate structured models can provide analysis opportunities and benefits for decision making.