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This article deals with the queuing-inventory system, composed of c junior servers, a senior server, two finite waiting halls, and an infinite orbit. On occasion, junior servers encounter challenges during customer service. In these instances, they approach the senior server for guidance in resolving the issue. Suppose the senior server is engaged with another junior server. The approaching junior servers await their turn in a finite waiting area with a capacity of c for consultation. Concerning this, we study the performance of junior servers approaching the senior server in the retrial queuing-inventory model with the two finite waiting halls dedicated to the primary customers and the junior servers for consultation. We formulate a level-dependent QBD process and solve its steady-state probability vector using Neuts and Rao’s truncation method. The stability condition of the system is derived and the R matrix is computed. The optimum total cost has been obtained, and the sensitivity analyses, which include the expected total cost, the waiting time of customers in the waiting hall and orbit, the number of busy servers, and a fraction of the successful retrial rate of the model, are computed numerically.

Data are produced every single instant in the modern era of technological breakthroughs we live in today and is correctly termed as the lifeblood of today's world; whether it is Google or Meta, everyone depends on data to survive. But, with the immense surge in technological boom comes several backlashes that tend to pull it down; one similar instance is the data morphing or modification of the data unethically. In many jurisdictions, the phenomenon of data morphing is considered a severe offense, subject to lifelong imprisonment. There are several cases where data are altered to encrypt reliable details. Recently, in March 2023, Silicon Valley Bank collapsed following unrest prompted by increasing rates. Silicon Valley Bank ran out of money as entrepreneurial investors pulled investments to maintain their businesses afloat in a frigid backdrop for IPOs and individual financing. The bank's collapse was the biggest since the financial meltdown of 2008 and the second-largest commercial catastrophe in American history. By confirming the "Silicon Valley Bank" stock price data, we will delve further into the actual condition of whether there has been any data morphing in the data put forward by the Silicon Valley Bank. To accomplish the very same, we applied a very well-known statistical paradigm, Benford's Law and have cross-validated the results using comparable statistics, like Zipf's Law, to corroborate the findings. Benford's Law has several temporal proximities, known as conformal ranges, which provide a closer examination of the extent of data morphing that has occurred in the data presented by the various organizations. In this research for validating the stock price data, we have considered the opening, closing, and highest prices of stocks for a time frame of 36 years, between 1987 and 2023. Though it is worth mentioning that the data used for this research are coarse-grained, still since the validation is subjected to a larger time horizon of 36 years; Benford's Law and the similar statistics used in this article can point out any irregularities, which can result in some insight into the situation and into whether there has been any data morphing in the Stock Price data presented by SVB or not. This research has clearly shown that the stock price variations of the SVB diverge much from the permissible ranges, which can give a conclusive direction on further investigations in this issue by the responsible authorities. In addition, readers of this article must note that the conclusion formed about the topic discussed in this article is objective and entirely based on statistical analysis and factual figures presented by the Silicon Valley Bank Group. [ABSTRACT FROM AUTHOR]