The economies of oil producing nations depend heavily on oil price dynamics. These dynamics are the determinants of budgetary sizes and capital project allocations in nations with oil. As a result, it has gained attention even among mathematicians: Cai and Newt  , Krugman  especially with the downturn dynamics of 2015; Lee and Huh  . Because to mathematicians, if nation A derives proceeds in a space X when the dynamics are positively increasing and sufficient for instance, the dynamics can be represented.1 Practically, if T is an arbitrary operator and X is a proceed space for A, one can write that
1The converse is also true.
Now, given additional information on (1) above, vital considerations can be made. Suppose a price shock occurs when X is complete. Then (1) is a transformation from Banach space to any space. Consequently, understanding the nature of inverse maps that reverse Y to X could be the solution of certain interesting problems. The direct interpretation is that to do with the needed policy maps that can take (1) to completeness once again.
In the queuing literature, it is well known for N-homogenous jobs that the stationary probability of maintaining these jobs in a uni-server production center is given by
The parameter is the occupation rate of the center, Medhi  . Unfortunately, homogeneity of jobs is unrealistic, Krishnamoorthy  . Suppose X is isomorphic to a job space Z. Suddenly, an oil price shock2 occurs in the neighborhood of X. Trivially, the completeness property of Z will alter similar to that of X a.s. Consequently, the extended job space is nowhere dense in Z. Thus, working under homogenous assumptions in is simplistic a.s. A re-consideration of the job size bracket is necessary for a complete discussion and analysis in . Moreover, the understanding of needed maps that takes to Z is equivalent to that which takes X to Y given that the later space is a normed space.
In the past, a lot of studies considered the stationary behavior of jobs in and Z identical. Nowadays, there are re-considerations proving otherwise. For instance, Krishnamoorthi and Sreenivan  , Kumar and Sharma  and more recently, Som and Kumar  . On managing queuing systems in , Ke and Pearn  studied an M/M/1 queuing system with server breakdowns and vacations where the arrival rate varies according to the server status and the vacation norm determined by the number of arrivals during the vacation period. Jayachitra and Albert  studied an Erlangian model under server breakdowns and multiple vacations and provided a cost model to determine the optimal operating policy at minimum cost. What is inherent in most of these management models is that the optimal criterion is proved from the service process. Essentially, optimal criterion from the number of jobs in the system is scarce in the literature.
2Similar to that of 2015 that takes oil price from 100+ USD to the neighborhood of 20+ USD.
3From principal component analysis of several factors affecting a steady state system from the exterior, policies, occupation rates and constraint size have the largest Eigen values.
Our aim is to present a methodology that studies the problem of strategies in imperfect production centers from the stationary number of jobs in the system. The most important gain is the generalization of known basic results. This extends the capacity of known results to centers with jobs of distinct characteristics. For instance, in service centers with normal jobs and constraint jobs, less time spending on jobs and delaying jobs, difficult to process jobs and easy to process jobs, etc. In this respect, our work is purely for operational research purpose geared towards best practices in centers with distinct job characteristics. We wish to provide the understanding of principal components of imperfect centers and develop optimal criteria under which production is maximized.
It turns out that3 the problem herein is that of how best the coupling of system policies (T), occupation rates ( ) and available constraints c can be tackled in .
2. Preliminary Results
Lemma 1 An operationally useful policy map is necessarily compact and infinite dimensional in .
It is enough to show that an infinite dimensional compact operator cannot have a close range in a complete normed space.
Let and be arbitrary normed spaces where is complete. Suppose that
is infinite dimensional and compact.
Suppose to the contrary that is closed. Then is complete.
where is the unit ball of jobs in Z.
It follows directly from (3) that
Since T is compact and is bounded, then is compact.
Given that ; is nowhere dense in . Hence, is of the first Baire category.
This contradiction completes the prove. □
Lemma 2 Let be a job sequence in . If has a convergent point in , then the measurable policy map such that is a strong job policy.
Proof. It suffices to show that if and are normed job spaces, then for any and a point given that and , then strongly.
Let and . We show that . Suppose that and define by
Clearly is linear. Similarly, since T is compact is compact. Thus, is bounded.
Given that , we have . Hence, .
Now to show the last component of the lemma, suppose that does not converge to . Then ( ) has a sub sequence ( ) such that for some . Since , then ( ) is bounded. Given that T is compact, then ( ) has a Cauchy sub-sequence.
This contradiction completes the proof. □
Lemma 3 A job policy on is compact if a corresponding job policy is compact.
Proof. Suppose that Z is a normed space and is a Banach space.
Furthermore, suppose that T is compact. Let ( ) be a sequence in . If is a functional such that for any , we have
Thus, ’s are uniformly bounded. In addition,
Thus, ’s are equicontinuous and so is compact. This implies that there exist a sub sequence
which converges uniformly on . Hence T* is compact. □
Lemma 4 Given ; then
Proof. Denote by the probability generating function (PGF) for the new jobs when there are fixed constraint jobs in a system such that
And in view of (2), we have
The lemma holds upon differentiating (15) at . □
4This is the only case the classical M/M/1 model depicts in its expectation.
It is interesting to note that if in (11) above, then . Consequently, there is only one job group in the system (homogenous).4 In this case, the result goes to that of a classical production center with homogenous jobs as expected.
Lemma 5 For a finite capacity imperfect production center with two distinct job groups and , we have
Proof. In view of (2) and for , it can be shown that the PGF is
Differentiating (17) w.r.t z, we have
upon substituting z = 1 in (18) above. Finally, the lemma holds if (20) is rearranged and simplified. □
Corollary 6 From the numerical results (Tables 1-6 in the appendix), it is clear that
Lemma 7 (First Criterion) A maximizer of the group is the solution for the policy map-constraint-occupation rate problem such that
Proof. We seek a solution for such that
where is a real valued semi-linear continuous function with respect to all its arguments. (22) is equivalent to
By Lemma 1, T is necessarily compact. Let T be a fixed point of G(.). Then (23) reduces to
Consider a differentiable arc in the plane such that points on .
By multivariate chain rule on the left hand side of (24), we have
Assuming that solves (21) and combining (24) and (25) and rearranging, we have
So5 that the couple differential equations
constitute in general a solution for for when T is a fixed point of . □
Lemma 8 (Second Criterion) Any continuously differentiable solution for satisfying the first criterion above must coincide with the original solution along the base curve .
Proof. Since (27) and (28) are coupled systems, only in rare cases analytic solution exists in closed form. However, if we specify an initial value for , and , then the existence of a unique solution pair and is guaranteed. A solution that did not pass through the origin leading to cannot be a solution for since it is nowhere differentiable around . □
Lemma 9 (Optimality Criterion) Suppose so that the constraint dependent occupation rate . A solution that passes through the origin for is optimal a.s.
Proof. Given that , we have . By the numerical approximation (Tables 1-6 in the appendix) . Given that is dependent, it is then trivial. □
3. Scope for Future Work
There is a scope in extending our results to some special cases of the problem solved in this work. For instance, when the function is independent of N or when and are linear or even non-linear combination of and T and N. The author are grateful to all literature sources used.
There is no competing interest of any kind within the authorship of this work.
SS: Drafted the entire manuscript, provided the introductory chapter (Section 1) and proved Lemmas 1, 2, 3, 5, 7, 8 and 9 together with Corollary 6.
HM: Participated in the sequence alignment of the manuscript and provided the numerical simulations.
MLM: Participated in the design of the manuscript and proved lemma 4 under my supervision.
All authors read and approved the final manuscript.
The authors are grateful to Dr. Babangida A. Albaba; the current Rector of the Katsina State Institute of Technology and Management (KSITM) for spearing time to go through the entire thesis leading to this manuscript and for making valuable suggestions.
For a numerical approximation, we study the model in (11) under various sizes of constraint numbers c and varying occupation rate for . The following numerical results are obtained.
Table 1. E[N] when c = 0.
Table 2. E[N] when c = 10.
Table 3. E[N] when c = 23.
Table 4. E[N] when r = 0.5.
Table 5. E[N] when r = 0.75.
Table 6. E[N] when r = 0.9.
 Krishnamoorthy, A. and Sreenivasan, C. (2012) An M/M/2 Queuing System with Heterogeneous Servers including One with Working Vacation. International Journal of Stochastic Analysis, 2012, Article ID: 145867. https://doi.org/10.1155/2012/145867
 Kumar, R. and Sharma, S.K. (2014) Two Heterogeneous Server Markovian Queueing Model with Discouraged Arrivals, Reneging and Retention of Reneged Customers. International Journal of Operations Research, 11, 64-68.
 Som, B.K. and Kumar, R. (2017) A Heterogeneous Queuing System with Reverse Balking and Reneging. Journal of Industrial and Production Engineering, 34, 1-5.
 Ke, J.C. and Pearn, W.L. (2004) Optimal Management Policy for Heterogeneous Arrival Queueing Systems with Server Breakdowns and Vacations. Quality Technology Quantitative Management, 1, 149-162. https://doi.org/10.1080/16843703.2004.11673069
 Jayachitra, P. and Albert, A.J. (2014) Optimal Management Policy for Heterogeneous Arrival M/EK/1 Queuing System with Server Breakdowns and Multiple Vacations. Advances in Theoretical and Applied Mathematics, 9, 87-95.